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Title: College Algebra
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Creator: Stitz, Carl
Zeager, Jeff
Publication Date: 08-10-2009
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Subjects / Keywords: Cartesian Coordinates, Graphs of Equations, Function Notation, Function Arithmetic, Graphs of Functions, Transformations, Inequalities, Regression, Factor Theorem, Remainder Theorem, Graphing Calculators, Complex Zeros, Rational Inequalities, Circles, Parabolas, Ellipses, Summation Notation, Mathematical Induction, OGT+ ISBN: 9781616101473
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Abstract: Introductory text for college algebra survey course featuring worked examples. Contents: 1) Relations and Functions. 2) Linear and Quadratic Functions. 3) Polynomial Functions. 4) Rational Functions. 5) Further Topics in Functions. 6) Exponential and Logarithmic Functions. 7) Hooked on Conics. 8) Systems of Equations and Matrices. 9) Sequences and the Binomial Theorem. *** WebAssign ancillaries are available.***
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CollegeAlgebra Version 0 : 9 by CarlStitz,Ph.D.JeffZeager,Ph.D. LakelandCommunityCollegeLorainCountyCommunityCollege August10,2009

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TableofContents 1RelationsandFunctions1 1.1TheCartesianCoordinatePlane.............................1 1.1.1DistanceinthePlane...............................6 1.1.2Exercises......................................10 1.1.3Answers.......................................12 1.2Relations..........................................14 1.2.1Exercises......................................18 1.2.2Answers.......................................20 1.3GraphsofEquations....................................22 1.3.1Exercises......................................27 1.3.2Answers.......................................28 1.4IntroductiontoFunctions.................................31 1.4.1Exercises......................................37 1.4.2Answers.......................................40 1.5FunctionNotation.....................................41 1.5.1Exercises......................................47 1.5.2Answers.......................................49 1.6FunctionArithmetic....................................50 1.6.1Exercises......................................55 1.6.2Answers.......................................57 1.7GraphsofFunctions....................................58 1.7.1GeneralFunctionBehavior............................64 1.7.2Exercises......................................71 1.7.3Answers.......................................74 1.8Transformations......................................76 1.8.1Exercises......................................96 1.8.2Answers.......................................98 2LinearandQuadraticFunctions101 2.1LinearFunctions......................................101 2.1.1Exercises......................................113 2.1.2Answers.......................................116

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ivTableofContents 2.2AbsoluteValueFunctions.................................117 2.2.1Exercises......................................125 2.2.2Answers.......................................126 2.3QuadraticFunctions....................................129 2.3.1Exercises......................................138 2.3.2Answers.......................................140 2.4Inequalities.........................................143 2.4.1Exercises......................................156 2.4.2Answers.......................................157 2.5Regression..........................................159 2.5.1Exercises......................................164 2.5.2Answers.......................................167 3PolynomialFunctions169 3.1GraphsofPolynomials...................................169 3.1.1Exercises......................................180 3.1.2Answers.......................................183 3.2TheFactorTheoremandTheRemainderTheorem...................187 3.2.1Exercises......................................195 3.2.2Answers.......................................196 3.3RealZerosofPolynomials.................................197 3.3.1ForThoseWishingtouseaGraphingCalculator................198 3.3.2ForThoseWishingNOTtouseaGraphingCalculator.............201 3.3.3Exercises......................................207 3.3.4Answers.......................................208 3.4ComplexZerosandtheFundamentalTheoremofAlgebra...............209 3.4.1Exercises......................................217 3.4.2Answers.......................................218 4RationalFunctions219 4.1IntroductiontoRationalFunctions............................219 4.1.1Exercises......................................230 4.1.2Answers.......................................232 4.2GraphsofRationalFunctions...............................234 4.2.1Exercises......................................247 4.2.2Answers.......................................249 4.3RationalInequalitiesandApplications..........................253 4.3.1Exercises......................................262 4.3.2Answers.......................................264

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TableofContentsv 5FurtherTopicsinFunctions265 5.1FunctionComposition...................................265 5.1.1Exercises......................................275 5.1.2Answers.......................................277 5.2InverseFunctions......................................279 5.2.1Exercises......................................295 5.2.2Answers.......................................296 5.3OtherAlgebraicFunctions.................................297 5.3.1Exercises......................................307 5.3.2Answers.......................................310 6ExponentialandLogarithmicFunctions315 6.1IntroductiontoExponentialandLogarithmicFunctions................315 6.1.1Exercises......................................328 6.1.2Answers.......................................330 6.2PropertiesofLogarithms..................................332 6.2.1Exercises......................................340 6.2.2Answers.......................................342 6.3ExponentialEquationsandInequalities.........................343 6.3.1Exercises......................................351 6.3.2Answers.......................................352 6.4LogarithmicEquationsandInequalities.........................353 6.4.1Exercises......................................360 6.4.2Answers.......................................362 6.5ApplicationsofExponentialandLogarithmicFunctions................363 6.5.1ApplicationsofExponentialFunctions......................363 6.5.2ApplicationsofLogarithms............................371 6.5.3Exercises......................................376 6.5.4Answers.......................................380 7HookedonConics383 7.1IntroductiontoConics...................................383 7.2Circles............................................386 7.2.1Exercises......................................390 7.2.2Answers.......................................391 7.3Parabolas..........................................392 7.3.1Exercises......................................400 7.3.2Answers.......................................401 7.4Ellipses...........................................402 7.4.1Exercises......................................411 7.4.2Answers.......................................412 7.5Hyperbolas.........................................414 7.5.1Exercises......................................425

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viTableofContents 7.5.2Answers.......................................427 8SystemsofEquationsandMatrices429 8.1SystemsofLinearEquations:GaussianElimination..................429 8.1.1Exercises......................................442 8.1.2Answers.......................................443 8.2SystemsofLinearEquations:AugmentedMatrices...................444 8.2.1Exercises......................................451 8.2.2Answers.......................................453 8.3MatrixArithmetic.....................................454 8.3.1Exercises......................................467 8.3.2Answers.......................................470 8.4SystemsofLinearEquations:MatrixInverses......................471 8.4.1Exercises......................................483 8.4.2Answers.......................................485 8.5DeterminantsandCramer'sRule.............................486 8.5.1DenitionandPropertiesoftheDeterminant..................486 8.5.2Cramer'sRuleandMatrixAdjoints.......................490 8.5.3Exercises......................................495 8.5.4Answers.......................................498 8.6PartialFractionDecomposition..............................499 8.6.1Exercises......................................507 8.6.2Answers.......................................508 8.7SystemsofNon-LinearEquationsandInequalities...................509 8.7.1Exercises......................................518 8.7.2Answers.......................................520 9SequencesandtheBinomialTheorem523 9.1Sequences..........................................523 9.1.1Exercises......................................531 9.1.2Answers.......................................533 9.2SummationNotation....................................534 9.2.1Exercises......................................542 9.2.2Answers.......................................543 9.3MathematicalInduction..................................544 9.3.1Exercises......................................549 9.3.2SelectedAnswers..................................550 9.4TheBinomialTheorem..................................552 9.4.1Exercises......................................561 9.4.2Answers.......................................562 Index 563

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Chapter1 RelationsandFunctions 1.1TheCartesianCoordinatePlane InordertovisualizethepureexcitementthatisAlgebra,weneedtouniteAlgebraandGeometry. Simplyput,wemustndawaytodrawalgebraicthings.Let'sstartwithpossiblythegreatest mathematicalachievementofalltime:the CartesianCoordinatePlane 1 Imaginetworeal numberlinescrossingatarightangleat0asbelow. x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 Thehorizontalnumberlineisusuallycalledthe x -axis whiletheverticalnumberlineisusually calledthe y -axis 2 Aswiththeusualnumberline,weimaginetheseaxesextendingoindenitely inbothdirections.Havingtwonumberlinesallowsustolocatethepositionofpointsoofthe numberlinesaswellaspointsonthelinesthemselves. 1 SonamedinhonorofReneDescartes 2 Thelabelscanvarydependingonthecontextofapplication.

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2RelationsandFunctions Forexample,considerthepoint P belowontheleft.Tousethenumbersontheaxestolabelthis point,weimaginedroppingaverticallinefromthe x -axisto P andextendingahorizontalline fromthe y -axisto P .Wethendescribethepoint P usingthe orderedpair ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4.Therst numberintheorderedpairiscalledthe abscissa or x -coordinate andthesecondiscalledthe ordinate or y -coordinate 3 Takentogether,theorderedpair ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4comprisethe Cartesian coordinates ofthepoint P .Inpractice,thedistinctionbetweenapointanditscoordinatesis blurred;forexample,weoftenspeakof`thepoint ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(4.'Wecanthinkof ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4asinstructions onhowtoreach P fromtheoriginbymoving2unitstotherightand4unitsdownwards.Notice thattheorderintheordered pairisimportant )]TJ/F15 10.9091 Tf 12.71 0 Td [(ifwewishtoplotthepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 2,wewould movetotheleft4unitsfromtheoriginandthenmoveupwards2units,asbelowontheright. x y P )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 x y P ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 ; 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 Example 1.1.1 Plotthefollowingpoints: A ; 8, B )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.681 4.295 Td [(5 2 ; 3 C )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 : 8 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(3, D : 5 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1, E ; 0, F ; 5, G )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 ; 0, H ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(9, O ; 0. 4 Solution. Toplotthesepoints,westartattheoriginandmovetotherightifthe x -coordinateis positive;totheleftifitisnegative.Next,wemoveupifthe y -coordinateispositiveordownifit isnegative.Ifthe x -coordinateis0,westartattheoriginandmovealongthe y -axisonly.Ifthe y -coordinateis0wemovealongthe x -axisonly. 3 Again,thenamesofthecoordinatescanvarydependingonthecontextoftheapplication.If,forexample,the horizontalaxisrepresentedtimewemightchoosetocallitthe t -axis.Therstnumberintheorderedpairwould thenbethe t -coordinate. 4 Theletter O isalmostalwaysreservedfortheorigin.

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1.1TheCartesianCoordinatePlane3 x y A ; 8 B )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(5 2 ; 3 C )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 : 8 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 D : 5 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 E ; 0 F ; 5 G )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 ; 0 H ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 O ; 0 )]TJ/F34 7.9701 Tf 6.587 0 Td [(9 )]TJ/F34 7.9701 Tf 6.587 0 Td [(8 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123456789 )]TJ/F34 7.9701 Tf 6.587 0 Td [(9 )]TJ/F34 7.9701 Tf 6.587 0 Td [(8 )]TJ/F34 7.9701 Tf 6.587 0 Td [(7 )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 5 6 7 8 9 WhenwespeakoftheCartesianCoordinatePlane,wemeanthesetofallpossibleorderedpairs x;y as x and y takevaluesfromtherealnumbers.Belowisasummaryofimportantfactsabout Cartesiancoordinates. ImportantFactsabouttheCartesianCoordinatePlane a;b and c;d representthesamepointintheplaneifandonlyif a = c and b = d x;y liesonthe x -axisifandonlyif y =0. x;y liesonthe y -axisifandonlyif x =0. Theoriginisthepoint ; 0.Itistheonlypointcommontobothaxes.

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4RelationsandFunctions Theaxesdividetheplaneintofourregionscalled quadrants .TheyarelabeledwithRoman numeralsandproceedcounterclockwisearoundtheplane: x y QuadrantI x> 0, y> 0 QuadrantII x< 0, y> 0 QuadrantIII x< 0, y< 0 QuadrantIV x> 0, y< 0 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 Forexample, ; 2liesinQuadrantI, )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 2inQuadrantII, )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2inQuadrantIII,and ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 inQuadrantIV.Ifapointotherthantheoriginhappenstolieontheaxes,wetypicallyreferto thepointaslyingonthepositiveornegative x -axisif y =0oronthepositiveornegative y -axis if x =0.Forexample, ; 4liesonthepositive y -axiswhereas )]TJ/F15 10.9091 Tf 8.485 0 Td [(117 ; 0liesonthenegative x -axis.Suchpointsdonotbelongtoanyofthefourquadrants. Oneofthemostimportantconceptsinallofmathematicsis symmetry 5 Therearemanytypesof symmetryinmathematics,butthreeofthemcanbediscussedeasilyusingCartesianCoordinates. Definition 1.1 Twopoints a;b and c;d intheplanearesaidtobe symmetricaboutthe x -axis if a = c and b = )]TJ/F53 10.9091 Tf 8.485 0 Td [(d symmetricaboutthe y -axis if a = )]TJ/F53 10.9091 Tf 8.485 0 Td [(c and b = d symmetricabouttheorigin if a = )]TJ/F53 10.9091 Tf 8.485 0 Td [(c and b = )]TJ/F53 10.9091 Tf 8.485 0 Td [(d 5 AccordingtoCarl.Jethinkssymmetryisoverrated.

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1.1TheCartesianCoordinatePlane5 Schematically, 0 x y P x;y Q )]TJ/F53 10.9091 Tf 8.485 0 Td [(x;y S x; )]TJ/F53 10.9091 Tf 8.485 0 Td [(y R )]TJ/F53 10.9091 Tf 8.485 0 Td [(x; )]TJ/F53 10.9091 Tf 8.485 0 Td [(y Intheabovegure, P and S aresymmetricaboutthe x -axis,asare Q and R ; P and Q are symmetricaboutthe y -axis,asare R and S ;and P and R aresymmetricabouttheorigin,asare Q and S Example 1.1.2 Let P bethepoint )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 ; 3.Findthepointswhicharesymmetricto P aboutthe: 1. x -axis2. y -axis3.origin Checkyouranswerbygraphing. Solution. ThegureafterDenition1.1givesusagoodwaytothinkaboutndingsymmetric pointsintermsoftakingtheoppositesofthe x -and/or y -coordinatesof P )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3. 1.Tondthepointsymmetricaboutthe x -axis,wereplacethe y -coordinatewithitsopposite toget )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3. 2.Tondthepointsymmetricaboutthe y -axis,wereplacethe x -coordinatewithitsopposite toget ; 3. 3.Tondthepointsymmetricabouttheorigin,wereplacethe x and y -coordinateswiththeir oppositestoget ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3. x y P )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1123 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3

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6RelationsandFunctions Onewaytovisualizetheprocessesinthepreviousexampleiswiththeconceptof reections .If westartwithourpoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3andpretendthe x -axisisamirror,thenthereectionof )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3 acrossthe x -axiswouldlieat )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.Ifwepretendthe y -axisisamirror,thereectionof )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3 acrossthataxiswouldbe ; 3.Ifwereectacrossthe x -axisandthenthe y -axis,wewouldgo from )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3to )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3thento ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3,andsowewouldendupatthepointsymmetricto )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3 abouttheorigin.Wesummarizeandgeneralizethisprocessbelow. Reections Toreectapoint x;y aboutthe: x -axis,replace y with )]TJ/F53 10.9091 Tf 8.485 0 Td [(y y -axis,replace x with )]TJ/F53 10.9091 Tf 8.485 0 Td [(x origin,replace x with )]TJ/F53 10.9091 Tf 8.484 0 Td [(x and y with )]TJ/F53 10.9091 Tf 8.484 0 Td [(y 1.1.1DistanceinthePlane Anotherimportantconceptingeometryisthenotionoflength.IfwearegoingtouniteAlgebra andGeometryusingtheCartesianPlane,thenweneedtodevelopanalgebraicunderstandingof whatdistanceintheplanemeans.Supposewehavetwopoints, P x 1 ;y 1 and Q x 2 ;y 2 ; inthe plane.Bythe distance d between P and Q ,wemeanthelengthofthelinesegmentjoining P with Q .Remember,givenanytwodistinctpointsintheplane,thereisauniquelinecontainingboth points.Ourgoalnowistocreateanalgebraicformulatocomputethedistancebetweenthesetwo points.Considerthegenericsituationbelowontheleft. P x 1 ;y 1 Q x 2 ;y 2 d P x 1 ;y 1 Q x 2 ;y 2 d x 2 ;y 1 Withalittlemoreimagination,wecanenvisionarighttrianglewhosehypotenusehaslength d as drawnaboveontheright.Fromthelattergure,weseethatthelengthsofthelegsofthetriangle are j x 2 )]TJ/F53 10.9091 Tf 10.91 0 Td [(x 1 j and j y 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 1 j sothePythagoreanTheorem givesus j x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 1 j 2 + j y 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 1 j 2 = d 2 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 1 2 + y 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 1 2 = d 2 Doyourememberwhywecanreplacetheabsolutevaluenotationwithparentheses?Byextracting thesquarerootofbothsidesofthesecondequationandusingthefactthatdistanceisnever negative,weget

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1.1TheCartesianCoordinatePlane7 Equation 1.1 TheDistanceFormula: Thedistance d betweenthepoints P x 1 ;y 1 and Q x 2 ;y 2 is: d = q x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 1 2 + y 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 1 2 Itisnotalwaysthecasethatthepoints P and Q lendthemselvestoconstructingsuchatriangle. Ifthepoints P and Q arearrangedverticallyorhorizontally,ordescribetheexactsamepoint,we cannotusetheabovegeometricargumenttoderivethedistanceformula.Itislefttothereaderto verifyEquation1.1forthesecases. Example 1.1.3 Findandsimplifythedistancebetween P )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 ; 3and Q ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3. Solution. d = q x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 1 2 + y 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 1 2 = p )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 = p 9+36 =3 p 5 So,thedistanceis3 p 5. Example 1.1.4 Findallofthepointswith x -coordinate1whichare4unitsfromthepoint ; 2. Solution. Weshallsoonseethatthepointswewishtondareontheline x =1,butfornow we'lljustviewthemaspointsoftheform ;y .Visually, ;y ; 2 x y distanceis4units 23 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 Werequirethatthedistancefrom ; 2to ;y be4.TheDistanceFormula,Equation1.1,yields

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8RelationsandFunctions d = q x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 1 2 + y 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 1 2 4= p )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 4= p 4+ y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 4 2 = p 4+ y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 2 squaringbothsides 16=4+ y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 12= y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 =12 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2= p 12extractingthesquareroot y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2= 2 p 3 y =2 2 p 3 Weobtaintwoanswers: ; 2+2 p 3and ; 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 p 3 : Thereaderisencouragedtothinkabout whytherearetwoanswers. Relatedtondingthedistancebetweentwopointsistheproblemofndingthe midpoint ofthe linesegmentconnectingtwopoints.Giventwopoints, P x 1 ;y 1 and Q x 2 ;y 2 ,the midpoint M of P and Q isdenedtobethepointonthelinesegmentconnecting P and Q whosedistancefrom P isequaltoitsdistancefrom Q P x 1 ;y 1 Q x 2 ;y 2 M Ifwethinkofreaching M bygoing`halfwayover'and`halfwayup'wegetthefollowingformula. Equation 1.2 TheMidpointFormula: Themidpoint M ofthelinesegmentconnecting P x 1 ;y 1 and Q x 2 ;y 2 is: M = x 1 + x 2 2 ; y 1 + y 2 2 Ifwelet d denotethedistancebetween P and Q ,weleaveitasanexercisetoshowthatthedistance between P and M is d= 2whichisthesameasthedistancebetween M and Q .Thissucesto showthatEquation1.2givesthecoordinatesofthemidpoint.

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1.1TheCartesianCoordinatePlane9 Example 1.1.5 Findthemidpointofthelinesegmentconnecting P )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3and Q ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3. Solution. M = x 1 + x 2 2 ; y 1 + y 2 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+1 2 ; 3+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(1 2 ; 0 2 = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(1 2 ; 0 Themidpointis )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(1 2 ; 0 Aninterestingapplication 6 ofthemidpointformulafollows. Example 1.1.6 Provethatthepoints a;b and b;a aresymmetricabouttheline y = x Solution. By`symmetricabouttheline y = x ',wemeanthatifamirrorwereplacedalongthe line y = x ,thepoints a;b and b;a wouldbemirrorimagesofoneanother.Youshouldcompare andcontrastthiswiththeothertypesofsymmetrypresentedbackinDenition1.1.Schematically, a;b b;a y = x Fromthegure,weseethatthisproblemamountstoshowingthatthemidpointofthelinesegment connecting a;b and b;a liesontheline y = x .ApplyingEquation1.2yields M = a + b 2 ; b + a 2 = a + b 2 ; a + b 2 Sincethe x and y coordinatesofthispointarethesame,wendthatthemidpointliesontheline y = x ,asrequired. 6 Thisisakeyconceptinthedevelopmentofinversefunctions.SeeSection5.2

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10RelationsandFunctions 1.1.2Exercises 1.Plotandlabelthepoints A )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(7, B : 3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2, C ; p 10, D ; 8, E )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 : 5 ; 0, F )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 ; 4, G : 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 : 8and H ; 5intheCartesianCoordinatePlanegivenbelow. x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(9 )]TJ/F34 7.9701 Tf 6.587 0 Td [(8 )]TJ/F34 7.9701 Tf 6.587 0 Td [(7 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123456789 )]TJ/F34 7.9701 Tf 6.586 0 Td [(9 )]TJ/F34 7.9701 Tf 6.586 0 Td [(8 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 5 6 7 8 9 2.ForeachpointgiveninExercise1above Identifythequadrantoraxisin/onwhichthepointlies. Findthepointsymmetrictothegivenpointaboutthe x -axis. Findthepointsymmetrictothegivenpointaboutthe y -axis. Findthepointsymmetrictothegivenpointabouttheorigin.

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1.1TheCartesianCoordinatePlane11 3.Foreachofthefollowingpairsofpoints,ndthedistance d betweenthemandndthe midpoint M ofthelinesegmentconnectingthem. a ; 2, )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 5 b ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(10, )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 2 c 1 2 ; 4 3 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 d )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(2 3 ; 3 2 7 3 ; 2 e )]TJ/F54 10.9091 Tf 5 0.188 Td [(p 2 ; p 3 )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ 8.485 9.024 Td [(p 8 ; )]TJ 8.485 9.024 Td [(p 12 f ; 0, x;y 4.Findallofthepointsoftheform x; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1whichare4unitsfromthepoint ; 2. 5.Findallofthepointsonthe y -axiswhichare5unitsfromthepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 ; 3. 6.Findallofthepointsonthe x -axiswhichare2unitsfromthepoint )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 1. 7.Findallofthepointsoftheform x; )]TJ/F53 10.9091 Tf 8.485 0 Td [(x whichare1unitfromtheorigin. 8.Let'sassumeforamomentthatwearestandingattheoriginandthepositive y -axispoints dueNorthwhilethepositive x -axispointsdueEast.OurSasquatch-o-metertellsusthat Sasquatchis3milesWestand4milesSouthofourcurrentposition.Whatarethecoordinates ofhisposition?Howfarawayishefromus?Ifheruns7milesdueEastwhatwouldhisnew positionbe? 9.VerifytheDistanceFormula1.1forthecaseswhen: aThepointsarearrangedvertically.Hint:Use P a;y 1 and Q a;y 2 bThepointsarearrangedhorizontally.Hint:Use P x 1 ;b and Q x 2 ;b cThepointsareactuallythesamepoint.Youshouldn'tneedahintforthisone. 10.VerifytheMidpointFormulabyshowingthedistancebetween P x 1 ;y 1 and M andthe distancebetween M and Q x 2 ;y 2 arebothhalfofthedistancebetween P and Q 11.Showthatthepoints A )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1, B ; 0and C ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(3aretheverticesofarighttriangle. 12.Findapoint D x;y suchthatthepoints A )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1, B ; 0, C ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3and D arethecorners ofasquare.Justifyyouranswer. 13.Theworldisnotat. 7 ThustheCartesianPlanecannotpossiblybetheendofthestory. DiscusswithyourclassmateshowyouwouldextendCartesianCoordinatestorepresentthe threedimensionalworld.WhatwouldtheDistanceandMidpointformulaslooklike,assuming thoseconceptsmakesenseatall? 7 Therearethosewhodisagreewiththisstatement.LookthemupontheInternetsometimewhenyou'rebored.

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12RelationsandFunctions 1.1.3Answers 1.Therequiredpoints A )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(7, B : 3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2, C ; p 10, D ; 8, E )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 : 5 ; 0, F )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 ; 4, G : 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 : 8,and H ; 5areplottedintheCartesianCoordinatePlanebelow. x y A )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 B : 3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 C ; p 10 D ; 8 E )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 : 5 ; 0 F )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 ; 4 G : 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 : 8 H ; 5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(9 )]TJ/F34 7.9701 Tf 6.586 0 Td [(8 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1123456789 )]TJ/F34 7.9701 Tf 6.587 0 Td [(9 )]TJ/F34 7.9701 Tf 6.587 0 Td [(8 )]TJ/F34 7.9701 Tf 6.587 0 Td [(7 )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 5 6 7 8 9

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1.1TheCartesianCoordinatePlane13 2.aThepoint A )]TJ/F15 9.9626 Tf 7.748 0 Td [(3 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(7is inQuadrantIII symmetricabout x -axiswith )]TJ/F15 9.9626 Tf 7.749 0 Td [(3 ; 7 symmetricabout y -axiswith ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(7 symmetricaboutoriginwith ; 7 bThepoint B : 3 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(2is inQuadrantIV symmetricabout x -axiswith : 3 ; 2 symmetricabout y -axiswith )]TJ/F15 9.9626 Tf 7.749 0 Td [(1 : 3 ; )]TJ/F15 9.9626 Tf 7.748 0 Td [(2 symmetricaboutoriginwith )]TJ/F15 9.9626 Tf 7.749 0 Td [(1 : 3 ; 2 cThepoint C ; p 10is inQuadrantI symmetricabout x -axiswith ; )]TJ 7.749 8.241 Td [(p 10 symmetricabout y -axiswith )]TJ/F53 9.9626 Tf 7.749 0 Td [(; p 10 symmetricaboutoriginwith )]TJ/F37 7.9701 Tf 6.586 0 Td [(; )]TJ 6.586 6.598 Td [(p 10 dThepoint D ; 8is onthepositive y -axis symmetricabout x -axiswith ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(8 symmetricabout y -axiswith ; 8 symmetricaboutoriginwith ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(8 eThepoint E )]TJ/F15 9.9626 Tf 7.749 0 Td [(5 : 5 ; 0is onthenegative x -axis symmetricabout x -axiswith )]TJ/F15 9.9626 Tf 7.749 0 Td [(5 : 5 ; 0 symmetricabout y -axiswith : 5 ; 0 symmetricaboutoriginwith : 5 ; 0 fThepoint F )]TJ/F15 9.9626 Tf 7.749 0 Td [(8 ; 4is inQuadrantII symmetricabout x -axiswith )]TJ/F15 9.9626 Tf 7.749 0 Td [(8 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(4 symmetricabout y -axiswith ; 4 symmetricaboutoriginwith ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(4 gThepoint G : 2 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(7 : 8is inQuadrantIV symmetricabout x -axiswith : 2 ; 7 : 8 symmetricabout y -axiswith )]TJ/F34 7.9701 Tf 6.586 0 Td [(9 : 2 ; )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 : 8 symmetricaboutoriginwith )]TJ/F15 9.9626 Tf 7.749 0 Td [(9 : 2 ; 7 : 8 hThepoint H ; 5is inQuadrantI symmetricabout x -axiswith ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(5 symmetricabout y -axiswith )]TJ/F15 9.9626 Tf 7.749 0 Td [(7 ; 5 symmetricaboutoriginwith )]TJ/F15 9.9626 Tf 7.749 0 Td [(7 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(5 3.a d =5, M = )]TJ/F15 9.9626 Tf 7.749 0 Td [(1 ; 7 2 b d =4 p 10, M = ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(4 c d = p 26, M = 1 ; 3 2 d d = p 37 2 M = 5 6 ; 7 4 e d =3 p 5, M = )]TJ 8.944 14.981 Td [(p 2 2 ; )]TJ 8.944 14.981 Td [(p 3 2 f d = p x 2 + y 2 M = x 2 ; y 2 4.+ p 7 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(2, )]TJ 9.962 8.241 Td [(p 7 ; )]TJ/F15 9.9626 Tf 7.748 0 Td [(2 5. ; 3 6. )]TJ/F15 9.9626 Tf 7.749 0 Td [(1+ p 3 ; 0, )]TJ/F15 9.9626 Tf 7.749 0 Td [(1 )]TJ 9.963 8.241 Td [(p 3 ; 0 7. p 2 2 ; )]TJ/F13 6.9738 Tf 8.944 9.724 Td [(p 2 2 )]TJ/F13 6.9738 Tf 8.944 9.724 Td [(p 2 2 ; p 2 2 8. )]TJ/F15 9.9626 Tf 7.749 0 Td [(3 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(4,5miles, ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(4

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14RelationsandFunctions 1.2Relations Wenowturnourattentiontosetsofpointsintheplane. Definition 1.2 A relation isasetofpointsintheplane. Throughoutthistextwewillseemanydierentwaystodescriberelations.Inthissectionwewill focusourattentionondescribingrelationsgraphically,bymeansofthelistorrostermethodand algebraically.Dependingonthesituation,onemethodmaybeeasierormoreconvenienttouse thananother.Considerthesetofpointsbelow )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 ; 3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 Thesethreepointsconstitutearelation.Letuscallthisrelation R .Above,wehavea graphical descriptionof R .Althoughitisquitepleasingtotheeye,itisn'tthemostportablewaytodescribe R .The list or roster methodofdescribing R simplylistsallofthepointswhichbelongto R Hence,wewrite: R = f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 ; ; 3 ; ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 g : 1 Therostermethodcanbeextendedtodescribe innitelymanypoints,asthenextexampleillustrates. Example 1.2.1 Graphthefollowingrelations. 1. A = f ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1 ; ; 2 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 ; 2 g 2. HLS 1 = f x; 3: )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 4 g 3. HLS 2 = f x; 3: )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x< 4 g 4. V = f ;y : y isarealnumber g 1 Weuse`setbraces' fg toindicatethatthepointsinthelistallbelongtothesameset,inthiscase, R .

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1.2Relations 15 Solution. 1.Tograph A ,wesimplyplotallofthepointswhichbelongto A ,asshownbelowontheleft. 2.Don'tletthenotationinthispartfoolyou.Thenameofthisrelationis HLS 1 ,justlikethe nameoftherelationinpart1was R .Thelettersandnumbersarejustpartofitsname,just likethenumbersandlettersofthephrase`KingGeorgeIII'werepartofGeorge'sname.The nexthurdletoovercomeisthedescriptionof HLS 1 itself )]TJ/F15 10.9091 Tf 12.664 0 Td [(avariableandsomeseemingly extraneouspunctuationhavefoundtheirwayintoournicelittlerosternotation!Theway tomakesenseoftheconstruction f x; 3: )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 4 g istoverbalizethesetbraces fg as`thesetof'andthecolon:as`suchthat'.Inwords, f x; 3: )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 4 g is:`theset ofpoints x; 3suchthat )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 4.'Thepurposeofthevariable x inthiscaseisto describeinnitelymanypoints.Allofthesepointshavethesame y -coordinate,3,butthe x -coordinateisallowedtovarybetween )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and4,inclusive.Someofthepointswhichbelong to HLS 1 includesomefriendlypointslike: )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3, )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 3, ; 3, ; 3, ; 3, ; 3,and ; 3.However, HLS 1 alsocontainsthepoints : 829 ; 3, )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.681 4.295 Td [(5 6 ; 3 p ; 3,andsoon.Itis impossibletolistallofthesepoints,whichiswhythevariable x isused.Plottingseveral friendlyrepresentativepointsshouldconvinceyouthat HLS 1 describesthehorizontalline segmentfromthepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3uptoandincludingthepoint ; 3. x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11234 1 2 3 4 Thegraphof A x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11234 1 2 3 4 Thegraphof HLS 1 3. HLS 2 ishauntinglysimilarto HLS 1 .Infact,theonlydierencebetweenthetwoisthat insteadof` )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 4'wehave` )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x< 4'.Thismeansthatwestillgetahorizontalline segmentwhichincludes )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3andextendsto ; 3, butdoesnotinclude ; 3becauseof thestrictinequality x< 4.Howdowedenotethisonourgraph?Itisacommonmistaketo makethegraphstartat )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3endat ; 3aspicturedbelowontheleft.Theproblemwith thisgraphisthatweareforgettingaboutthepointslike : 1 ; 3, : 5 ; 3, : 9 ; 3, : 99 ; 3, andsoforth.Thereisnorealnumberthatcomes`immediatelybefore'4,andsotodescribe thesetofpointswewant,wedrawthehorizontallinesegmentstartingat )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3anddraw an`opencircle'at ; 3asdepictedbelowontheright.

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16RelationsandFunctions x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(11234 1 2 3 4 ThisisNOTthecorrectgraphof HLS 2 x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11234 1 2 3 4 Thegraphof HLS 2 4.Ourlastexample, V ,describesthesetofpoints ;y suchthat y isarealnumber.Allof thesepointshavean x -coordinateof3,butthe y -coordinateisfreetobewhateveritwants tobe,withoutrestriction.Plottingafew`friendly'pointsof V shouldconvinceyouthatall thepointsof V lieonaverticallinewhichcrossesthe x -axisat x =3.Sincethereisno restrictiononthe y -coordinate,weputarrowsontheendoftheportionofthelinewedraw toindicateitextendsindenitelyinbothdirections.Thegraphof V isbelowontheleft. x y 1234 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 Thegraphof V x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 Thegraphof y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 Therelation V inthepreviousexampleleadsustoournalwaytodescriberelations: algebraically .Wecansimplydescribethepointsin V asthosepointswhichsatisfytheequation x =3.Mostlikely,youhaveseenequationslikethisbefore.Dependingonthecontext,` x =3' couldmeanwehavesolvedanequationfor x andarrivedatthesolution x =3.Inthiscase,however,` x =3'describesasetofpointsintheplanewhose x -coordinateis3.Similarly,theequation y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2inthiscontextcorrespondstoallpointsintheplanewhose y -coordinateis )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Sincethere arenorestrictionsonthe x -coordinatelisted,wewouldgraphtherelation y = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2asthehorizontal lineaboveontheright.Ingeneral,wehavethefollowing.

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1.2Relations 17 EquationsofVerticalandHorizontalLines Thegraphoftheequation x = a isa verticalline through a; 0. Thegraphoftheequation y = b isa horizontalline through ;b Inthenextsection,andinmanymoreafterthat,weshallexplorethegraphsofequationsingreat detail. 2 Fornow,weshalluseournalexampletoillustratehowrelationscanbeusedtodescribe entireregionsintheplane. Example 1.2.2 Graphtherelation: R = f x;y :1
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18RelationsandFunctions 1.2.1Exercises 1.Graphthefollowingrelations. a f )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 9, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 4, )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1, ; 0, ; 1, ; 4, ; 9 g b f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 2, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, ; 5, ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 g c )]TJ/F53 10.9091 Tf 11.364 -8.837 Td [(n; 4 )]TJ/F53 10.9091 Tf 10.909 0 Td [(n 2 : n =0 ; 1 ; 2 d )]TJ/F34 7.9701 Tf 12.753 -4.542 Td [(6 k ;k : k = 1 ; 2 ; 3 ; 4 ; 5 ; 6 2.Graphthefollowingrelations. a f x; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2: x> )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 g b f ;y : y 5 g c f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ;y : )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 ;y< 4 g h f x;y : )]TJ 8.485 9.024 Td [(p 2 x 2 3 ;
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1.2Relations 19 e x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(112345 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 5 Thegraphofrelation E f x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(112345 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 Thegraphofrelation F 4.Graphthefollowinglines. a x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2b y =3 5.Whatisanothernamefortheline x =0?For y =0? 6.Somerelationsarefairlyeasytodescribeinwordsorwiththerostermethodbutarerather dicult,ifnotimpossible,tograph.Discusswithyourclassmateshowyoumightgraphthe followingrelations.Pleasenotethatinthenotationbelowweareusingthe ellipsis ,..., todenotethatthelistdoesnotend,butrather,continuestofollowtheestablishedpattern indenitely.Forthersttworelations,givetwoexamplesofpointswhichbelongtothe relationandtwopointswhichdonotbelongtotherelation. a f x;y : x isanoddinteger,and y isaneveninteger. g b f x; 1: x isanirrationalnumber g c f ; 0 ; ; 1 ; ; 2 ; ; 3 ; ; 4 ; ; 5 ;::: g d f :::; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 9 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 ; 4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; ; 0 ; ; 1 ; ; 4 ; ; 9 ;::: g

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20RelationsandFunctions 1.2.2Answers 1.a x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123 1 2 3 4 5 6 7 8 9 b x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1123 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 5 c x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112 1 2 3 4 d x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123456 )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 5 6 2.a x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 b x y 123 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 5 c x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 d x y 123 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3

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1.2Relations 21 e x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123 1 2 3 4 f x y 123 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 g x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123 1 2 3 4 h x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(11 1 2 3 4 5 6 7 3.a A = f )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 2 ; ; 3 ; ; 4 g b B = f x;y : x> )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 g c C = f x;y : y 0 g d D = f x;y : )]TJ/F15 10.9091 Tf 8.484 0 Td [(3
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22RelationsandFunctions 1.3GraphsofEquations Intheprevioussection,wesaidthatonecoulddescriberelationsalgebraicallyusingequations.In thissection,webegintoexplorethistopicingreaterdetail.Themainideaofthissectionis TheFundamentalGraphingPrinciple Thegraphofanequationisthesetofpointswhichsatisfytheequation.Thatis,apoint x;y is onthegraphofanequationifandonlyif x and y satisfytheequation. Example 1.3.1 Determineif ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1isonthegraphof x 2 + y 3 =1. Solution. Tocheck,wesubstitute x =2and y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1intotheequationandseeiftheequationis satised 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 ? =1 3 6 =1 Hence, ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1is not onthegraphof x 2 + y 3 =1. Wecouldspendhoursrandomlyguessingandcheckingtoseeifpointsareonthegraphofthe equation.Amoresystematicapproachisoutlinedinthefollowingexample. Example 1.3.2 Graph x 2 + y 3 =1. Solution. Toecientlygeneratepointsonthegraphofthisequation,werstsolvefor y x 2 + y 3 =1 y 3 =1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 3 p y 3 = 3 p 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 y = 3 p 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 Wenowsubstituteavalueinfor x ,determinethecorrespondingvalue y ,andplottheresulting point, x;y .Forexample,for x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(3,wesubstitute y = 3 p 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = 3 p 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 = 3 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(8= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; sothepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2isonthegraph.Continuinginthismanner,wegenerateatableofpoints whichareonthegraphoftheequation.Thesepointsarethenplottedintheplaneasshownbelow.

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1.3GraphsofEquations23 x y x;y )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F35 5.9776 Tf 11.515 5.069 Td [(3 p 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F35 5.9776 Tf 11.515 5.069 Td [(3 p 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 0 1 ; 1 1 0 ; 0 2 )]TJ/F35 5.9776 Tf 11.515 5.069 Td [(3 p 3 ; )]TJ/F35 5.9776 Tf 11.516 5.069 Td [(3 p 3 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 Remember,thesepointsconstituteonlyasmall sampling ofthepointsonthegraphofthis equation.Togetabetterideaoftheshapeofthegraph,wecouldplotmorepointsuntilwefeel comfortable`connectingthedots.'Doingsowouldresultinacurvesimilartotheonepictured belowonthefarleft. x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 Don'tworryofyoudon'tgetallofthelittlebendsandcurvesjustright )]TJ/F15 10.9091 Tf 12.446 0 Td [(Calculusiswherethe artofprecisegraphingtakescenterstage.Fornow,wewillsettlewithournaive`plugandplot' approachtographing.Ifyoufeellikeallofthistediouscomputationandplottingisbeneathyou, thenyoucanreachforagraphingcalculator,inputtheformulaasshownabove,andgraph. Ofallofthepointsonthegraphofanequation,theplaceswherethegraphcrossestheaxeshold specialsignicance.Thesearecalledthe intercepts ofthegraph.Interceptscomeintwodistinct varieties: x -interceptsand y -intercepts.Theyaredenedbelow. Definition 1.3 Supposethegraphofanequationisgiven. Apointatwhichagraphmeetsthe x -axisiscalledan x -intercept ofthegraph. Apointatwhichagraphmeetsthe y -axisiscalledan y -intercept ofthegraph. Inourpreviousexamplethegraphhadtwo x -intercepts, )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 0and ; 0,andone y -intercept, ; 1.Thegraphofanequationcanhaveanynumberofintercepts,includingnoneatall!Since

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24RelationsandFunctions x -interceptslieonthe x -axis,wecanndthembysetting y =0intheequation.Similarly,since y -interceptslieonthe y -axis,wecanndthembysetting x =0intheequation.Keepinmind, interceptsare points andthereforemustbewrittenasorderedpairs.Tosummarize, Stepsforndingtheinterceptsofthegraphofanequation Givenanequationinvolving x and y : the x -interceptsalwayshavetheform x; 0;tondthe x -interceptsofthegraph,set y =0 andsolvefor x y -interceptsalwayshavetheform ;y ;tondthe y -interceptsofthegraph,set x =0and solvefor y Anotherfactwhichyoumayhavenoticedaboutthegraphinthepreviousexampleisthatitseems tobesymmetricaboutthe y -axis.Toactuallyprovethisanalytically,weassume x;y isageneric pointonthegraphoftheequation.Thatis,weassume x 2 + y 3 =1.AswelearnedinSection1.1, thepointsymmetricto x;y aboutthe y -axisis )]TJ/F53 10.9091 Tf 8.485 0 Td [(x;y .Toshowthegraphissymmetricabout the y -axis,weneedtoshowthat )]TJ/F53 10.9091 Tf 8.485 0 Td [(x;y isonthegraphwhenever x;y is.Inotherwords,we needtoshow )]TJ/F53 10.9091 Tf 8.485 0 Td [(x;y satisestheequation x 2 + y 3 =1whenever x;y does.Substitutinggives )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 + y 3 ? =1 x 2 + y 3 X =1 Whenwesubstituted )]TJ/F53 10.9091 Tf 8.485 0 Td [(x;y intotheequation x 2 + y 3 =1,weobtainedtheoriginalequationback whenwesimplied.Thismeans )]TJ/F53 10.9091 Tf 8.485 0 Td [(x;y satisestheequationandhenceisonthegraph.Inthis way,wecancheckwhetherthegraphofagivenequationpossessesanyofthesymmetriesdiscussed inSection1.1.Theresultsaresummarizedbelow. Stepsfortestingifthegraphofanequationpossessessymmetry Totestthegraphofanequationforsymmetry Aboutthe y -axis:Substitute )]TJ/F53 10.9091 Tf 8.485 0 Td [(x;y intotheequationandsimplify.Iftheresultisequivalent totheoriginalequation,thegraphissymmetricaboutthe y -axis. Aboutthe x -axis:Substitute x; )]TJ/F53 10.9091 Tf 8.485 0 Td [(y intotheequationandsimplify.Iftheresultisequivalenttotheoriginalequation,thegraphissymmetricaboutthe x -axis. Abouttheorigin:Substitute )]TJ/F53 10.9091 Tf 8.485 0 Td [(x; )]TJ/F53 10.9091 Tf 8.485 0 Td [(y intotheequationandsimplify.Iftheresultis equivalenttotheoriginalequation,thegraphissymmetricabouttheorigin. Interceptsandsymmetryaretwotoolswhichcanhelpussketchthegraphofanequationanalytically,asevidencedinthenextexample. Example 1.3.3 Findthe x -and y -interceptsifanyofthegraphof x )]TJ/F15 10.9091 Tf 11.088 0 Td [(2 2 + y 2 =1.Testfor symmetry.Plotadditionalpointsasneededtocompletethegraph.

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1.3GraphsofEquations25 Solution. Tolookfor x -intercepts,weset y =0andsolve: x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 + y 2 =1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 +0 2 =1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 =1 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 = p 1extractsquareroots x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2= 1 x =2 1 x =3 ; 1 Weget two answersfor x whichcorrespondto two x -intercepts: ; 0and ; 0.Turningour attentionto y -intercepts,weset x =0andsolve: x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 + y 2 =1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 + y 2 =1 4+ y 2 =1 y 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 SincethereisnorealnumberwhichsquarestoanegativenumberDoyourememberwhy?,we areforcedtoconcludethatthegraphhas no y -intercepts. Plottingthedatawehavesofar,weget ; 0 ; 0 x y 1234 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 Movingalongtosymmetry,wecanimmediatelydismissthepossibilitythatthegraphissymmetric aboutthe y -axisortheorigin.Ifthegraphpossessedeitherofthesesymmetries,thenthefact that ; 0isonthegraphwouldmean )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 0wouldhavetobeonthegraph.Why?Since )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0wouldbeanother x -interceptandwe'vefoundallofthese,thegraphcan'thave y -axisor originsymmetry.Theonlysymmetrylefttotestissymmetryaboutthe x -axis.Tothatend,we substitute x; )]TJ/F53 10.9091 Tf 8.485 0 Td [(y intotheequationandsimplify x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 + y 2 =1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 + )]TJ/F53 10.9091 Tf 8.485 0 Td [(y 2 ? =1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 + y 2 X =1 Sincewehaveobtainedouroriginalequation,weknowthegraphissymmetricaboutthe x -axis. Thismeanswecancutour`plugandplot'timeinhalf:whateverhappensbelowthe x -axisis reectedabovethe x -axis,andvice-versa.Proceedingaswedidinthepreviousexample,weobtain

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26RelationsandFunctions x y 1234 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 Acoupleofremarksareinorder.First,itisentirelypossibletochooseavaluefor x whichdoes notcorrespondtoapointonthegraph.Forexample,inthepreviousexample,ifwesolvefor y as isourcustom,weget: y = p 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 : Uponsubstituting x =0intotheequation,wewouldobtain y = p 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [( )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 = p 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4= p )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; whichisnotarealnumber.Thismeanstherearenopointsonthegraphwithan x -coordinate of0.Whenthishappens,wemoveonandtryanotherpoint.Thisisanotherdrawbackofthe `plug-and-plot'approachtographingequations.Luckily,wewilldevotemuchoftheremainder ofthisbookdevelopingtechniqueswhichallowustographentirefamiliesofequationsquickly. 1 Second,itisinstructivetoshowwhatwouldhavehappenedhadwetestedtheequationinthelast exampleforsymmetryaboutthe y -axis.Substituting )]TJ/F53 10.9091 Tf 8.485 0 Td [(x;y intotheequationyields x )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 2 + y 2 =1 )]TJ/F53 10.9091 Tf 8.484 0 Td [(x )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 2 + y 2 ? =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x +2 2 + y 2 ? =1 x +2 2 + y 2 ? =1 : Thislastequationdoesnot appear tobeequivalenttoouroriginalequation.However,to prove itisnotsymmetricaboutthe y -axis,weneedtondapoint x;y onthegraphwhosereection )]TJ/F53 10.9091 Tf 8.485 0 Td [(x;y isnot.Our x -intercept ; 0tsthisbillnicely,sinceifwesubstitute )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 0intothe equationweget x )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 2 + y 2 ? =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 +0 2 6 =1 9 6 =1 : Thisprovesthat )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0isnotonthegraph. 1 Withouttheuseofacalculator,ifyoucanbelieveit!

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1.3GraphsofEquations27 1.3.1Exercises 1.Foreachequationgivenbelow Findthe x -and y -interceptsofthegraph,ifanyexist. FollowingtheprocedureinExample1.3.2,createatableofsamplepointsonthegraph oftheequation. Plotthesamplepointsandcreatearoughsketchofthegraphoftheequation. Testforsymmetry.Iftheequationappearstofailanyofthesymmetrytests,nda pointonthegraphoftheequationwhosereectionfailstobeonthegraphaswasdone attheendofExample1.3.3 a y = x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x b y = x 2 +1 c y = p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 d3 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y =7 e x 3 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 f x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 2 =1 2.TheprocedureswhichwehaveoutlinedintheExamplesofthissectionandusedintheexercisesgivenaboveallrelyonthefactthattheequationswerewell-behaved".Noteverything inMathematicsisquitesotame,asthefollowingequationswillshowyou.Discusswithyour classmateshowyoumightapproachgraphingtheseequations.Whatdicultiesarisewhen tryingtoapplythevarioustestsandproceduresgiveninthissection?Formoreinformation, includingpicturesofthecurves,eachcurvenameisalinktoitspageatwww.wikipedia.org. Foramuchlongerlistoffascinatingcurves,clickhere a x 3 + y 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 xy =0FoliumofDescartes b x 4 = x 2 + y 2 KampyleofEudoxus c y 2 = x 3 +3 x 2 Tschirnhausencubic d x 2 + y 2 2 = x 3 + y 3 Crookedegg

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28RelationsandFunctions 1.3.2Answers 1.a y = x 3 )]TJ/F53 10.9091 Tf 10.91 0 Td [(x x -intercepts: )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; ; 0 ; ; 1 y -intercept: ; 0 x y x;y )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 0 0 ; 0 1 0 ; 0 2 6 ; 6 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 Thegraphisnotsymmetricaboutthe x -axis.e.g. ; 6isonthegraphbut ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(6isnot Thegraphisnotsymmetricaboutthe y -axis.e.g. ; 6isonthegraphbut )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 6isnot Thegraphissymmetricaboutthe origin. b y = x 2 +1 Thegraphhasno x -intercepts y -intercept: ; 1 x y x;y )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 2 0 1 ; 1 1 2 ; 2 2 5 ; 5 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 1 2 3 4 5 Thegraphisnotsymmetricaboutthe x -axise.g. ; 5isonthegraphbut ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(5isnot Thegraphissymmetricaboutthe y -axis Thegraphisnotsymmetricaboutthe origine.g. ; 5isonthegraphbut )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(5isnot

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1.3GraphsofEquations29 c y = p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x -intercept: ; 0 Thegraphhasno y -intercepts x y x;y 2 0 ; 0 3 1 ; 1 6 2 ; 2 11 3 ; 3 x y 1234567891011 1 2 3 Thegraphisnotsymmetricaboutthe x -axise.g. ; 1isonthegraphbut ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1isnot Thegraphisnotsymmetricaboutthe y -axise.g. ; 1isonthegraphbut )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1isnot Thegraphisnotsymmetricaboutthe origine.g. ; 1isonthegraphbut )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1isnot d3 x )]TJ/F53 10.9091 Tf 11.45 0 Td [(y =7Re-writeas y =3 x )]TJ/F15 10.9091 Tf 11.45 0 Td [(7to createthechart. x -intercept: 7 3 ; 0 y -intercept: ; 7 x y x;y )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(13 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 2 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(13 )]TJ/F35 5.9776 Tf 5.756 0 Td [(12 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.756 0 Td [(10 )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 Thegraphisnotsymmetricaboutthe x -axise.g. ; 2isonthegraphbut ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2isnot Thegraphisnotsymmetricaboutthe y -axise.g. ; 2isonthegraphbut )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 2isnot Thegraphisnotsymmetricaboutthe origine.g. ; 2isonthegraphbut )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2isnot

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30RelationsandFunctions e x 3 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4Re-writeas y = )]TJ/F15 10.9091 Tf 12.436 7.38 Td [(4 x 3 tocreatethechart. Thegraphhasno x -intercepts Thegraphhasno y -intercepts x y x;y )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 4 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 32 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 ; 32 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(32 1 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(32 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 2 ; )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(112 )]TJ/F35 5.9776 Tf 5.756 0 Td [(32 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 4 32 Thegraphisnotsymmetricaboutthe x -axise.g. ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4isonthegraphbut ; 4isnot Thegraphisnotsymmetricaboutthe y -axise.g. ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4isonthegraphbut )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4isnot Thegraphissymmetricaboutthe origin f x 2 )]TJ/F53 10.9091 Tf 11.109 0 Td [(y 2 =1Re-writeas y = p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 tocreatethechart. x -intercepts: )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; ; 0 Thegraphhasno y -intercepts x y x;y )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 p 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; p 8 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 p 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; p 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 1 0 ; 0 2 p 3 ; p 3 3 p 8 ; p 8 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 Thegraphissymmetricaboutthe x -axis Thegraphissymmetricaboutthe y -axis Thegraphissymmetricaboutthe origin

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1.4IntroductiontoFunctions31 1.4IntroductiontoFunctions OneofthecoreconceptsinCollegeAlgebraisthe function .Therearemanywaystodescribea functionandwebeginbydeningafunctionasaspecialkindofrelation. Definition 1.4 Arelationinwhicheach x -coordinateismatchedwithonlyone y -coordinateis saidtodescribe y asa function of x Example 1.4.1 Whichofthefollowingrelationsdescribe y asafunctionof x ? 1. R 1 = f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 ; ; 3 ; ; 4 ; ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 g 2. R 2 = f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 ; ; 3 ; ; 3 ; ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 g Solution. Aquickscanofthepointsin R 1 revealsthatthe x -coordinate1ismatchedwith two dierent y -coordinates:namely3and4.Hencein R 1 y is not afunctionof x .Onthe otherhand,every x -coordinatein R 2 occursonlyoncewhichmeanseach x -coordinatehasonlyone corresponding y -coordinate.So, R 2 does represent y asafunctionof x Notethatinthepreviousexample,therelation R 2 containedtwodierentpointswiththesame y -coordinates,namely ; 3and ; 3.Remember,inordertosay y isafunctionof x ,wejust needtoensurethesame x -coordinateisn'tusedinmorethanonepoint. 1 Toseewhatthefunctionconceptmeansgeometrically,wegraph R 1 and R 2 intheplane. x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1123 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 Thegraphof R 1 x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1123 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 Thegraphof R 2 Thefactthatthe x -coordinate1ismatchedwithtwodierent y -coordinatesin R 1 presentsitself graphicallyasthepoints ; 3and ; 4lyingonthesameverticalline, x =1.Ifweturnour attentiontothegraphof R 2 ,weseethatnotwopointsoftherelationlieonthesameverticalline. Wecangeneralizethisideaasfollows Theorem 1.1 TheVerticalLineTest: Asetofpointsintheplanerepresents y asafunction of x ifandonlyifnotwopointslieonthesameverticalline. 1 Wewillhaveoccasionlaterinthetexttoconcernourselveswiththeconceptof x beingafunctionof y .Inthis case, R 1 represents x asafunctionof y ; R 2 doesnot.

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32RelationsandFunctions ItisworthtakingsometimetomeditateontheVerticalLineTest;itwillchecktoseehowwell youunderstandtheconceptof`function'aswellastheconceptof`graph'. Example 1.4.2 UsetheVerticalLineTesttodeterminewhichofthefollowingrelationsdescribes y asafunctionof x x y 123 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 Thegraphof R x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(11 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 Thegraphof S Solution. Lookingatthegraphof R ,wecaneasilyimagineaverticallinecrossingthegraph morethanonce.Hence, R does not represent y asafunctionof x .However,inthegraphof S everyverticallinecrossesthegraphatmostonce,andso S does represent y asafunctionof x Intheprevioustest,wesaythatthegraphoftherelation R fails theVerticalLineTest,whereas thegraphof S passes theVerticalLineTest.Notethatinthegraphof R thereareinnitelymany verticallineswhichcrossthegraphmorethanonce.However,tofailtheVerticalLineTest,allyou needis one verticallinethattsthebill,asthenextexampleillustrates. Example 1.4.3 UsetheVerticalLineTesttodeterminewhichofthefollowingrelationsdescribes y asafunctionof x x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(11 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 Thegraphof S 1 x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(11 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 Thegraphof S 2

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1.4IntroductiontoFunctions33 Solution. Both S 1 and S 2 areslightmodicationstotherelation S inthepreviousexamplewhose graphwedeterminedpassedtheVerticalLineTest.Inboth S 1 and S 2 ,itistheadditionofthe point ; 2whichthreatenstocausetrouble.In S 1 ,thereisapointonthecurvewith x -coordinate 1justbelow ; 2,whichmeansthatboth ; 2andthispointonthecurvelieontheverticalline x =1.Seethepicturebelow.Hence,thegraphof S 1 failstheVerticalLineTest,so y is not a functionof x here.However,in S 2 noticethatthepointwith x -coordinate1onthecurvehasbeen omitted,leavingan`opencircle'there.Hence,theverticalline x =1crossesthegraphof S 2 only atthepoint ; 2.Indeed,anyverticallinewillcrossthegraphatmostonce,sowehavethatthe graphof S 2 passestheVerticalLineTest.Thusitdescribes y asafunctionof x x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 S 1 andtheline x =1 Supposearelation F describes y asafunctionof x .Thesetsof x -and y -coordinatesaregiven specialnames. Definition 1.5 Suppose F isarelationwhichdescribes y asafunctionof x Thesetofthe x -coordinatesofthepointsin F iscalledthe domain of F Thesetofthe y -coordinatesofthepointsin F iscalledthe range of F Wedemonstratendingthedomainandrangeoffunctionsgiventouseithergraphicallyorviathe rostermethodinthefollowingexample. Example 1.4.4 Findthedomainandrangeofthefollowingfunctions 1. F = f )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 2 ; ; 1 ; ; 2 ; ; 2 g 2. G isthefunctiongraphedbelow:

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34RelationsandFunctions x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(11 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 Thegraphof G Solution. Thedomainof F isthesetofthe x -coordinatesofthepointsin F : f)]TJ/F15 10.9091 Tf 13.94 0 Td [(3 ; 0 ; 4 ; 5 g and therangeof F isthesetofthe y -coordinates: f 1 ; 2 g : 2 Todeterminethedomainandrangeof G ,weneedtodeterminewhich x and y valuesoccuras coordinatesofpointsonthegivengraph.Tondthedomain,itmaybehelpfultoimaginecollapsing thecurvetothe x -axisanddeterminingtheportionofthe x -axisthatgetscovered.Thisiscalled projecting thecurvetothe x -axis.Beforewestartprojecting,weneedtopayattentiontotwo subtlenotationsonthegraph:thearrowheadonthelowerleftcornerofthegraphindicatesthatthe graphcontinuestocurvedownwardstotheleftforevermore;andtheopencircleat ; 3indicates thatthepoint ; 3isn'tonthegraph,butallpointsonthecurveleadinguptothatpointareon thecurve. projectdown projectup x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(11 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 Thegraphof G x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(11 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 Thegraphof G 2 Whenlistingnumbersinaset,welisteachnumberonlyonce,inincreasingorder.

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1.4IntroductiontoFunctions35 Weseefromthegurethatifweprojectthegraphof G tothe x -axis,wegetallrealnumbersless than1.Usingintervalnotation,wewritethedomainof G is ; 1.Todeterminetherangeof G ,weprojectthecurvetothe y -axisasfollows: projectleft projectright x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(11 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 Thegraphof G x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(11 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 Thegraphof G Notethateventhoughthereisanopencircleat ; 3,westillincludethe y valueof3inourrange, sincethepoint )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 3isonthegraphof G .Weseethattherangeof G isallrealnumbersless thanorequalto4,or,inintervalnotation: ; 4]. Allfunctionsarerelations,butnotallrelationsarefunctions.Thustheequationswhichdescribed therelationsinSection1.2mayormaynotdescribe y asafunctionof x .Thealgebraicrepresentation offunctionsispossiblythemostimportantwaytoviewthemsoweneedaprocessfordetermining whetherornotanequationofarelationrepresentsafunction.Wedelaythediscussionofnding thedomainofafunctiongivenalgebraicallyuntilSection1.5. Example 1.4.5 Determinewhichequationsrepresent y asafunctionof x : 1. x 3 + y 2 =1 2. x 2 + y 3 =1 3. x 2 y =1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 y Solution. Foreachoftheseequations,wesolvefor y anddeterminewhethereachchoiceof x will determineonlyonecorrespondingvalueof y 1. x 3 + y 2 =1 y 2 =1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 p y 2 = p 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 extractsquareroots y = p 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3

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36RelationsandFunctions Ifwesubstitute x =0intoourequationfor y ,weget: y = p 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 3 = 1,sothat ; 1 and ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1areonthegraphofthisequation.Hence,thisequationdoes not represent y as afunctionof x 2. x 2 + y 3 =1 y 3 =1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 3 p y 3 = 3 p 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 y = 3 p 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 Foreverychoiceof x ,theequation y = 3 p 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 returnsonly one valueof y .Hence,this equationdescribes y asafunctionof x 3. x 2 y =1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 y x 2 y +3 y =1 y )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 +3 =1factor y = 1 x 2 +3 Foreachchoiceof x ,thereisonlyonevaluefor y ,sothisequationdescribes y asafunctionof x Ofcourse,wecouldalwaysuseourgraphingcalculatortoverifyourresponsestotheprevious example.Forexample,ifwewantedtoverifythattherstequationdoesnotrepresent y asa functionof x ,wecouldentertheequationfor y intothecalculatorasindicatedbelowandgraph. Notethatweneedtoenterbothsolutions )]TJ/F15 10.9091 Tf 11.63 0 Td [(thepositiveandthenegativesquareroot )]TJ/F15 10.9091 Tf 11.63 0 Td [(for y .The resultinggraphclearlyfailstheVerticalLineTest,sodoesnotrepresent y asafunctionof x

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1.4IntroductiontoFunctions37 1.4.1Exercises 1.Determinewhichofthefollowingrelationsrepresent y asafunctionof x .Findthedomain andrangeofthoserelationswhicharefunctions. a f )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 9, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 4, )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1, ; 0, ; 1, ; 4, ; 9 g b f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 2, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, ; 5, ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 g c f x;y : x isanoddinteger,and y isaneveninteger g d f x; 1: x isanirrationalnumber g e f ; 0, ; 1, ; 2, ; 3, ; 4, ; 5 ; ... g f f :::; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 9, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 4, )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 1, ; 0, ; 1, ; 4, ; 9 ; ... g g f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ;y : )]TJ/F15 10.9091 Tf 8.485 0 Td [(3
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38RelationsandFunctions e x y 123456789 1 2 3 f x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 1 2 3 4 g x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 h x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 3.Determinewhichofthefollowingrelationsrepresent y asafunctionof x a y = x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x b y = p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 c x 3 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 d x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 2 =1 e y = x x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 f y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 4.Explainwhytheheight h ofaSasquatchisafunctionofitsage N inyears.Giventhata Sasquatchis2feettallatbirth,experiencesgrowthspurtsatages3,23and57,andlivesto beabout150yearsoldwithamaximumheightof9feet,sketcharoughgraphoftheheight function. 5.Explainwhythepopulation P ofSasquatchinagivenareaisafunctionoftime t .What wouldbetherangeofthisfunction? 6.Explainwhytherelationbetweenyourclassmatesandtheiremailaddressesmaynotbea function.WhataboutphonenumbersandSocialSecurityNumbers? 7.TheprocessgiveninExample1.4.5fordeterminingwhetheranequationofarelationrepresents y asafunctionof x breaksdownifwecannotsolvetheequationfor y intermsof x .

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1.4IntroductiontoFunctions39 However,thatdoesnotpreventusfromprovingthatanequationwhichfailstorepresent y asafunctionof x actuallyfailstodoso.Whatwereallyneedistwopointswiththesame x -coordinateanddierent y -coordinateswhichbothsatisfytheequationsothatthegraph oftherelationwouldfailtheVerticalLineTest1.1.Discusswithyourclassmateshowyou mightndsuchpointsfortherelationsgivenbelow. a x 3 + y 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 xy =0 b x 4 = x 2 + y 2 c y 2 = x 3 +3 x 2 d x 2 + y 2 2 = x 3 + y 3

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40RelationsandFunctions 1.4.2Answers 1.aFunction domain= f)]TJ/F15 10.9091 Tf 13.939 0 Td [(3, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2, )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,0,1,2,3 g range= f 0,1,4,9 g bNotafunction cNotafunction dFunction domain= f x : x isirrational g range= f 1 g eFunction domain= f x : x =2 n forsomewhole number n g range= f y : y isanywholenumber g fFunction domain= f x : x isanyinteger g range= f y : y = n 2 forsomeinteger n g gNotafunction hFunction domain=[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 4 range= f 3 g 2.aFunction domain= f)]TJ/F15 10.9091 Tf 13.939 0 Td [(4, )]TJ/F15 10.9091 Tf 8.485 0 Td [(3, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2, )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,0,1 g range= f)]TJ/F15 10.9091 Tf 13.94 0 Td [(1,0,1,2,3,4 g bNotafunction cFunction domain= ; 1 range=[1 ; 1 dNotafunction eFunction domain=[2 ; 1 range=[0 ; 1 fFunction domain= ; 1 range= ; 4] gNotafunction hFunction domain=[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 ; 3 range= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ [0 ; 4 3.aFunction bFunction cFunction dNotafunction eFunction fNotafunction

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1.5FunctionNotation41 1.5FunctionNotation InDenition1.4,wedescribedafunctionasaspecialkindofrelation )]TJ/F15 10.9091 Tf 13.292 0 Td [(oneinwhicheach x coordinateismatchedwithonlyone y -coordinate.Inthissection,wefocusmoreonthe process bywhichthe x ismatchedwiththe y .Ifwethinkofthedomainofafunctionasasetof inputs andtherangeasasetof outputs ,wecanthinkofafunction f asaprocessbywhicheachinput x ismatchedwithonlyoneoutput y .Sincetheoutputiscompletelydeterminedbytheinput x andtheprocess f ,wesymbolizetheoutputwith functionnotation :` f x ',read` f of x .'Inthis case,theparenthesesheredonotindicatemultiplication,astheydoelsewhereinalgebra.This couldcauseconfusionifthecontextisnotclear.Inotherwords, f x isthe output whichresults byapplyingthe process f tothe input x .Thisrelationshipistypicallyvisualizedusingadiagram similartotheonebelow. f x Domain Inputs y = f x Range Outputs Thevalueof y iscompletelydependentonthechoiceof x .Forthisreason, x isoftencalledthe independentvariable ,or argument of f ,whereas y isoftencalledthe dependentvariable Asweshallsee,theprocessofafunction f isusuallydescribedusinganalgebraicformula.For example,supposeafunction f takesarealnumberandperformsthefollowingtwosteps,insequence 1.multiplyby3 2.add4 Ifwechoose5asourinput,instep1wemultiplyby3toget=15.Instep2,weadd4to ourresultfromstep1whichyields15+4=19.Usingfunctionnotation,wewouldwrite f =19 toindicatethattheresultofapplyingtheprocess f totheinput5givestheoutput19.Ingeneral, ifweuse x fortheinput,applyingstep1produces3 x .Followingwithstep2produces3 x +4as ournaloutput.Henceforaninput x ,wegettheoutput f x =3 x +4.Noticethattocheckour formulaforthecase x =5,we replace theoccurrenceof x intheformulafor f x with5toget f =3+4=15+4=19,asrequired.

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42RelationsandFunctions Example 1.5.1 Supposeafunction g isdescribedbyapplyingthefollowingsteps,insequence 1.add4 2.multiplyby3 Determine g andndanexpressionfor g x Solution. Startingwith5,step1gives5+4=9.Continuingwithstep2,weget=27.To ndaformulafor g x ,westartwithourinput x .Step1produces x +4.Wenowwishtomultiply thisentirequantityby3,soweuseaparentheses:3 x +4=3 x +12.Hence, g x =3 x +12.We cancheckourformulabyreplacing x with5toget g =3+12=15+12=27 X MostofthefunctionswewillencounterinCollegeAlgebrawillbedescribedusingformulaslike theoneswedevelopedfor f x and g x above.Evaluatingformulasusingthisfunctionnotation isakeyskillforsuccessinthisandmanyothermathcourses. Example 1.5.2 For f x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 +3 x +4,ndandsimplify 1. f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, f f 2. f x ,2 f x 3. f x +2, f x +2, f x + f Solution. 1.Tond f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,wereplaceeveryoccurrenceof x intheexpression f x with )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 +3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3+4 =0 Similarly, f = )]TJ/F15 10.9091 Tf 8.485 0 Td [( 2 +3+4=4,and f = )]TJ/F15 10.9091 Tf 8.485 0 Td [( 2 +3+4= )]TJ/F15 10.9091 Tf 8.485 0 Td [(4+6+4=6. 2.Tond f x ,wereplaceeveryoccurrenceof x withthequantity2 x f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [( x 2 +3 x +4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [( x 2 + x +4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 x 2 +6 x +4 Theexpression2 f x meanswemultiplytheexpression f x by2 2 f x =2 )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 +3 x +4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 2 +6 x +8

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1.5FunctionNotation43 Notethedierencebetweentheanswersfor f x and2 f x .For f x ,wearemultiplying the input by2;for2 f x ,wearemultiplyingthe output by2.Aswesee,wegetentirely dierentresults.Alsonotethepracticeofusingparentheseswhensubstitutingonealgebraic expressionintoanother;wehighlyrecommendthispracticeasitwillreducecarelesserrors. 3.Tond f x +2,wereplaceeveryoccurrenceof x withthequantity x +2 f x +2= )]TJ/F15 10.9091 Tf 8.485 0 Td [( x +2 2 +3 x +2+4 = )]TJ/F55 10.9091 Tf 10.303 8.837 Td [()]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 +4 x +4 + x +6+4 = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4+3 x +6+4 = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x +6 Tond f x +2,weadd2totheexpressionfor f x f x +2= )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 +3 x +4 +2 = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 +3 x +6 Onceagain,weseethereisadramaticdierencebetweenmodifyingtheinputandmodifying theoutput.Finally,in f x + f weareaddingthevalue f totheexpression f x Fromourworkabove,wesee f =6sothat f x + f = )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 +3 x +4 +6 = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 +3 x +10 Noticethat f x +2, f x +2and f x + f arethree dierent expressions.Eventhough functionnotationusesparentheses,asdoesmultiplication,thereisno general`distributive property'offunctionnotation. Supposewewishtond r for r x = 2 x x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 .Substitutiongives r = 2 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 = 6 0 ; whichisundened.Thenumber3isnotanallowableinputtothefunction r ;inotherwords,3is notinthedomainof r .Whichotherrealnumbersareforbiddeninthisformula?Wethinkback toarithmetic.Thereason r isundenedisbecausesubstitutionresultsinadivisionby0.To determinewhichothernumbersresultinsuchatransgression,wesetthedenominatorequalto0 andsolve x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9=0 x 2 =9 p x 2 = p 9extractsquareroots x = 3

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44RelationsandFunctions Aslongaswesubstitutenumbersotherthan3and )]TJ/F15 10.9091 Tf 8.485 0 Td [(3,theexpression r x isarealnumber.Hence, wewriteourdomaininintervalnotationas ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 3 [ ; 1 .Whenaformulafora functionisgiven,weassumethefunctionisvalidforallrealnumberswhichmakearithmeticsense whensubstitutedintotheformula.Thissetofnumbersisoftencalledthe implieddomain 1 of thefunction.Atthisstage,thereareonlytwomathematicalsinsweneedtoavoid:divisionby0 andextractingevenrootsofnegativenumbers.Thefollowingexampleillustratestheseconcepts. Example 1.5.3 Findthedomain 2 ofthefollowingfunctions. 1. f x = 2 1 )]TJ/F15 10.9091 Tf 18.772 7.38 Td [(4 x x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2. g x = p 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3. h x = 5 p 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4. r x = 4 6 )]TJ 10.909 8.57 Td [(p x +3 5. I x = 3 x 2 x Solution. 1.Intheexpressionfor f ,therearetwodenominators.Weneedtomakesureneitherofthemis 0.Tothatend,weseteachdenominatorequalto0andsolve.Forthe`small'denominator, weget x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3=0or x =3.Forthe`large'denominator 1 )]TJ/F15 10.9091 Tf 18.772 7.38 Td [(4 x x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 =0 1= 4 x x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3= 4 x x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3cleardenominators x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3=4 x )]TJ/F15 10.9091 Tf 8.485 0 Td [(3=3 x )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= x Sowegettworealnumberswhichmakedenominators0,namely x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and x =3.Our domainisallrealnumbers except )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and3: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 3 [ ; 1 1 or,`implicitdomain' 2 Theword`implied'is,well,implied.

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1.5FunctionNotation45 2.Thepotentialdisasterfor g isiftheradicand 3 isnegative.Toavoidthis,weset4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 0 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 0 4 3 x 4 3 x Hence,aslongas x 4 3 ,theexpression4 )]TJ/F15 10.9091 Tf 11.61 0 Td [(3 x 0,andtheformula g x returnsareal number.Ourdomainis )]TJ/F54 10.9091 Tf 5 -8.836 Td [( ; 4 3 3.Theformulafor h x ishauntinglyclosetothatof g x withonekeydierence )]TJ/F15 10.9091 Tf 12.671 0 Td [(whereas theexpressionfor g x includesanevenindexedrootnamelyasquareroot,theformula for h x involvesanoddindexedrootthefthroot.Sinceoddrootsofrealnumberseven negativerealnumbersarerealnumbers,thereisnorestrictionontheinputsto h .Hence, thedomainis ; 1 4.Tondthedomainof r ,wenoticethatwehavetwopotentiallyhazardousissues:notonly dowehaveadenominator,wehaveasquarerootinthatdenominator.Tosatisfythesquare root,wesettheradicand x +3 0so x )]TJ/F15 10.9091 Tf 20 0 Td [(3.Settingthedenominatorequaltozerogives 6 )]TJ 10.909 8.57 Td [(p x +3=0 6= p x +3 6 2 = )]TJ/F54 10.9091 Tf 5 -0.266 Td [(p x +3 2 36= x +3 33= x Sincewesquaredbothsidesinthecourseofsolvingthisequation,weneedtocheckour answer.Sureenough,when x =33,6 )]TJ 11.351 8.57 Td [(p x +3=6 )]TJ 11.351 9.025 Td [(p 36=0,andso x =33willcause problemsinthedenominator.Atlastwecanndthedomainof r :weneed x )]TJ/F15 10.9091 Tf 21.319 0 Td [(3,but x 6 =33.Ournalansweris[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 33 [ ; 1 5.It'stemptingtosimplify I x = 3 x 2 x =3 x ,and,sincetherearenolongeranydenominators, claimthattherearenolongeranyrestrictions.However,insimplifying I x ,weareassuming x 6 =0,since 0 0 isundened. 4 Proceedingasbefore,wendthedomainof I tobeallreal numbersexcept0: ; 0 [ ; 1 Itisworthreiteratingtheimportanceofndingthedomainofafunction before simplifying,as evidencedbythefunction I inthepreviousexample.Eventhoughtheformula I x simpliesto 3 x ,itwouldbeinaccuratetowrite I x =3 x withoutaddingthestipulationthat x 6 =0.Itwould beanalogoustonotreportingtaxableincomeorsomeothersinofomission. 3 The`radicand'istheexpression`inside'theradical. 4 Moreprecisely,thefraction 0 0 isan`indeterminantform'.MuchtimewillbespentinCalculuswrestlingwith suchcreatures.

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46RelationsandFunctions Ournextexampleshowshowafunctioncanbeusedtomodelreal-worldphenomena. Example 1.5.4 Theheight h infeetofamodelrocketabovetheground t secondsafterliftois givenby h t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 t 2 +100 t; if0 t 20 0 ; if t> 20 Findandinterpret h and h Solution. Thereareafewqualitiesof h whichmaybeo-putting.Therstisthat,unlike previousexamples,theindependentvariableis t ,not x .Inthiscontext, t ischosenbecauseit representstime.Thesecondisthatthefunctionisbrokenupintotworules:oneformulaforvalues of t between0and20inclusive,andanotherforvaluesof t greaterthan20.Tond h ,werst noticethat10isbetween0and20soweusetherstformulalisted: h t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 t 2 +100 t .Hence, h = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 2 +100=500.Intermsofthemodelrocket,thismeansthat10secondsafter lifto,themodelrocketis500feetabovetheground.Tond h ,wenotethat60isgreater than20,soweusetherule h t =0.Thisfunctionreturnsavalueof0regardlessofwhatvalueis substitutedinfor t ,so h =0.Thismeansthat60secondsafterlifto,therocketis0feetabove theground;inotherwords,aminuteafterlifto,therockethasalreadyreturnedtoearth. Thetypeoffunctioninthepreviousexampleiscalleda piecewise-dened function,or`piecewise' functionforshort.Manyreal-worldphenomenae.g.postalrates, 5 incometaxformulas 6 are modeledbysuchfunctions.Alsonotethatthedomainof h intheaboveexampleisrestrictedto t 0.Forexample, h )]TJ/F15 10.9091 Tf 8.485 0 Td [(3isnotdenedbecause t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3doesn'tsatisfyanyoftheconditionsinany ofthefunction'spieces.Thereisnoinherentarithmeticreasonwhichpreventsusfromcalculating, say, )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 +100 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3,it'sjustthatinthisappliedsetting, t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3ismeaningless.Inthiscase, wesay h hasan applieddomain 7 of[0 ; 1 5 SeetheUnitedStatesPostalServicewebsitehttp://www.usps.com/prices/rst-class-mail-prices.htm 6 SeetheInternalRevenueService'swebsitehttp://www.irs.gov/pub/irs-pdf/i1040tt.pdf 7 or,`explicitdomain'

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1.5FunctionNotation47 1.5.1Exercises 1.Suppose f isafunctionthattakesarealnumber x andperformsthefollowingthreestepsin theordergiven:squareroot;subtract13;makethequantitythedenominatorof afractionwithnumerator4.Findanexpressionfor f x andnditsdomain. 2.Suppose g isafunctionthattakesarealnumber x andperformsthefollowingthreestepsin theordergiven:subtract13;squareroot;makethequantitythedenominatorof afractionwithnumerator4.Findanexpressionfor g x andnditsdomain. 3.Suppose h isafunctionthattakesarealnumber x andperformsthefollowingthreestepsin theordergiven:squareroot;makethequantitythedenominatorofafractionwith numerator4;subtract13.Findanexpressionfor h x andnditsdomain. 4.Suppose k isafunctionthattakesarealnumber x andperformsthefollowingthreestepsin theordergiven:makethequantitythedenominatorofafractionwithnumerator4; squareroot;subtract13.Findanexpressionfor k x andnditsdomain. 5.For f x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x +2,ndandsimplifythefollowing: a f b f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 c f )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(3 2 d f x e4 f x f f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x g f x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 h f x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 i f )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 6.RepeatExercise5abovefor f x = 2 x 3 7.Findtheimplieddomainofthefunction. a f x = x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 x 3 +56 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(19 b s t = t t )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 c Q r = p r r )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 d b = p )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 e y = 3 r y y )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 f A x = p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7+ p 9 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x g g v = 1 4 )]TJ/F15 10.9091 Tf 14.583 7.38 Td [(1 v 2 h u w = w )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 5 )]TJ 10.909 7.857 Td [(p w

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48RelationsandFunctions 8.Let f x = 8 > < > : x 2 if x )]TJ/F15 10.9091 Tf 20 0 Td [(1 p 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 if )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 Computethefollowingfunctionvalues. a f b f )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 c f d f e f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 f f )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 999 9.ThepopulationofSasquatchinPortageCountycanbemodeledbythefunction P t = 150 t t +15 ,where t =0representstheyear1803.Whatistheapplieddomainof P ?Whatrange makessense"forthisfunction?Whatdoes P represent?Whatdoes P represent? 10.Recallthatthe integers isthesetofnumbers Z = f :::; )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; 1 ; 2 ; 3 ;::: g 8 The greatestintegerof x b x c ,isdenedtobethelargestinteger k with k x aFind b 0 : 785 c b 117 c b)]TJ/F15 10.9091 Tf 13.334 0 Td [(2 : 001 c ,and b +6 c bDiscusswithyourclassmateshow b x c maybedescribedasapiece-wisedenedfunction. HINT: Thereareinnitelymanypieces! cIs b a + b c = b a c + b b c alwaystrue?Whatif a or b isaninteger?Testsomevalues,make aconjecture,andexplainyourresult. 11.Wehavethroughourexamplestriedtoconvinceyouthat,ingeneral, f a + b 6 = f a + f b .Ithasbeenourexperiencethatstudentsrefusetobelieveussowe'lltryagainwitha dierentapproach.Withthehelpofyourclassmates,ndafunction f forwhichthefollowing propertiesarealwaystrue. a f = f )]TJ/F15 10.9091 Tf 8.484 0 Td [(1+1= f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+ f b f = f +3= f + f c f )]TJ/F15 10.9091 Tf 8.484 0 Td [(6= f )]TJ/F15 10.9091 Tf 10.909 0 Td [(6= f )]TJ/F53 10.9091 Tf 10.909 0 Td [(f d f a + b = f a + f b regardlessofwhattwonumberswegiveyoufor a and b Howmanyfunctionsdidyoundthatfailedtosatisfytheconditionsabove?Did f x = x 2 work?Whatabout f x = p x or f x =3 x +7or f x = 1 x ?Didyoundanattribute commontothosefunctionsthatdidsucceed?Youshouldhave,becausethereisonlyone extremelyspecialfamilyoffunctionsthatactuallyworkshere.Thuswereturntoourprevious statement, ingeneral f a + b 6 = f a + f b 8 Theuseoftheletter Z fortheintegersisostensiblybecausetheGermanword zahlen means`tocount.'

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1.5FunctionNotation49 1.5.2Answers 1. f x = 4 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 Domain:[0 ; 169 [ ; 1 2. g x = 4 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 Domain: ; 1 3. h x = 4 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 Domain: ; 1 4. k x = r 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 Domain: ; 1 5.a2 b6 c )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(1 4 d16 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x +2 e4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x +8 f x 2 +3 x +2 g x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(11 x +30 h x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 2 +2 6.a 2 27 b )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 c 16 27 d 1 32 x 3 e 8 x 3 f )]TJ/F15 10.9091 Tf 12.436 7.38 Td [(2 x 3 g 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 3 = 2 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 2 +48 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(64 h 2 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4= 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 x 3 i 2 x 6 7.a ; 1 b ; 8 [ ; 1 c[0 ; 8 [ ; 1 d ; 1 e ; 8 [ ; 1 f[7 ; 9] g )]TJ/F54 10.9091 Tf 5 -8.837 Td [( ; )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(1 2 [ )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F34 7.9701 Tf 9.681 4.296 Td [(1 2 ; 0 [ )]TJ/F15 10.9091 Tf 5 -8.837 Td [(0 ; 1 2 [ )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 ; 1 h[0 ; 25 [ ; 1 8.a f =4 b f )]TJ/F15 10.9091 Tf 8.485 0 Td [(3=9 c f =0 d f =1 e f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=1 f f )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 999=0 : 001999 9.Theapplieddomainof P is[0 ; 1 .Therangeissomesubsetofthenaturalnumbersbecause wecannothavefractionalSasquatch.Thiswasabitofatrickquestionandwe'lladdressthe notionofmathematicalmodelingmorethoroughlyinlaterchapters. P =0meansthat therewerenoSasquatchinPortageCountyin1803. P 139 : 77wouldmeantherewere 139or140SasquatchinPortageCountyin2008. 10.a b 0 : 785 c =0, b 117 c =117, b)]TJ/F15 10.9091 Tf 13.334 0 Td [(2 : 001 c = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3,and b +6 c =9

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50RelationsandFunctions 1.6FunctionArithmetic Intheprevioussectionweusedthenewlydenedfunctionnotationtomakesenseofexpressions suchas` f x +2'and`2 f x 'foragivenfunction f .Itwouldseemnatural,then,thatfunctions shouldhavetheirownarithmeticwhichisconsistentwiththearithmeticofrealnumbers.The followingdenitionsallowustoadd,subtract,multiplyanddividefunctionsusingthearithmetic wealreadyknowforrealnumbers. FunctionArithmetic Suppose f and g arefunctionsand x isanelementcommontothedomainsof f and g The sum of f and g ,denoted f + g ,isthefunctiondenedbytheformula: f + g x = f x + g x The dierence of f and g ,denoted f )]TJ/F53 10.9091 Tf 10.909 0 Td [(g ,isthefunctiondenedbytheformula: f )]TJ/F53 10.9091 Tf 10.909 0 Td [(g x = f x )]TJ/F53 10.9091 Tf 10.909 0 Td [(g x The product of f and g ,denoted fg ,isthefunctiondenedbytheformula: fg x = f x g x The quotient of f and g ,denoted f g ,isthefunctiondenedbytheformula: f g x = f x g x ; provided g x 6 =0. Inotherwords,toaddtwofunctions,weaddtheiroutputs;tosubtracttwofunctions,wesubtract theiroutputs,andsoon.Notethatwhiletheformula f + g x = f x + g x lookssuspiciously likesomekindofdistributiveproperty,itisnothingofthesort;theadditiononthelefthandside oftheequationis function addition,andweareusingthisequationto dene theoutputofthe newfunction f + g asthesumoftherealnumberoutputsfrom f and g 1 Example 1.6.1 Let f x =6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x and g x =3 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x .Findandsimplifyexpressionsfor 1. f + g x 2. g )]TJ/F53 10.9091 Tf 10.909 0 Td [(f x 3. fg x 4. g f x 1 Theauthoriswellawarethatthispointispedantic,andlostonmostreaders.

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1.6FunctionArithmetic51 Inaddition,ndthedomainofeachofthesefunctions. Solution. 1. f + g x isdenedtobe f x + g x .Tothatend,weget f + g x = f x + g x = )]TJ/F15 10.9091 Tf 5 -8.836 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x + 3 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x =6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +3 )]TJ/F15 10.9091 Tf 12.494 7.38 Td [(1 x = 6 x 3 x )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(2 x 2 x + 3 x x )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x getcommondenominators = 6 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 +3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x Tondthedomainof f + g wedoso before wesimplify,thatis,atthestep )]TJ/F15 10.9091 Tf 5 -8.836 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x + 3 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x Wesee x 6 =0,buteverythingelseisne.Hence,thedomainis ; 0 [ ; 1 2. g )]TJ/F53 10.9091 Tf 10.909 0 Td [(f x isdenedtobe g x )]TJ/F53 10.9091 Tf 10.909 0 Td [(f x .Tothatend,weget g )]TJ/F53 10.9091 Tf 10.91 0 Td [(f x = g x )]TJ/F53 10.9091 Tf 10.909 0 Td [(f x = 3 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x )]TJ/F55 10.9091 Tf 10.909 8.836 Td [()]TJ/F15 10.9091 Tf 5 -8.836 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x =3 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +2 x = 3 x x )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(6 x 3 x + 2 x 2 x getcommondenominators = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x 3 +2 x 2 +3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x Lookingattheexpressionfor g )]TJ/F53 10.9091 Tf 10.909 0 Td [(f beforewesimplied 3 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x )]TJ/F55 10.9091 Tf 10.909 8.836 Td [()]TJ/F15 10.9091 Tf 5 -8.836 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x wesee,asbefore, x 6 =0istheonlyrestriction.Thedomainis ; 0 [ ; 1 .

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52RelationsandFunctions 3. fg x isdenedtobe f x g x .Substitutingyields fg x = f x g x = )]TJ/F15 10.9091 Tf 5 -8.837 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x = )]TJ/F15 10.9091 Tf 5 -8.836 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x = 2 x x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 1 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x factor = 2 x x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 1 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x cancel =2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 =2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(9 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x +1 =18 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x +2 Todeterminethedomain,wecheckthestepjustafterwesubstituted )]TJ/F15 10.9091 Tf 5 -8.836 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x whichgivesus,asbefore,thedomain: ; 0 [ ; 1 4. g f x isdenedtobe g x f x .Thuswehave g f x = g x f x = 3 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x 6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x = 3 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x 6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x x x simplifycomplexfractions = 3 )]TJ/F15 10.9091 Tf 12.494 7.38 Td [(1 x x x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 x x = 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 x x

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1.6FunctionArithmetic53 = 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 x 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 factor = : 1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 x 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 cancel = 1 2 x 2 Tondthedomain,weconsidertherststepaftersubstitution: 3 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x 6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x Toavoiddivisionbyzerointhe`little'fraction, 1 x ,weneed x 6 =0.Forthe`big'fractionwe set6 x 2 )]TJ/F15 10.9091 Tf 10.561 0 Td [(2 x =0andsolve:2 x x )]TJ/F15 10.9091 Tf 10.561 0 Td [(1=0andget x =0 ; 1 3 .Thuswemustexclude x = 1 3 as well,resultinginadomainof ; 0 [ )]TJ/F15 10.9091 Tf 5 -8.836 Td [(0 ; 1 3 [ )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 3 ; 1 Weclosethissectionwithconceptofthe dierencequotient ofafunction.Itisacriticaltool forCalculusandalsoagreatwaytopracticefunctionnotation. 2 Definition 1.6 Givenafunction, f ,the dierencequotient of f istheexpression: f x + h )]TJ/F53 10.9091 Tf 10.909 0 Td [(f x h Example 1.6.2 Findandsimplifythedierencequotientsforthefollowingfunctions 1. f x = x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 2. g x = 3 2 x +1 Solution. 1.Tond f x + h ,wereplaceeveryoccurrenceof x intheformula f x = x 2 )]TJ/F53 10.9091 Tf 10.643 0 Td [(x )]TJ/F15 10.9091 Tf 10.643 0 Td [(2withthe quantity x + h toget f x + h = x + h 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( x + h )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 = x 2 +2 xh + h 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(h )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 : Sothedierencequotientis f x + h )]TJ/F53 10.9091 Tf 10.909 0 Td [(f x h = )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 +2 xh + h 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(h )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F55 10.9091 Tf 10.909 8.836 Td [()]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 h 2 YoumayneedtobrushuponyourIntermediateAlgebraskills,aswell.

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54RelationsandFunctions = x 2 +2 xh + h 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(h )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 + x +2 h = 2 xh + h 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(h h = h x + h )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 h factor = h x + h )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 h cancel =2 x + h )]TJ/F15 10.9091 Tf 10.909 0 Td [(1. 2.Tond g x + h ,wereplaceeveryoccurrenceof x intheformula g x = 3 2 x +1 withthe quantity x + h g x + h = 3 2 x + h +1 = 3 2 x +2 h +1 ; whichyields g x + h )]TJ/F53 10.9091 Tf 10.909 0 Td [(g x h = 3 2 x +2 h +1 )]TJ/F15 10.9091 Tf 24.616 7.38 Td [(3 2 x +1 h = 3 2 x +2 h +1 )]TJ/F15 10.9091 Tf 24.616 7.38 Td [(3 2 x +1 h x +2 h +1 x +1 x +2 h +1 x +1 = 3 x +1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x +2 h +1 h x +2 h +1 x +1 = 6 x +3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 h )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 h x +2 h +1 x +1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 h h x +2 h +1 x +1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 h h x +2 h +1 x +1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x +2 h +1 x +1 ForreasonswhichwillbecomeclearinCalculus,wedonotexpandthedenominator.

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1.6FunctionArithmetic55 1.6.1Exercises 1.Let f x = p x g x = x +10and h x = 1 x aComputethefollowingfunctionvalues. i. f + g ii. g )]TJ/F53 10.9091 Tf 10.909 0 Td [(h iii. fh iv. h g bFindthedomainofthefollowingfunctionsthensimplifytheirexpressions. i. f + g x ii. g )]TJ/F53 10.9091 Tf 10.909 0 Td [(h x iii. fh x iv. h g x v. g h x vi. h )]TJ/F53 10.9091 Tf 10.909 0 Td [(f x 2.Let f x =3 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1, g x =2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2and h x = 3 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x aComputethefollowingfunctionvalues. i. f + g ii. g )]TJ/F53 10.9091 Tf 10.909 0 Td [(h iii. fh iv. h g )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 bFindthedomainofthefollowingfunctionsthensimplifytheirexpressions. i. f )]TJ/F53 10.9091 Tf 10.909 0 Td [(g x ii. gh x iii. f g x iv. f h x 3.Findandsimplifythedierencequotient f x + h )]TJ/F53 10.9091 Tf 10.91 0 Td [(f x h forthefollowingfunctions. a f x =2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 b f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x +5 c f x =6 d f x =3 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x e f x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 f f x = x 3 +1 g f x = 2 x h f x = 3 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x i f x = x x )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 j f x = p x 3 3 Rationalizethenumerator.Itwon'tlook`simplied'perse,butworkthroughuntilyoucancancelthe`h'.

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56RelationsandFunctions k f x = mx + b where m 6 =0l f x = ax 2 + bx + c where a 6 =0

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1.6FunctionArithmetic57 1.6.2Answers 1.ai. f + g =16 ii. g )]TJ/F53 10.9091 Tf 10.909 0 Td [(h = 118 7 iii. fh = 1 5 iv. h g = 1 39 bi. f + g x = p x + x +10 Domain:[0 ; 1 ii. g )]TJ/F53 10.9091 Tf 10.909 0 Td [(h x = x +10 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(1 x Domain: ; 0 [ ; 1 iii. fh x = 1 p x Domain: ; 1 iv. h g x = 1 x x +10 Domain: ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 ; 0 [ ; 1 v. g h x = x x +10 Domain: ; 0 [ ; 1 vi. h )]TJ/F53 10.9091 Tf 10.909 0 Td [(f x = 1 x )]TJ 10.909 7.858 Td [(p x Domain: ; 1 2.ai. f + g =23 ii. g )]TJ/F53 10.9091 Tf 10.909 0 Td [(h = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 iii. fh = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(3 2 iv. h g )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= 1 3 bi. f )]TJ/F53 10.9091 Tf 10.909 0 Td [(g x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 2 +3 x +3 p x +1 Domain:[0 ; 1 ii. gh x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 Domain: ; 2 [ ; 1 iii. f g x = 3 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 Domain:[0 ; 2 [ ; 1 iv. f h x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x p x + 1 3 x +2 p x )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(2 3 Domain:[0 ; 2 [ ; 1 3.a2 b )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 c0 d6 x +3 h )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 e )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(h +2 f3 x 2 +3 xh + h 2 g )]TJ/F15 10.9091 Tf 27.24 7.381 Td [(2 x x + h h 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x + h )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x + h )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 j 1 p x + h + p x k m l2 ax + ah + b

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58RelationsandFunctions 1.7GraphsofFunctions InSection1.4wedenedafunctionasaspecialtypeofrelation;oneinwhicheach x -coordinate wasmatchedwithonlyone y -coordinate.Wespentmostofourtimeinthatsectionlookingat functionsgraphicallybecausetheywere,afterall,justsetsofpointsintheplane.TheninSection 1.5wedescribedafunctionasaprocessanddenedthenotationnecessarytoworkwithfunctions algebraically.Sonowit'stimetolookatfunctionsgraphicallyagain,onlythistimewe'lldosowith thenotationdenedinSection1.5.Westartwithwhatshouldnotbeasurprisingconnection. TheFundamentalGraphingPrincipleforFunctions Thegraphofafunction f isthesetofpointswhichsatisfytheequation y = f x .Thatis,the point x;y isonthegraphof f ifandonlyif y = f x Example 1.7.1 Graph f x = x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6. Solution. Tograph f ,wegraphtheequation y = f x .Tothisend,weusethetechniques outlinedinSection1.3.Specically,wecheckforintercepts,testforsymmetry,andplotadditional pointsasneeded.Tondthe x -intercepts,weset y =0.Since y = f x ,thismeans f x =0. f x = x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 0= x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 0= x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x +2factor x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3=0or x +2=0 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3 Soweget )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0and ; 0as x -intercepts.Tondthe y -intercept,weset x =0.Usingfunction notation,thisisthesameasnding f and f =0 2 )]TJ/F15 10.9091 Tf 11.531 0 Td [(0 )]TJ/F15 10.9091 Tf 11.531 0 Td [(6= )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 : Thusthe y -interceptis ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(6.Asfarassymmetryisconcerned,wecantellfromtheinterceptsthatthegraphpossesses noneofthethreesymmetriesdiscussedthusfar.Youshouldverifythis.Wecanmakeatable analogoustotheoneswemadeinSection1.3,plotthepointsandconnectthedotsinasomewhat pleasingfashiontogetthegraphbelowontheright. x f x x;f x )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 0 ; 0 4 6 ; 6 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7

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1.7GraphsofFunctions59 Graphingpiecewise-denedfunctionsisabitmoreofachallenge. Example 1.7.2 Graph: f x = 4 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 if x< 1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ; if x 1 Solution. Weproceedasbefore:ndingintercepts,testingforsymmetryandthenplotting additionalpointsasneeded.Tondthe x -intercepts,asbefore,weset f x =0.Thetwististhat wehavetwoformulasfor f x .For x< 1,weusetheformula f x =4 )]TJ/F53 10.9091 Tf 11.305 0 Td [(x 2 .Setting f x =0 gives0=4 )]TJ/F53 10.9091 Tf 11.06 0 Td [(x 2 ,sothat x = 2.However,ofthesetwoanswers,only x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2tsinthedomain x< 1forthispiece.Thismeanstheonly x -interceptforthe x< 1regionofthe x -axisis )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0. For x 1, f x = x )]TJ/F15 10.9091 Tf 11.332 0 Td [(3.Setting f x =0gives0= x )]TJ/F15 10.9091 Tf 11.332 0 Td [(3,or x =3.Since x =3satisesthe inequality x 1,weget ; 0asanother x -intercept.Next,weseekthe y -intercept.Noticethat x =0fallsinthedomain x< 1.Thus f =4 )]TJ/F15 10.9091 Tf 11.536 0 Td [(0 2 =4yieldsthe y -intercept ; 4.Asfar assymmetryisconcerned,youcancheckthattheequation y =4 )]TJ/F53 10.9091 Tf 11.528 0 Td [(x 2 issymmetricaboutthe y -axis;unfortunately,thisequationanditssymmetryisvalidonlyfor x< 1.Youcanalsoverify y = x )]TJ/F15 10.9091 Tf 10.642 0 Td [(3possessesnoneofthesymmetriesdiscussedintheSection1.3.Whenplottingadditional points,itisimportanttokeepinmindtherestrictionson x foreachpieceofthefunction.The stickingpointforthisfunctionis x =1,sincethisiswheretheequationschange.When x =1,we usetheformula f x = x )]TJ/F15 10.9091 Tf 10.398 0 Td [(3,sothepointonthegraph ;f is ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.However,forallvalues lessthan1,weusetheformula f x =4 )]TJ/F53 10.9091 Tf 10.731 0 Td [(x 2 .AswehavediscussedearlierinSection1.2,thereis norealnumberwhichimmediatelyprecedes x =1onthenumberline.Thusforthevalues x =0 : 9, x =0 : 99, x =0 : 999,andsoon,wendthecorresponding y valuesusingtheformula f x =4 )]TJ/F53 10.9091 Tf 9.931 0 Td [(x 2 Makingatableasbefore,weseethatasthe x valuessneakupto x =1inthisfashion,the f x valuesinchcloserandcloser 1 to4 )]TJ/F15 10.9091 Tf 11.022 0 Td [(1 2 =3.Toindicatethisgraphically,weuseanopencircleat thepoint ; 3.Puttingallofthisinformationtogetherandplottingadditionalpoints,weget x f x x;f x 0 : 9 3 : 19 : 9 ; 3 : 19 0 : 99 3 : 02 : 99 ; 3 : 02 0 : 999 3 : 002 : 999 ; 3 : 002 x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1123 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 1 We'vejuststeppedintoCalculushere!

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60RelationsandFunctions Intheprevioustwoexamples,the x -coordinatesofthe x -interceptsofthegraphof y = f x were foundbysolving f x =0.Forthisreason,theyarecalledthe zeros of f Definition 1.7 The zeros ofafunction f arethesolutionstotheequation f x =0.Inother words, x isazeroof f ifandonlyif x; 0isan x -interceptofthegraphof y = f x OfthethreesymmetriesdiscussedinSection1.3,onlytwoareofsignicancetofunctions:symmetry aboutthe y -axisandsymmetryabouttheorigin. 2 Recallthatwecantestwhetherthegraphofan equationissymmetricaboutthe y -axisbyreplacing x with )]TJ/F53 10.9091 Tf 8.485 0 Td [(x andcheckingtoseeifanequivalent equationresults.Ifwearegraphingtheequation y = f x ,substituting )]TJ/F53 10.9091 Tf 8.484 0 Td [(x for x resultsinthe equation y = f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x .Inorderforthisequationtobeequivalenttotheoriginalequation y = f x weneed f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = f x .Inasimilarfashion,werecallthattotestanequation'sgraphforsymmetry abouttheorigin,wereplace x and y with )]TJ/F53 10.9091 Tf 8.485 0 Td [(x and )]TJ/F53 10.9091 Tf 8.485 0 Td [(y ,respectively.Doingthissubstitutioninthe equation y = f x resultsin )]TJ/F53 10.9091 Tf 8.485 0 Td [(y = f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x .Solvingthelatterequationfor y gives y = )]TJ/F53 10.9091 Tf 8.485 0 Td [(f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x .In orderforthisequationtobeequivalenttotheoriginalequation y = f x weneed )]TJ/F53 10.9091 Tf 8.485 0 Td [(f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = f x or,equivalently, f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(f x .Theseresultsaresummarizedbelow. Stepsfortestingifthegraphofafunctionpossessessymmetry Thegraphofafunction f issymmetric: Aboutthe y -axisifandonlyif f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = f x forall x inthedomainof f Abouttheoriginifandonlyif f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = )]TJ/F53 10.9091 Tf 8.484 0 Td [(f x forall x inthedomainof f Forreasonswhichwon'tbecomeclearuntilwestudypolynomials,wecallafunction even ifits graphissymmetricaboutthe y -axisor odd ifitsgraphissymmetricabouttheorigin.Apartfrom averyspecializedfamilyoffunctionswhicharebothevenandodd, 3 functionsfallintooneofthree distinctcategories:even,odd,orneitherevennorodd. Example 1.7.3 Analyticallydetermineifthefollowingfunctionsareeven,odd,orneithereven norodd.Verifyyourresultwithagraphingcalculator. 1. f x = 5 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 2. g x = 5 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 3. h x = 5 x 2 )]TJ/F53 10.9091 Tf 10.91 0 Td [(x 3 4. i x = 5 x 2 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 5. j x = x 2 )]TJ/F53 10.9091 Tf 17.169 7.38 Td [(x 100 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Solution. Therststepinalloftheseproblemsistoreplace x with )]TJ/F53 10.9091 Tf 8.485 0 Td [(x andsimplify. 2 Whyarewesodismissiveaboutsymmetryaboutthe x -axisforgraphsoffunctions? 3 Anyideas?

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1.7GraphsofFunctions61 1. f x = 5 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = 5 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = 5 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = f x Hence, f is even .Thegraphingcalculatorfurnishesthefollowing: Thissuggests 4 thegraphof f issymmetricaboutthe y -axis,asexpected. 2. g x = 5 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 g )]TJ/F53 10.9091 Tf 8.484 0 Td [(x = 5 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 g )]TJ/F53 10.9091 Tf 8.484 0 Td [(x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 Itdoesn'tappearthat g )]TJ/F53 10.9091 Tf 8.485 0 Td [(x isequivalentto g x .Toprovethis,wecheckwithan x value. Aftersometrialanderror,weseethat g =5whereas g )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(5.Thisprovesthat g is noteven,butitdoesn'truleoutthepossibilitythat g isodd.Whynot?Tocheckif g is odd,wecompare g )]TJ/F53 10.9091 Tf 8.485 0 Td [(x with )]TJ/F53 10.9091 Tf 8.485 0 Td [(g x )]TJ/F53 10.9091 Tf 8.485 0 Td [(g x = )]TJ/F15 10.9091 Tf 18.713 7.381 Td [(5 x 2 )]TJ/F53 10.9091 Tf 10.91 0 Td [(x 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F53 10.9091 Tf 8.485 0 Td [(g x = g )]TJ/F53 10.9091 Tf 8.485 0 Td [(x Hence, g isodd.Graphically, 4 `Suggests'isabouttheextentofwhatitcando.

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62RelationsandFunctions Thecalculatorindicatesthegraphof g issymmetricabouttheorigin,asexpected. 3. h x = 5 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 h )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = 5 )]TJ/F53 10.9091 Tf 8.484 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 3 h )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x 2+ x 3 Onceagain, h )]TJ/F53 10.9091 Tf 8.484 0 Td [(x doesn'tappeartobeequivalentto h x .Wecheckwithan x value,for example, h =5but h )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(5 3 .Thisprovesthat h isnotevenanditalsoshows h isnot odd.Why?Graphically, Thegraphof h appearstobeneithersymmetricaboutthe y -axisnortheorigin. 4. i x = 5 x 2 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 i )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = 5 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 3 i )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x + x 3 Theexpression i )]TJ/F53 10.9091 Tf 8.484 0 Td [(x doesn'tappeartobeequivalentto i x .However,aftercheckingsome x values,forexample x =1yields i =5and i )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=5,itappearsthat i )]TJ/F53 10.9091 Tf 8.484 0 Td [(x does,in fact,equal i x .However,whilethissuggests i iseven,itdoesn't prove it.Itdoes,however,

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1.7GraphsofFunctions63 prove i isnotodd.Toprove i )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = i x ,weneedtomanipulateourexpressionsfor i x and i )]TJ/F53 10.9091 Tf 8.485 0 Td [(x andshowtheyareequivalent.Aclueastohowtoproceedisinthenumerators: intheformulafor i x ,thenumeratoris5 x andin i )]TJ/F53 10.9091 Tf 8.485 0 Td [(x thenumeratoris )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x .Tore-write i x withanumeratorof )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x ,weneedtomultiplyitsnumeratorby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Tokeepthevalue ofthefractionthesame,weneedtomultiplythedenominatorby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1aswell.Thus i x = 5 x 2 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 x + x 3 Hence, i x = i )]TJ/F53 10.9091 Tf 8.485 0 Td [(x ,so i iseven.Thecalculatorsupportsourconclusion. 5. j x = x 2 )]TJ/F53 10.9091 Tf 17.169 7.38 Td [(x 100 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 j )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 )]TJ 12.926 7.38 Td [()]TJ/F53 10.9091 Tf 8.485 0 Td [(x 100 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 j )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = x 2 + x 100 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Theexpressionfor j )]TJ/F53 10.9091 Tf 8.485 0 Td [(x doesn'tseemtobeequivalentto j x ,sowecheckusing x =1to get j = )]TJ/F34 7.9701 Tf 13.914 4.296 Td [(1 100 and j )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= 1 100 .Thisrulesout j beingeven.However,itdoesn'truleout j beingodd.Examining )]TJ/F53 10.9091 Tf 8.485 0 Td [(j x gives j x = x 2 )]TJ/F53 10.9091 Tf 17.169 7.38 Td [(x 100 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F53 10.9091 Tf 8.485 0 Td [(j x = )]TJ/F55 10.9091 Tf 10.303 12.11 Td [( x 2 )]TJ/F53 10.9091 Tf 17.169 7.38 Td [(x 100 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F53 10.9091 Tf 8.485 0 Td [(j x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 + x 100 +1 Theexpression )]TJ/F53 10.9091 Tf 8.485 0 Td [(j x doesn'tseemtomatch j )]TJ/F53 10.9091 Tf 8.485 0 Td [(x either.Testing x =2gives j = 149 50 and j )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= 151 50 ,so j isnotodd,either.Thecalculatorgives:

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64RelationsandFunctions Thecalculatorsuggeststhatthegraphof j issymmetricaboutthe y -axiswhichwouldimply that j iseven.However,wehaveproventhatisnotthecase. Therearetwolessonstobelearnedfromthelastexample.Therstisthatsamplingfunction valuesatparticular x valuesisnotenoughtoprovethatafunctionisevenorodd )]TJ/F15 10.9091 Tf 12.559 0 Td [(despitethe factthat j )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= )]TJ/F53 10.9091 Tf 8.485 0 Td [(j j turnedoutnottobeodd.Secondly,whilethecalculatormay suggest mathematicaltruths,itisthealgebrawhich proves mathematicaltruths. 5 1.7.1GeneralFunctionBehavior Thelasttopicwewishtoaddressinthissectionisgeneralfunctionbehavior.Asyoushallseein thenextseveralchapters,eachfamilyoffunctionshasitsownuniqueattributesandwewillstudy themallingreatdetail.Thepurposeofthissection'sdiscussion,then,istolaythefoundationfor thatfurtherstudybyinvestigatingaspectsoffunctionbehaviorwhichapplytoallfunctions.To start,wewillexaminetheconceptsof increasing decreasing ,and constant .Beforedening theconceptsalgebraically,itisinstructivetorstlookatthemgraphically.Considerthegraphof thefunction f givenonthenextpage. Readingfromlefttoright,thegraph`starts'atthepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3and`ends'atthepoint ; 5 : 5.If weimaginewalkingfromlefttorightonthegraph,between )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3and )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 4 : 5,wearewalking `uphill';thenbetween )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 4 : 5and ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(8,wearewalking`downhill';andbetween ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(8and ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(6,wearewalking`uphill'oncemore.From ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(6to ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(6,we`levelo',andthenresume walking`uphill'from ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(6to6 ; 5 : 5.Inotherwords,forthe x valuesbetween )]TJ/F15 10.9091 Tf 8.485 0 Td [(4and )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 inclusive,the y -coordinatesonthegrapharegettinglarger,or increasing ,aswemovefromleft toright.Since y = f x ,the y valuesonthegrapharethefunctionvalues,andwesaythatthe function f is increasing ontheinterval[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2].Analogously,wesaythat f is decreasing onthe interval[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3]increasingoncemoreontheinterval[3 ; 4], constant on[4 ; 5],andnallyincreasing onceagainon[5 ; 6].Itisextremelyimportanttonoticethatthebehaviorincreasing,decreasing orconstantoccursonanintervalonthe x -axis.Whenwesaythatthefunction f isincreasing on[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(2]wedonotmentiontheactual y valuesthat f attainsalongtheway.Thus,wereport where thebehavioroccurs,nottowhatextentthebehavioroccurs. 6 Alsonoticethatwedonot saythatafunctionisincreasing,decreasingorconstantatasingle x value.Infact,wewouldrun 5 Or,inotherwords,don'trelytooheavilyonthemachine! 6 ThenotionsofhowquicklyorhowslowlyafunctionincreasesordecreasesareexploredinCalculus.

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1.7GraphsofFunctions65 intoserioustroubleinourpreviousexampleifwetriedtodosobecause x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2iscontainedinan intervalonwhich f wasincreasingandoneonwhichitisdecreasing.There'smoreonthisissue andmanyothersintheexercises. )]TJ/F15 9.9626 Tf 7.748 0 Td [(4 ; )]TJ/F15 9.9626 Tf 7.748 0 Td [(3 )]TJ/F15 9.9626 Tf 7.748 0 Td [(2 ; 4 : 5 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(8 ; 5 : 5 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(6 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(6 x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11234567 )]TJ/F34 7.9701 Tf 6.586 0 Td [(9 )]TJ/F34 7.9701 Tf 6.586 0 Td [(8 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 1 2 3 4 5 6 7 Thegraphof y = f x We'renowreadyforthemoreformalalgebraicdenitionsofwhatitmeansforafunctiontobe increasing,decreasingorconstant. Definition 1.8 Suppose f isafunctiondenedonaninterval I .Wesay f is: increasing on I ifandonlyif f a f b forallrealnumbers a b in I with a
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66RelationsandFunctions ; 5 : 5 )]TJ/F15 10.9091 Tf 13.276 0 Td [(but )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 4 : 5shouldgetsomesortofconsolationprizeforbeing`thetopofthehill' between x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4and x =3.Wesaythatthefunction f hasa localmaximum 7 atthepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 4 : 5,becausethe y -coordinate4 : 5isthelargest y -valuehence,functionvalueonthecurve `near' 8 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Similarly,wesaythatthefunction f hasa localminimum 9 atthepoint ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(8, sincethe y -coordinate )]TJ/F15 10.9091 Tf 8.485 0 Td [(8isthesmallestfunctionvaluenear x =3.Althoughitistemptingto saythatlocalextrema 10 occurwhenthefunctionchangesfromincreasingtodecreasingorvice versa,itisnotapreciseenoughwaytodenetheconceptsfortheneedsofCalculus.Attheriskof beingpedantic,wewillpresentthetraditionaldenitionsandthoroughlyvetthepathologiesthey induceintheexercises.Wehaveonelastobservationtomakebeforeweproceedtothealgebraic denitionsandlookatafairlytame,yethelpful,example. Ifwelookattheentiregraph,weseethelargest y valuehencethelargestfunctionvalueis5 : 5 at x =6.Inthiscase,wesaythe maximum 11 of f is5 : 5;similarly,the minimum 12 of f is )]TJ/F15 10.9091 Tf 8.485 0 Td [(8. Weformalizetheseconceptsinthefollowingdenitions. Definition 1.9 Suppose f isafunctionwith f a = b Wesay f hasa localmaximum atthepoint a;b ifandonlyifthereisanopeninterval I containing a forwhich f a f x forall x in I dierentthan a .Thevalue f a = b is called`alocalmaximumvalueof f 'inthiscase. Wesay f hasa localminimum atthepoint a;b ifandonlyifthereisanopeninterval I containing a forwhich f a f x forall x in I dierentthan a .Thevalue f a = b is called`alocalminimumvalueof f 'inthiscase. Thevalue b iscalledthe maximum of f if b f x forall x inthedomainof f Thevalue b iscalledthe minimum of f if b f x forall x inthedomainof f It'simportanttonotethatnoteveryfunctionwillhaveallofthesefeatures.Indeed,itispossible tohaveafunctionwithnolocalorabsoluteextremaatall!Anyideasofwhatsuchafunction's graphwouldhavetolooklike?Weshallseeintheexercisesexamplesoffunctionswhichhaveone ortwo,butnotall,ofthesefeatures,somethathaveinstancesofeachtypeofextremumandsome functionsthatseemtodefycommonsense.Inallcases,though,weshalladheretothealgebraic denitionsaboveasweexplorethewonderfuldiversityofgraphsthatfunctionsprovidetous. Hereisthe`tame'examplewhichwaspromisedearlier.Itsummarizesalloftheconceptspresented inthissectionaswellassomefromprevioussectionssoyoushouldspendsometimethinking deeplyaboutitbeforeproceedingtotheexercises. 7 Alsocalled`relativemaximum'. 8 Wewillmakethismorepreciseinamoment. 9 Alsocalleda`relativeminimum'. 10 `Maxima'isthepluralof`maximum'and`mimima'isthepluralof`minimum'.`Extrema'isthepluralof `extremum'whichcombinesmaximumandminimum. 11 Sometimescalledthe`absolute'or`global'maximum. 12 Again,`absolute'or`global'minimumcanbeused.

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1.7GraphsofFunctions67 Example 1.7.4 Giventhegraphof y = f x below,answerallofthefollowingquestions. )]TJ/F15 9.9626 Tf 7.749 0 Td [(2 ; 0 ; 0 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(3 )]TJ/F15 9.9626 Tf 7.749 0 Td [(4 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(3 ; 3 x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 1.Findthedomainof f 2.Findtherangeof f 3.Determine f 4.Listthe x -intercepts,ifanyexist. 5.Listthe y -intercepts,ifanyexist. 6.Findthezerosof f 7.Solve f x < 0. 8.Determinethenumberofsolutionstothe equation f x =1. 9.Listtheintervalsonwhich f isincreasing. 10.Listtheintervalsonwhich f isdecreasing. 11.Listthelocalmaximums,ifanyexist. 12.Listthelocalminimums,ifanyexist. 13.Findthemaximum,ifitexists. 14.Findtheminimum,ifitexists. 15.Does f appeartobeeven,odd,orneither? Solution. 1.Tondthedomainof f ,weproceedasinSection1.4.Byprojectingthegraphtothe x -axis, weseetheportionofthe x -axiswhichcorrespondstoapointonthegraphiseverythingfrom )]TJ/F15 10.9091 Tf 8.485 0 Td [(4to4,inclusive.Hence,thedomainis[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 4]. 2.Tondtherange,weprojectthegraphtothe y -axis.Weseethatthe y valuesfrom )]TJ/F15 10.9091 Tf 8.485 0 Td [(3to 3,inclusive,constitutetherangeof f .Hence,ouransweris[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 3]. 3.Sincethegraphof f isthegraphoftheequation y = f x f isthe y -coordinateofthe pointwhichcorrespondsto x =2.Sincethepoint ; 0isonthegraph,wehave f =0.

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68RelationsandFunctions 4.The x -interceptsarethepointsonthegraphwith y -coordinate0,namely )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0and ; 0. 5.The y -interceptisthepointonthegraphwith x -coordinate0,namely ; 3. 6.Thezerosof f arethe x -coordinatesofthe x -interceptsofthegraphof y = f x whichare x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 2. 7.Tosolve f x < 0,welookforthe x valuesofthepointsonthegraphwherethe y -coordinate islessthan0.Graphically,wearelookingwherethegraphis below the x -axis.Thishappens forthe x valuesfrom )]TJ/F15 10.9091 Tf 8.485 0 Td [(4to )]TJ/F15 10.9091 Tf 8.485 0 Td [(2andagainfrom2to4.Soouransweris[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 [ ; 4]. 8.Tondwhere f x =1,welookforpointsonthegraphwherethe y -coordinateis1.Even thoughthesepointsaren'tspecied,weseethatthecurvehastwopointswitha y valueof 1,asseeninthegraphbelow.Thatmeanstherearetwosolutionsto f x =1. x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 9.Aswemovefromlefttoright,thegraphrisesfrom )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3to ; 3.Thismeans f is increasingontheinterval[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 0].Remember,theanswerhereisanintervalonthe x -axis. 10.Aswemovefromlefttoright,thegraphfallsfrom ; 3to ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.Thismeans f isdecreasing ontheinterval[0 ; 4].Remember,theanswerhereisanintervalonthe x -axis. 11.Thefunctionhasitsonlylocalmaximumat ; 3. 12.Therearenolocalminimums.Whydon't )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3and ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3count?Let'sconsiderthe point )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3foramoment.Recallthat,inthedenitionoflocalminimum,thereneedsto beanopeninterval I whichcontains x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4suchthat f )]TJ/F15 10.9091 Tf 8.485 0 Td [(4
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1.7GraphsofFunctions69 foralocalminimum,wecannotclaimthat f hasoneat )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.Thepoint ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(3failsfor thesamereason )]TJ/F15 10.9091 Tf 12.122 0 Td [(noopenintervalaround x =4stayswithinthedomainof f 13.Themaximumvalueof f isthelargest y -coordinatewhichis3. 14.Theminimumvalueof f isthesmallest y -coordinatewhichis )]TJ/F15 10.9091 Tf 8.484 0 Td [(3. 15.Thegraphappearstobesymmetricaboutthe y -axis.Thissuggests 13 that f iseven. Withfewexceptions,wewillnotdeveloptechniquesinCollegeAlgebrawhichallowustodetermine theintervalsonwhichafunctionisincreasing,decreasingorconstantortondthelocalmaximums andlocalminimumsanalytically;thisisthebusinessofCalculus. 14 Whenwehaveneedtondsuch beasts,wewillresorttothecalculator.Mostgraphingcalculatorshave`Minimum'and`Maximum' featureswhichcanbeusedtoapproximatethesevalues,asdemonstratedbelow. Example 1.7.5 Let f x = 15 x x 2 +3 .Useagraphingcalculatortoapproximatetheintervalson which f isincreasingandthoseonwhichitisdecreasing.Approximateallextrema. Solution. Enteringthisfunctionintothecalculatorgives UsingtheMinimumandMaximumfeatures,weget 13 butdoesnotprove 14 Although,truthbetold,thereisonlyonestepofCalculusinvolved,followedbyseveralpagesofalgebra.

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70RelationsandFunctions Totwodecimalplaces, f appearstohaveitsonlylocalminimumat )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 73 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 : 33anditsonly localmaximumat ; 73 ; 4 : 33.Giventhesymmetryabouttheoriginsuggestedbythegraph,the relationbetweenthesepointsshouldn'tbetoosurprising.Thefunctionappearstobeincreasingon [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 73 ; 1 : 73]anddecreasingon ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 73] [ [1 : 73 ; 1 .Thismakes )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 : 33theabsoluteminimum and4 : 33theabsolutemaximum. Example 1.7.6 Findthepointsonthegraphof y = x )]TJ/F15 10.9091 Tf 9.408 0 Td [(3 2 whichareclosesttotheorigin.Round youranswerstotwodecimalplaces. Solution. Supposeapoint x;y isonthegraphof y = x )]TJ/F15 10.9091 Tf 10.308 0 Td [(3 2 .Itsdistancetotheorigin, ; 0, isgivenby d = p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 2 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 2 = p x 2 + y 2 = q x 2 +[ x )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 2 ] 2 Since y = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 = p x 2 + x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 4 Givenavaluefor x ,theformula d = p x 2 + x )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 4 isthedistancefrom ; 0tothepoint x;y onthecurve y = x )]TJ/F15 10.9091 Tf 11.758 0 Td [(3 2 .Whatwehavedened,then,isafunction d x whichwewishto minimizeoverallvaluesof x .ToaccomplishthistaskanalyticallywouldrequireCalculussoas we'vementionedbefore,wecanuseagraphingcalculatortondanapproximatesolution.Using thecalculator,weenterthefunction d x asshownbelowandgraph. UsingtheMinimumfeature,weseeaboveontherightthattheabsoluteminimumoccursnear x =2.Roundingtotwodecimalplaces,wegetthattheminimumdistanceoccurswhen x =2 : 00. Tondthe y valueontheparabolaassociatedwith x =2 : 00,wesubstitute2 : 00intotheequation toget y = x )]TJ/F15 10.9091 Tf 11.145 0 Td [(3 2 = : 00 )]TJ/F15 10.9091 Tf 11.144 0 Td [(3 2 =1 : 00.So,ournalansweris : 00 ; 1 : 00 : 15 Whatdoesthe y valuelistedonthecalculatorscreenmeaninthisproblem? 15 Itseemssillytolistanalansweras : 00 ; 1 : 00.Indeed,Calculusconrmsthatthe exact answertothis problemis,infact, ; 1.Asyouarewellawarebynow,theauthorisapedant,andassuch,usesthedecimalplaces toremindthereaderthat any resultgarneredfromacalculatorinthisfashionisanapproximation,andshouldbe treatedassuch.

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1.7GraphsofFunctions71 1.7.2Exercises 1.Sketchthegraphsofthefollowingfunctions.Statethedomainofthefunction,identifyany interceptsandtestforsymmetry. a f x = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 3 b f x = p 5 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x c f x = 3 p x d f x = 1 x 2 +1 2.Analyticallydetermineifthefollowingfunctionsareeven,oddorneither. a f x =7 x b f x =7 x +2 c f x = 1 x 3 d f x =4 e f x =0 f f x = x 6 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 4 + x 2 +9 g f x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 5 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 + x h f x = x 4 + x 3 + x 2 + x +1 i f x = p 5 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x j f x = x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.91 0 Td [(6 3.Useyourgraphingcalculatortoapproximatethelocalandabsoluteextremaofthefollowing functions.Approximatetheintervalsonwhichthefunctionisincreasingandthoseonwhich itisdecreasing.Roundyouranswerstotwodecimalplaces. a f x = x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(24 x 2 +28 x +48 b f x = x 2 = 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 c f x = p 9 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 d f x = x p 9 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 4.Sketchthegraphsofthefollowingpiecewise-denedfunctions. a f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4if x< 0 3 x if x 0 b f x = p x +4if )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 x< 5 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1if x 5 c f x = 8 > < > : x 2 if x )]TJ/F15 10.9091 Tf 20 0 Td [(2 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x if )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 > < > > : 1 x if )]TJ/F15 10.9091 Tf 8.485 0 Td [(6
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72RelationsandFunctions 6.Useyourgraphingcalculatortoshowthatthefollowingfunctionsdonothaveanyextrema, neitherlocalnorabsolute. a f x = x 3 + x )]TJ/F15 10.9091 Tf 10.909 0 Td [(12b f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x +2 7.InExercise9inSection1.5,wesawthatthepopulationofSasquatchinPortageCounty couldbemodeledbythefunction P t = 150 t t +15 ,where t =0representstheyear1803.Use yourgraphingcalculatortoanalyzethegeneralfunctionbehaviorof P .Willthereeverbea timewhen200SasquatchroamPortageCounty? 8.OneofthemostimportantaspectsoftheCartesianCoordinatePlaneisitsabilitytoput AlgebraintogeometrictermsandGeometryintoalgebraicterms.We'vespentmostofthis chapterlookingatthisveryphenomenonandnowyoushouldspendsometimewithyour classmatesreviewingwhatwe'vedone.WhatmajorresultsdowehavethattieAlgebraand Geometrytogether?WhatconceptsfromGeometryhavewenotyetdescribedalgebraically? WhattopicsfromIntermediateAlgebrahavewenotyetdiscussedgeometrically? 9.It'snowtimetothoroughlyvetthepathologiesinduced"bytheprecisedenitionsoflocal maximumandlocalminimum.We'lldothisbyprovidingyouandyourclassmatesaseries ofexercisestodiscuss.YouwillneedtoreferbacktoDenition1.8Increasing,Decreasing andConstantandDenition1.9MaximumandMinimumduringthediscussion. aConsiderthegraphofthefunction f givenbelow. x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 i.Showthat f hasalocalmaximumbutnotalocalminimumatthepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1. ii.Showthat f hasalocalminimumbutnotalocalmaximumatthepoint ; 1. iii.Showthat f hasalocalmaximumANDalocalminimumatthepoint ; 1. iv.Showthat f isconstantontheinterval[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1]andthushasbothalocalmaximum ANDalocalminimumateverypoint x;f x where )]TJ/F15 10.9091 Tf 8.485 0 Td [(1
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1.7GraphsofFunctions73 thisgraph?Alsondtheintervalsonwhich g isincreasingandthoseonwhich g is decreasing. x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 5 cWesaidearlierinthesectionthatitisnotgoodenoughtosaylocalextremaexist whereafunctionchangesfromincreasingtodecreasingorviceversa.Asaprevious exerciseshowed,wecouldhavelocalextremawhenafunctionisconstantsonowwe needtoexaminesomefunctionswhosegraphsdoindeedchangedirection.Considerthe functionsgraphedbelow.Noticethatallfourofthemchangedirectionatanopencircle onthegraph.Examineeachforlocalextrema.Whatistheeectofplacingthedot" onthe y -axisaboveorbelowtheopencircle?Whatcouldyousayifnofunctionvalue wasassignedto x =0? i. x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 FunctionI ii. x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 FunctionII iii. x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 FunctionIII iv. x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112 1 2 3 4 5 FunctionIV

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74RelationsandFunctions 1.7.3Answers 1.a f x = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 3 Domain: ; 1 x -intercept: ; 0 y -intercept: )]TJ/F15 10.9091 Tf 5 -8.836 Td [(0 ; )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(2 3 Nosymmetry x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 b f x = p 5 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x Domain: ; 5] x -intercept: ; 0 y -intercept: ; p 5 Nosymmetry x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 1 2 3 c f x = 3 p x Domain: ; 1 x -intercept: ; 0 y -intercept: ; 0 Symmetryabouttheorigin x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345678 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 1 2 d f x = 1 x 2 +1 Domain: ; 1 No x -intercepts y -intercept: ; 1 Symmetryaboutthe y -axis x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112 1 2.a f x =7 x isodd b f x =7 x +2isneither c f x = 1 x 3 isodd d f x =4iseven e f x =0iseven and odd f f x = x 6 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 4 + x 2 +9iseven g f x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 5 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 + x isodd h f x = x 4 + x 3 + x 2 + x +1isneither i f x = p 5 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x isneither j f x = x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.91 0 Td [(6isneither 3.aNoabsolutemaximum Absoluteminimum f : 55 )]TJ/F15 10.9091 Tf 20 0 Td [(176 : 32 Localminimumat )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 : 84 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(91 : 32 Localmaximumat : 54 ; 55 : 73 Localminimumat : 55 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(176 : 32 Increasingon[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 : 84 ; 0 : 54] ; [4 : 55 ; 1 Decreasingon ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 : 84] ; [0 : 54 ; 4 : 55] bNoabsolutemaximum Noabsoluteminimum Localmaximumat ; 0 Localminimumat : 60 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 : 28 Increasingon ; 0] ; [1 : 60 ; 1 Decreasingon[0 ; 1 : 60]

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1.7GraphsofFunctions75 cAbsolutemaximum f =3 Absoluteminimum f 3=0 Localmaximumat ; 3 Nolocalminimum Increasingon[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 0] Decreasingon[0 ; 3] dAbsolutemaximum f : 12 4 : 50 Absoluteminimum f )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 : 12 )]TJ/F15 10.9091 Tf 20 0 Td [(4 : 50 Localmaximum : 12 ; 4 : 50 Localminimum )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 : 12 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 : 50 Increasingon[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 : 12 ; 2 : 12] Decreasingon[ )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 : 12] ; [2 : 12 ; 3] 4.a x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(11 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 b x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234567 1 2 3 c x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 1 2 3 4 5 6 d x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123456789 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 5.a x y . . . )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123456 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 1 2 3 4 5 6 Thegraphof f x = b x c bNotethat f : 1=1,but f )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 : 1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,andso f isneitherevennorodd.

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76RelationsandFunctions 1.8Transformations Inthissection,westudyhowthegraphsoffunctionschange,or transform ,whencertainspecialized modicationsaremadetotheirformulas.Thetransformationswewillstudyfallintothreebroad categories:shifts,reections,andscalings,andwewillpresenttheminthatorder.Supposethe graphbelowisthecompletegraphof f ; 1 ; 3 ; 3 ; 5 x y 12345 2 3 4 5 y = f x TheFundamentalGraphingPrincipleforFunctionssaysthatforapoint a;b tobeonthegraph, f a = b .Inparticular,weknow f =1, f =3, f =3and f =5.Supposewewantedto graphthefunctiondenedbytheformula g x = f x +2.Let'stakeaminutetoremindourselves ofwhat g isdoing.Westartwithaninput x tothefunction f andweobtaintheoutput f x Thefunction g takestheoutput f x andadds2toit.Inordertograph g ,weneedtographthe points x;g x .Howarewetondthevaluesfor g x withoutaformulafor f x ?Theansweris thatwedon'tneeda formula for f x ,wejustneedthe values of f x .Thevaluesof f x arethe y valuesonthegraphof y = f x .Forexample,usingthepointsindicatedonthegraphof f ,we canmakethefollowingtable. x x;f x f x g x = f x +2 x;g x 0 ; 1 1 3 ; 3 2 ; 3 3 5 ; 5 4 ; 3 3 5 ; 5 5 ; 5 5 7 ; 7 Ingeneral,if a;b isonthegraphof y = f x ,then f a = b ,so g a = f a +2= b +2.Hence, a;b +2isonthegraphof g .Inotherwords,toobtainthegraphof g ,weadd2tothe y -coordinate ofeachpointonthegraphof f .Geometrically,adding2tothe y -coordinateofapointmovesthe point2units above itspreviouslocation.Adding2toevery y -coordinateonagraph enmasse is usuallydescribedas`shiftingthegraphup2units'.Noticethatthegraphretainsthesamebasic shapeasbefore,itisjust2unitsaboveitsoriginallocation.Inotherwords,weconnectthefour pointswemovedinthesamemannerinwhichtheywereconnectedbefore.Wehavetheresults side-by-sidebelow.

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1.8Transformations77 ; 1 ; 3 ; 3 ; 5 x y 12345 2 3 4 5 6 7 y = f x shiftup2units )454()222()222()222()223()222()222()222()223()222()222()454(! add2toeach y -coordinate ; 3 ; 5 ; 5 ; 7 x y 12345 1 2 4 5 6 7 y = g x = f x +2 You'llnotethatthedomainof f andthedomainof g arethesame,namely[0 ; 5],butthatthe rangeof f is[1 ; 5]whiletherangeof g is[3 ; 7].Ingeneral,shiftingafunctionverticallylikethis willleavethedomainunchanged,butcouldverywellaecttherange.Youcaneasilyimaginewhat wouldhappenifwewantedtographthefunction j x = f x )]TJ/F15 10.9091 Tf 11.012 0 Td [(2.Insteadofadding2toeachof the y -coordinatesonthegraphof f ,we'dbesubtracting2.Geometrically,wewouldbemoving thegraphdown2units.Weleaveittothereadertoverifythatthedomainof j isthesameas f buttherangeof j is[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 3].Whatwehavediscussedisgeneralizedinthefollowingtheorem. Theorem 1.2 VerticalShifts. Suppose f isafunctionand k isapositivenumber. Tograph y = f x + k ,shiftthegraphof y = f x up k unitsby adding k tothe y -coordinates ofthepointsonthegraphof f Tograph y = f x )]TJ/F53 10.9091 Tf 11.053 0 Td [(k ,shiftthegraphof y = f x down k unitsby subtracting k from the y -coordinates ofthepointsonthegraphof f ThekeytounderstandingTheorem1.2and,indeed,allofthetheoremsinthissectioncomesfrom anunderstandingoftheFundamentalGraphingPrincipleforFunctions.If a;b isonthegraph of f ,then f a = b .Substituting x = a intotheequation y = f x + k gives y = f a + k = b + k Hence, a;b + k isonthegraphof y = f x + k ,andwehavetheresult.Inthelanguageof`inputs' and`outputs',Theorem1.2canbeparaphrasedasAddingto,orsubtractingfrom,the output of afunctioncausesthegraphtoshiftupordown,respectively".Sowhathappensifweaddtoor subtractfromthe input ofthefunction? Keepingwiththegraphof y = f x above,supposewewantedtograph g x = f x +2.Inother words,wearelookingtoseewhathappenswhenweadd2totheinputofthefunction. 1 Let'stry togenerateatableofvaluesof g basedonthoseweknowfor f .Wequicklyndthatweruninto somediculties. 1 Wehavespentalotoftimeinthistextshowingyouthat f x +2and f x +2are,ingeneral,wildlydierent algebraicanimals.Wewillseemomentarilythatthegeometryisalsodramaticallydierent.

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78RelationsandFunctions x x;f x f x g x = f x +2 x;g x 0 ; 1 1 f +2= f =3 ; 3 2 ; 3 3 f +2= f =3 ; 3 4 ; 3 3 f +2= f =? 5 ; 5 5 f +2= f =? Whenwesubstitute x =4intotheformula g x = f x +2,weareaskedtond f +2= f whichdoesn'texistbecausethedomainof f isonly[0 ; 5].Thesamethinghappenswhenwe attempttond g .Whatweneedhereisanewstrategy.Weknow,forinstance, f =1.To determinethecorrespondingpointonthegraphof g ,weneedtogureoutwhatvalueof x wemust substituteinto g x = f x +2sothatthequantity x +2,worksouttobe0.Solving x +2=0 gives x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,and g )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+2= f =1so )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1onthegraphof g .Tousethefact f =3,weset x +2=2toget x =0.Substitutinggives g = f +2= f =3.Continuing inthisfashion,weget x x +2 g x = f x +2 x;g x )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 g )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= f =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 0 2 g = f =3 ; 3 2 4 g = f =3 ; 3 3 5 g = f =5 ; 5 Insummary,thepoints ; 1, ; 3, ; 3and ; 5onthegraphof y = f x giverisetothe points )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1, ; 3, ; 3and ; 5onthegraphof y = g x ,respectively.Ingeneral,if a;b is onthegraphof y = f x ,then f a = b .Solving x +2= a gives x = a )]TJ/F15 10.9091 Tf 11.287 0 Td [(2sothat g a )]TJ/F15 10.9091 Tf 11.288 0 Td [(2= f a )]TJ/F15 10.9091 Tf 11.064 0 Td [(2+2= f a = b .Assuch, a )]TJ/F15 10.9091 Tf 11.064 0 Td [(2 ;b isonthegraphof y = g x .Thepoint a )]TJ/F15 10.9091 Tf 11.064 0 Td [(2 ;b is exactly2unitstothe left ofthepoint a;b sothegraphof y = g x isobtainedbyshiftingthe graph y = f x totheleft2units,aspicturedbelow. ; 1 ; 3 ; 3 ; 5 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 2 3 4 5 y = f x shiftleft2units )454()222()222()222()223()222()222()222()223()222()222()454(! subtract2fromeach x -coordinate )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 ; 1 ; 3 ; 3 ; 5 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 1 2 4 5 y = g x = f x +2 Notethatwhiletherangesof f and g arethesame,thedomainof g is[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3]whereasthedomain of f is[0 ; 5].Ingeneral,whenweshiftthegraphhorizontally,therangewillremainthesame,but thedomaincouldchange.Ifwesetouttograph j x = f x )]TJ/F15 10.9091 Tf 11.066 0 Td [(2,wewouldndourselves adding

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1.8Transformations79 2toallofthe x valuesofthepointsonthegraphof y = f x toeectashifttothe right 2units. Generalizing,wehavethefollowingresult. Theorem 1.3 HorizontalShifts. Suppose f isafunctionand h isapositivenumber. Tograph y = f x + h ,shiftthegraphof y = f x left h unitsby subtracting h fromthe x -coordinates ofthepointsonthegraphof f Tograph y = f x )]TJ/F53 10.9091 Tf 11.644 0 Td [(h ,shiftthegraphof y = f x right h unitsby adding h tothe x -coordinates ofthepointsonthegraphof f Inotherwords,Theorem1.3saysaddingtoorsubtractingfromthe input toafunctionamountsto shiftingthegraphleftorright,respectively.Theorems1.2and1.3presentathemewhichwillrun commonthroughoutthesection:changestothe outputs fromafunctionaectthe y -coordinates ofthegraph,resultinginsomekindofverticalchange;changestothe inputs toafunctionaect the x -coordinates ofthegraph,resultinginsomekindofhorizontalchange. Example 1.8.1 1.Graph f x = p x .Plotatleastthreepoints. 2.Useyourgraphin1tograph g x = p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1. 3.Useyourgraphin1tograph j x = p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1. 4.Useyourgraphin1tograph m x = p x +3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2. Solution. 1.Owingtothesquareroot,thedomainof f is x 0,or[0 ; 1 .Wechooseperfectsquaresto buildourtableandgraphbelow.Fromthegraphweverifythedomainof f is[0 ; 1 andthe rangeof f isalso[0 ; 1 x f x x;f x 0 0 ; 0 1 1 ; 1 4 2 ; 2 ; 0 ; 1 ; 2 x y 1234 1 2 y = f x = p x 2.Thedomainof g isthesameasthedomainof f ,sincetheonlyconditiononbothfunctions isthat x 0.Ifwecomparetheformulafor g x with f x ,weseethat g x = f x )]TJ/F15 10.9091 Tf 11.276 0 Td [(1. Inotherwords,wehavesubtracted1fromtheoutputofthefunction f .ByTheorem1.2, weknowthatinordertograph g ,weshiftthegraphof f down oneunitby subtracting 1fromeachofthe y -coordinatesofthepointsonthegraphof f .Applyingthistothethree pointswehavespeciedonthegraph,wemove ; 0to ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, ; 1to ; 0,and ; 2to ; 1.Therestofthepointsfollowsuit,andweconnectthemwiththesamebasicshapeas before.Weconrmthedomainof g is[0 ; 1 andndtherangeof g tobe[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 .

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80RelationsandFunctions ; 0 ; 1 ; 2 x y 1234 1 2 y = f x = p x shiftdown1unit )454()222()222()222()223()222()222()222()223()222()222()454(! subtract1fromeach y -coordinate ; )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ; 0 ; 1 x y 1234 1 2 y = g x = p x )]TJ/F34 7.9701 Tf 8.468 0 Td [(1 3.Solving x )]TJ/F15 10.9091 Tf 11.8 0 Td [(1 0gives x 1,sothedomainof j is[1 ; 1 .Tograph j ,wenotethat j x = f x )]TJ/F15 10.9091 Tf 11.456 0 Td [(1.Inotherwords,wearesubtracting1fromthe input of f .Accordingto Theorem1.3,thisinducesashifttothe right ofthegraphof f .We add 1tothe x -coordinates ofthepointsonthegraphof f andgettheresultbelow.Thegraphrearmsthedomainof j is[1 ; 1 andtellsusthattherangeis[0 ; 1 ; 0 ; 1 ; 2 x y 12345 1 2 y = f x = p x shiftright1unit )454()222()222()222()223()222()222()222()223()222()222()454(! add1toeach x -coordinate ; 0 ; 1 ; 2 x y 2345 1 2 y = j x = p x )]TJ/F34 7.9701 Tf 8.469 0 Td [(1 4.Tondthedomainof m ,wesolve x +3 0andget[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1 .Comparingtheformulasof f x and m x ,wehave m x = f x +3 )]TJ/F15 10.9091 Tf 10.384 0 Td [(2.Wehave3beingaddedtoaninput,indicating ahorizontalshift,and2beingsubtractedfromanoutput,indicatingaverticalshift.We leaveittothereadertoverifythat,inthisparticularcase,theorderinwhichweperform thesetransformationsisimmaterial;wewillarriveatthesamegraphregardlessastowhich transformationweapplyrst. 2 Wefollowtheconvention`inputsrst', 3 andtothatendwe rsttacklethehorizontalshift.Letting m 1 x = f x +3denotethisintermediatestep, Theorem1.3tellsusthatthegraphof y = m 1 x isthegraphof f shiftedtothe left 3units. Hence,we subtract 3fromeachofthe x -coordinatesofthepointsonthegraphof f ; 0 ; 1 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 1 2 y = f x = p x shiftleft3units )454()222()222()222()223()222()222()222()223()222()222()454(! subtract3fromeach x -coordinate )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 ; 0 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 ; 1 ; 2 x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 1 2 y = m 1 x = f x +3= p x +3 2 Weshallseeinthenextexamplethatorderisgenerallyimportantwhenapplyingmorethanonetransformation toagraph. 3 Wecouldequallyhavechosentheconvention`outputsrst'.

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1.8Transformations81 Since m x = f x +3 )]TJ/F15 10.9091 Tf 10.887 0 Td [(2and f x +3= m 1 x ,wehave m x = m 1 x )]TJ/F15 10.9091 Tf 10.887 0 Td [(2.Wecanapply Theorem1.2andobtainthegraphof m by subtracting 2fromthe y -coordinatesofeachof thepointsonthegraphof m 1 x .Thegraphveriesthatthedomainof m is[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1 and wendtherangeof m is[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 ; 0 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 ; 1 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 1 2 y = m 1 x = f x +3= p x +3 shiftdown2units )454()222()222()222()223()222()222()222()223()222()222()454(! subtract2fromeach y -coordinate )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 ; )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 ; )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; 0 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 1 2 y = m x = m 1 x )]TJ/F34 7.9701 Tf 8.468 0 Td [(2= p x +3 )]TJ/F34 7.9701 Tf 8.468 0 Td [(2 Keepinmindthatwecancheckouranswertoanyofthesekindsofproblemsbyshowingthatany ofthepointswe'vemovedlieonthegraphofournalanswer.Forexample,wecancheckthat )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2isonthegraphof m ,bycomputing m )]TJ/F15 10.9091 Tf 8.485 0 Td [(3= p )]TJ/F15 10.9091 Tf 8.485 0 Td [(3+3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2= p 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 X Wenowturnourattentiontoreections.WeknowfromSection1.1thattoreectapoint x;y acrossthe x -axis,wereplace y with )]TJ/F53 10.9091 Tf 8.485 0 Td [(y .If x;y isonthegraphof f ,then y = f x ,soreplacing y with )]TJ/F53 10.9091 Tf 8.485 0 Td [(y isthesameasreplacing f x with )]TJ/F53 10.9091 Tf 8.485 0 Td [(f x .Hence,thegraphof y = )]TJ/F53 10.9091 Tf 8.484 0 Td [(f x isthegraphof f reectedacrossthe x -axis.Similarly,thegraphof y = f )]TJ/F53 10.9091 Tf 8.484 0 Td [(x isthegraphof f reectedacrossthe y -axis.Returningtoinputsandoutputs,multiplyingtheoutputfromafunctionby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1reectsits graphacrossthe x -axis,whilemultiplyingtheinputtoafunctionby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1reectsthegraphacross the y -axis. 4 Theorem 1.4 Reections. Suppose f isafunction. Tograph y = )]TJ/F53 10.9091 Tf 8.485 0 Td [(f x ,reectthegraphof y = f x acrossthe x -axis bymultiplyingthe y -coordinates ofthepointsonthegraphof f by )]TJ/F15 10.9091 Tf 8.485 0 Td [(1. Tograph y = f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x ,reectthegraphof y = f x acrossthe y -axis bymultiplyingthe x -coordinates ofthepointsonthegraphof f by )]TJ/F15 10.9091 Tf 8.485 0 Td [(1. ApplyingTheroem1.4tothegraphof y = f x givenatthebeginningofthesection,wecangraph y = )]TJ/F53 10.9091 Tf 8.485 0 Td [(f x byreectingthegraphof f aboutthe x -axis 4 Theexpressions )]TJ/F64 8.9664 Tf 7.168 0 Td [(f x and f )]TJ/F64 8.9664 Tf 7.167 0 Td [(x shouldlookfamiliar-theyarethequantitiesweusedinSection1.7totestif afunctionwaseven,odd,orneither.Theinterestedreaderisinvitedtoexploretheroleofreectionsandsymmetry offunctions.Whathappensifyoureectanevenfunctionacrossthe y -axis?Whathappensifyoureectanodd functionacrossthe y -axis?Whataboutthe x -axis?

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82RelationsandFunctions ; 1 ; 3 ; 3 ; 5 x y 12345 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 2 3 4 5 y = f x reectacross x -axis )454()222()222()222()223()222()222()222()223()222()222()454(! multiplyeach y -coordinateby )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 ; )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 ; )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 x y 12345 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 1 2 3 4 5 y = )]TJ/F37 7.9701 Tf 6.586 0 Td [(f x Byreectingthegraphof f acrossthe y -axis,weobtainthegraphof y = f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x ; 1 ; 3 ; 3 ; 5 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(512345 2 3 4 5 y = f x reectacross y -axis )454()222()222()222()223()222()222()222()223()222()222()454(! multiplyeach x -coordinateby )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ; 1 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 ; 3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 ; 3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 ; 5 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(512345 2 3 4 5 y = f )]TJ/F37 7.9701 Tf 6.587 0 Td [(x Withtheadditionofreections,itisnowmoreimportantthanevertoconsiderthe order of transformations,asthenextexampleillustrates. Example 1.8.2 Let f x = p x .Usethegraphof f fromExample1.8.1tographthefollowing functionsbelow.Also,statetheirdomainsandranges. 1. g x = p )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2. j x = p 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3. m x =3 )]TJ 10.909 7.857 Td [(p x Solution. 1.Themeresightof p )]TJ/F53 10.9091 Tf 8.485 0 Td [(x usuallycausesalarm,ifnotpanic.Whenwediscusseddomains inSection1.5,weclearlybanishednegativesfromtheradicalsofevenroots.However,we mustrememberthat x isavariable,andassuch,thequantity )]TJ/F53 10.9091 Tf 8.485 0 Td [(x isn'talwaysnegative.For example,if x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4, )]TJ/F53 10.9091 Tf 8.485 0 Td [(x =4,thus p )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = p )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(4=2isperfectlywell-dened.Tondthe

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1.8Transformations83 domainanalytically,weset )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 0whichgives x 0,sothatthedomainof g is ; 0]. Since g x = f )]TJ/F53 10.9091 Tf 8.484 0 Td [(x ,Theorem1.4tellsusthegraphof g isthereectionofthegraphof f acrossthe y -axis.Wecanaccomplishthisbymultiplyingeach x -coordinateonthegraph of f by )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,sothatthepoints ; 0, ; 1,and ; 2moveto ; 0, )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 1,and )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 2, respectively.Graphically,weseethatthedomainof g is ; 0]andtherangeof g isthe sameastherangeof f ,namely[0 ; 1 ; 0 ; 1 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(41234 1 2 y = f x = p x reectacross y -axis )454()222()222()222()223()222()222()222()223()222()222()454(! multiplyeach x -coordinateby )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ; 0 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 ; 1 )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(41234 1 2 y = g x = f )]TJ/F37 7.9701 Tf 6.586 0 Td [(x = p )]TJ/F37 7.9701 Tf 6.586 0 Td [(x 2.Todeterminethedomainof j x = p 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x ,wesolve3 )]TJ/F53 10.9091 Tf 11.217 0 Td [(x 0andget x 3,or ; 3]. Todeterminewhichtransformationsweneedtoapplytothegraphof f toobtainthegraph of j ,werewrite j x = p )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +3= f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +3.Comparingthisformulawith f x = p x ,we seethatnotonlyarewemultiplyingtheinput x by )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,whichresultsinareectionacross the y -axis,butalsoweareadding3,whichindicatesahorizontalshifttotheleft.Doesit matterinwhichorderwedothetransformations?Ifso,whichorderisthecorrectorder? Let'sconsiderthepoint ; 2onthegraphof f .Werefertothediscussionleadingupto Theorem1.3.Weknow f =2andwishtondthepointon y = j x = f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +3which correspondsto ; 2.Weset )]TJ/F53 10.9091 Tf 8.484 0 Td [(x +3=4andsolve.Ourrststepistosubtract3fromboth sidestoget )]TJ/F53 10.9091 Tf 8.485 0 Td [(x =1.Subtracting3fromthe x -coordinate4isshiftingthepoint ; 2to theleft.From )]TJ/F53 10.9091 Tf 8.484 0 Td [(x =1,wethenmultiply 5 bothsidesby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1toget x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Multiplyingthe x -coordinateby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1correspondstoreectingthepointaboutthe y -axis.Hence,weperform thehorizontalshiftrst,thenfollowitwiththereectionaboutthe y -axis.Startingwith f x = p x ,welet j 1 x betheintermediatefunctionwhichshiftsthegraphof f 3unitsto theleft, j 1 x = f x +3. ; 0 ; 1 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(11234 1 2 y = f x = p x shiftleft3units )454()222()222()222()223()222()222()222()223()222()222()454(! subtract3fromeach x -coordinate )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 ; 0 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 ; 1 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 1 2 y = j 1 x = f x +3= p x +3 Toobtainthefunction j ,wereectthegraphof j 1 about y -axis.Theorem1.4tellsuswe have j x = j 1 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x .Puttingitalltogether,wehave j x = j 1 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +3= p )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +3, 5 Ordivide-itamountstothesamething.

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84RelationsandFunctions whichiswhatwewant. 6 Fromthegraph,weconrmthedomainof j is ; 3]andweget therangeis[0 ; 1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 ; 0 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 ; 1 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 1 2 y = j 1 x = p x +3 reectacross y -axis )454()222()222()222()223()222()222()222()223()222()222()454(! multiplyeach x -coordinateby )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; 0 ; 1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 2 y = j x = j 1 )]TJ/F37 7.9701 Tf 6.587 0 Td [(x = p )]TJ/F37 7.9701 Tf 6.586 0 Td [(x +3 3.Thedomainof m worksouttobethedomainof f ,[0 ; 1 .Rewriting m x = )]TJ 8.484 7.857 Td [(p x +3,we see m x = )]TJ/F53 10.9091 Tf 8.484 0 Td [(f x +3.Sincewearemultiplyingtheoutputof f by )]TJ/F15 10.9091 Tf 8.485 0 Td [(1andthenadding 3,weonceagainhavetwotransformationstodealwith:areectionacrossthe x -axisand averticalshift.Todeterminethecorrectorderinwhichtoapplythetransformations,we imaginetryingtodeterminethepointonthegraphof m whichcorrespondsto ; 2onthe graphof f .Sinceintheformulafor m x ,theinputto f isjust x ,wesubstitutetond m = )]TJ/F53 10.9091 Tf 8.485 0 Td [(f +3= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+3=1.Hence, ; 1isthecorrespondingpointonthegraphof m .Ifwecloselyexaminethearithmetic,weseethatwerstmultiply f by )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,which correspondstothereectionacrossthe x -axis,andthenweadd3,whichcorrespondsto theverticalshift.Ifwedeneanintermediatefunction m 1 x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(f x totakecareofthe reection,weget ; 0 ; 1 ; 2 x y 1234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 y = f x = p x reectacross x -axis )454()222()222()222()223()222()222()222()223()222()222()454(! multiplyeach y -coordinateby )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; 0 ; )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ; )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 x y 1234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 1 2 3 y = m 1 x = )]TJ/F37 7.9701 Tf 6.587 0 Td [(f x = )]TJ 6.587 5.745 Td [(p x Toshiftthegraphof m 1 up3units,weset m x = m 1 x +3.Since m 1 x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(f x ,when weputitalltogether,weget m x = m 1 x +3= )]TJ/F53 10.9091 Tf 8.485 0 Td [(f x +3= )]TJ 8.485 7.858 Td [(p x +3.Weseefromthe graphthattherangeof m is ; 3]. 6 Ifwehaddonethereectionrst,then j 1 x = f )]TJ/F64 8.9664 Tf 7.168 0 Td [(x .Followingthisbyashiftleftwouldgiveus j x = j 1 x +3= f )]TJ/F63 8.9664 Tf 7.167 0 Td [( x +3= f )]TJ/F64 8.9664 Tf 7.167 0 Td [(x )]TJ/F63 8.9664 Tf 9.405 0 Td [(3= p )]TJ/F64 8.9664 Tf 7.167 0 Td [(x )]TJ/F63 8.9664 Tf 9.216 0 Td [(3whichisn'twhatwewant.However,ifwedidthereectionrst andfolloweditbyashifttotheright3units,wewouldhavearrivedatthefunction j x .Weleaveittothereader toverifythedetails.

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1.8Transformations85 ; 0 ; )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 x y 1234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 1 2 3 shiftup3units )454()222()222()222()223()222()222()222()223()222()222()454(! add3toeach y -coordinate ; 3 ; 2 ; 1 x y 1234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 1 2 y = m 1 x = )]TJ 6.587 5.746 Td [(p xy = m x = m 1 x +3= )]TJ 6.587 5.746 Td [(p x +3 Wenowturnourattentiontoourlastclassoftransformations,scalings.Supposewewishtograph thefunction g x =2 f x where f x isthefunctionwhosegraphisgivenatthebeginningofthe section.Fromitsgraph,wecanbuildatableofvaluesfor g asbefore. ; 1 ; 3 ; 3 ; 5 x y 12345 2 3 4 5 y = f x x x;f x f x g x =2 f x x;g x 0 ; 1 1 2 ; 2 2 ; 3 3 6 ; 6 4 ; 3 3 6 ; 6 5 ; 5 5 10 ; 10 Ingeneral,if a;b isonthegraphof f ,then f a = b sothat g a =2 f a =2 b puts a; 2 b on thegraphof g .Inotherwords,toobtainthegraphof g ,wemultiplyallofthe y -coordinatesof thepointsonthegraphof f by2.Multiplyingallofthe y -coordinatesofallofthepointsonthe graphof f by2causeswhatisknownasa`verticalscaling 7 byafactorof2',andtheresultsare 7 Alsocalleda`verticalstretch',`verticalexpansion'or`verticaldilation'byafactorof2.

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86RelationsandFunctions givenbelow. ; 1 ; 3 ; 3 ; 5 x y 12345 2 3 4 5 6 7 8 9 10 y = f x verticalscalingbyafactorof2 )366()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()366(! multiplyeach y -coordinateby2 ; 2 ; 6 ; 6 ; 10 x y 12345 1 3 4 5 6 7 8 9 10 y =2 f x Ifwewishtograph y = 1 2 f x ,wemultiplytheallofthe y -coordinatesofthepointsonthegraph of f by 1 2 .Thiscreatesa`verticalscaling 8 byafactorof 1 2 'asseenbelow. ; 1 ; 3 ; 3 ; 5 x y 12345 2 3 4 5 y = f x verticalscalingbyafactorof 1 2 )366()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()366(! multiplyeach y -coordinateby 1 2 )]TJ/F34 7.9701 Tf 3.882 -6.416 Td [(0 ; 1 2 )]TJ/F34 7.9701 Tf 3.881 -6.416 Td [(2 ; 3 2 )]TJ/F34 7.9701 Tf 3.881 -6.416 Td [(4 ; 3 2 )]TJ/F34 7.9701 Tf 3.881 -6.416 Td [(5 ; 5 2 x y 12345 1 2 3 4 5 y = 1 2 f x Theseresultsaregeneralizedinthefollowingtheorem. Theorem 1.5 VerticalScalings. Suppose f isafunctionand a> 0.Tograph y = af x multiply allofthe y -coordinates ofthepointsonthegraphof f by a .Wesaythegraphof f hasbeen verticallyscaled byafactorof a If a> 1,wesaythegraphof f hasundergoneaverticalstretchexpansion,dilationbya factorof a If0
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1.8Transformations87 byafactorof2'makessenseifwearescalingsomethingbymultiplyingitby2.Similarly,we believewordslike`shrink',`compression'and`contraction'allindicatesomethinggettingsmaller, soifwescalesomethingbyafactorof 1 2 ,wewouldsayit`shrinksbyafactorof2'-not`shrinksby afactorof 1 2 .'Thisiswhywehavewrittenthedescriptions`stretchbyafactorof a 'and`shrinkby afactorof 1 a 'inthestatementofthetheorem.Second,intermsofinputsandoutputs,Theorem1.5 saysmultiplyingthe outputs fromafunctionbypositivenumber a causesthegraphtobevertically scaledbyafactorof a .Itisnaturaltoaskwhatwouldhappenifwemultiplythe inputs ofa functionbyapositivenumber.Thisleadsustoourlasttransformationofthesection. Referringtothegraphof f givenatthebeginningofthissection,supposewewanttograph g x = f x .Inotherwords,wearelookingtoseewhateectmultiplyingtheinputsto f by2 hasonitsgraph.Ifweattempttobuildatabledirectly,wequicklyrunintothesameproblemwe hadinourdiscussionleadinguptoTheorem1.3,asseeninthetableontheleftbelow.Wesolve thisprobleminthesamewaywesolvedthisproblembefore.Forexample,ifwewanttodetermine thepointon g whichcorrespondstothepoint ; 3onthegraphof f ,weset2 x =2sothat x =1. Substituting x =1into g x ,weobtain g = f 1= f =3,sothat ; 3isonthegraphof g .Continuinginthisfashion,weobtainthetableonthelowerright. x x;f x f x g x = f x x;g x 0 ; 1 1 f 0= f =1 ; 1 2 ; 3 3 f 2= f =3 ; 3 4 ; 3 3 f 4= f =? 5 ; 5 5 f 5= f =? x 2 x g x = f x x;g x 0 0 g = f =1 ; 0 1 2 g = f =3 ; 3 2 4 g = f =3 ; 3 5 2 5 g )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(5 2 = f =5 )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(5 2 ; 5 Ingeneral,if a;b isonthegraphof f ,then f a = b .Hence g )]TJ/F37 7.9701 Tf 6.195 -4.541 Td [(a 2 = f )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 a 2 = f a = b sothat )]TJ/F37 7.9701 Tf 6.196 -4.541 Td [(a 2 ;b isonthegraphof g .Inotherwords,tograph g wedividethe x -coordinatesofthepointson thegraphof f by2.Thisresultsinahorizontalscaling 9 byafactorof 1 2 ; 1 ; 3 ; 3 ; 5 x y 12345 2 3 4 5 y = f x horizontalscalingbyafactorof 1 2 )366()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()366(! multiplyeach x -coordinateby 1 2 ; 1 ; 3 ; 3 )]TJ/F35 5.9776 Tf 5.077 -3.158 Td [(5 2 ; 5 x y 12345 2 3 4 5 y = g x = f x If,ontheotherhand,wewishtograph y = f )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 2 x ,weendupmultiplyingthe x -coordinates ofthepointsonthegraphof f by2whichresultsinahorizontalscaling 10 byafactorof2,as demonstratedbelow. 9 Alsocalled`horizontalshrink,'`horizontalcompression'or`horizontalcontraction'byafactorof2. 10 Alsocalled`horizontalstretch,'`horizontalexpansion'or`horizontaldilation'byafactorof2.

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88RelationsandFunctions ; 1 ; 3 ; 3 ; 5 x y 12345678910 2 3 4 5 y = f x horizontalscalingbyafactorof2 )366()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()366(! multiplyeach x -coordinateby2 ; 1 ; 3 ; 3 ; 5 x y 12345678910 2 3 4 5 y = g x = f )]TJ/F35 5.9776 Tf 5.077 -3.158 Td [(1 2 x Wehavethefollowingtheorem. Theorem 1.6 HorizontalScalings. Suppose f isafunctionand b> 0.Tograph y = f bx divide allofthe x -coordinates ofthepointsonthegraphof f by b .Wesaythegraphof f has been horizontallyscaled byafactorof 1 b If0 1,wesaythegraphof f hasundergoneaverticalshrinkcompression,contraction byafactorof b Theorem1.6tellsusthatifwemultiplythe input toafunctionby b ,theresultinggraphisscaled horizontallybyafactorof 1 b sincethe x -valuesaredividedby b toproducecorrespondingpoints onthegraphof f bx .Thenextexampleexploreshowverticalandhorizontalscalingssometimes interactwitheachotherandwiththeothertransformationsintroducedinthissection. Example 1.8.3 Let f x = p x .Usethegraphof f fromExample1.8.1tographthefollowing functionsbelow.Also,statetheirdomainsandranges. 1. g x =3 p x 2. j x = p 9 x 3. m x =1 )]TJ/F55 10.9091 Tf 10.909 12.708 Td [(q x +3 2 Solution. 1.Firstwenotethatthedomainof g is[0 ; 1 fortheusualreason.Next,wehave g x =3 f x sobyTheorem1.5,weobtainthegraphof g bymultiplyingallofthe y -coordinatesofthe pointsonthegraphof f by3.Theresultisaverticalscalingofthegraphof f byafactorof 3.Wendtherangeof g isalso[0 ; 1 .

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1.8Transformations89 ; 0 ; 1 ; 2 x y 1234 1 2 3 4 5 6 y = f x = p x verticalscalebyafactorof3 )479()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()479(! multiplyeach y -coordinateby3 ; 0 ; 3 ; 6 x y 1234 1 2 3 4 5 6 y = g x =3 f x =3 p x 2.Todeterminethedomainof j ,wesolve9 x 0tond x 0.Ourdomainisonceagain [0 ; 1 .Werecognize j x = f x andbyTheorem1.6,weobtainthegraphof j bydividing the x -coordinatesofthepointsonthegraphof f by9.Fromthegraph,weseetherangeof j isalso[0 ; 1 ; 0 ; 1 ; 2 x y 1234 1 2 y = f x = p x horizontalscalebyafactorof 1 9 )479()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()479(! multiplyeach x -coordinateby 1 9 ; 0 )]TJ/F35 5.9776 Tf 5.076 -3.158 Td [(1 9 ; 1 )]TJ/F35 5.9776 Tf 5.076 -3.158 Td [(4 9 ; 2 x y 1234 1 2 y = j x = f x = p 9 x 3.Solving x +3 2 0gives x )]TJ/F15 10.9091 Tf 20.893 0 Td [(3,sothedomainof m is[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1 .Totakeadvantageofwhat weknowoftransformations,werewrite m x = )]TJ/F55 10.9091 Tf 8.485 12.708 Td [(q 1 2 x + 3 2 +1,or m x = )]TJ/F53 10.9091 Tf 8.484 0 Td [(f )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 2 x + 3 2 +1. Focusingontheinputsrst,wenotethattheinputto f intheformulafor m x is 1 2 x + 3 2 Multiplyingthe x by 1 2 correspondstoahorizontalstretchbyafactorof2,andaddingthe 3 2 correspondstoashifttotheleftby 3 2 .Asbefore,weresolvewhichtoperformrstby thinkingabouthowwewouldndthepointon m correspondingtoapointon f ,inthiscase, ; 2.Touse f =2,wesolve 1 2 x + 3 2 =4.Ourrststepistosubtractthe 3 2 thehorizontal shifttoobtain 1 2 x = 5 2 .Next,wemultiplyby2thehorizontalstretchandobtain x =5. Wedenetwointermediatefunctionstohandlersttheshift,thenthestretch.Inaccordance withTheorem1.3, m 1 x = f )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x + 3 2 = q x + 3 2 willshiftthegraphof f totheleft 3 2 units.

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90RelationsandFunctions ; 0 ; 1 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(312345 1 2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 y = f x = p x shiftleft 3 2 units )454()222()222()222()223()222()222()222()223()222()222()454(! subtract 3 2 fromeach x -coordinate )]TJ/F40 7.9701 Tf 3.881 -6.416 Td [()]TJ/F35 5.9776 Tf 7.782 3.259 Td [(3 2 ; 0 )]TJ/F40 7.9701 Tf 3.881 -6.416 Td [()]TJ/F35 5.9776 Tf 7.782 3.259 Td [(1 2 ; 1 )]TJ/F35 5.9776 Tf 5.077 -3.157 Td [(5 2 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(312345 2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 y = m 1 x = f )]TJ/F37 7.9701 Tf 3.882 -6.416 Td [(x + 3 2 = q x + 3 2 Next, m 2 x = m 1 )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 x = q 1 2 x + 3 2 will,accordingtoTheorem1.6,horizontallystretchthe graphof m 1 byafactorof2. )]TJ/F40 7.9701 Tf 3.882 -6.416 Td [()]TJ/F35 5.9776 Tf 7.782 3.259 Td [(3 2 ; 0 )]TJ/F40 7.9701 Tf 3.881 -6.416 Td [()]TJ/F35 5.9776 Tf 7.782 3.259 Td [(1 2 ; 1 )]TJ/F35 5.9776 Tf 5.076 -3.157 Td [(5 2 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(312345 2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 y = m 1 x = q x + 3 2 horizontalscalebyafactorof2 )470()223()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()471(! multiplyeach x -coordinateby2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 ; 0 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ; 1 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(212345 2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 y = m 2 x = m 1 )]TJ/F35 5.9776 Tf 5.077 -3.158 Td [(1 2 x = q 1 2 x + 3 2 Wenowexaminewhat'shappeningtotheoutputs.From m x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(f )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 x + 3 2 +1,weseethe outputfrom f isbeingmultipliedby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1areectionaboutthe x -axisandthena1isadded averticalshiftup1.Asbefore,wecandeterminethecorrectorderbylookingathowthe point ; 2ismoved.Wehavealreadydeterminedthattomakeuseoftheequation f =2, weneedtosubstitute x =5.Weget m = )]TJ/F53 10.9091 Tf 8.485 0 Td [(f )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 + 3 2 +1= )]TJ/F53 10.9091 Tf 8.485 0 Td [(f +1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(1. Weseethat f theoutputfrom f isrstmultipliedby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1thenthe1isaddedmeaningwe rstreectthegraphaboutthe x -axisthenshiftup1.Theorem1.4tellsus m 3 x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(m 2 x willhandlethereection. )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 ; 0 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ; 1 ; 2 x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(212345 2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 y = m 2 x = q 1 2 x + 3 2 reectacross x -axis )454()222()222()222()223()222()222()222()223()222()222()454(! multiplyeach y -coordinateby )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 ; 0 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ; )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(212345 1 2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 y = m 3 x = )]TJ/F37 7.9701 Tf 6.586 0 Td [(m 2 x = )]TJ/F43 7.9701 Tf 6.587 9.21 Td [(q 1 2 x + 3 2

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1.8Transformations91 Finally,tohandletheverticalshift,Theorem1.2gives m x = m 3 x +1,andweseethat therangeof m is ; 1]. )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 ; 0 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; 1 ; 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(212345 2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 y = m 2 x = q 1 2 x + 3 2 shiftup1unit )454()222()222()222()223()222()222()222()223()222()222()454(! add1toeach y -coordinate )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 ; 1 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ; 0 ; )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.757 0 Td [(212345 2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 y = m x = m 3 x +1= )]TJ/F43 7.9701 Tf 6.586 9.21 Td [(q 1 2 x + 3 2 +1 SomecommentsaboutExample1.8.3areinorder.First,recallingthepropertiesofradicalsfrom IntermediateAlgebra,weknowthatthefunctions g and j arethesame,since j and g havethe samedomainsand j x = p 9 x = p 9 p x =3 p x = g x .Weinvitethereadertoverifythatthe allofthepointsweplottedonthegraphof g lieonthegraphof j andvice-versa.Hence,for f x = p x ,averticalstretchbyafactorof3andahorizontalshrinkbyafactorof9resultin thesametransformation.Whilethiskindofphenomenonisnotuniversal,ithappenscommonly enoughwithsomeofthefamiliesoffunctionsstudiedinCollegeAlgebrathatitisworthyofnote. Secondly,tographthefunction m ,weappliedaseriesoffourtransformations.Whileitwouldhave beeneasierontheauthorstosimplyinformthereaderofwhichstepstotake,wehavestrivedto explainwhytheorderinwhichthetransformationswereappliedmadesense.Wegeneralizethe procedureinthetheorembelow. Theorem 1.7 Transformations. Suppose f isafunction.Tograph g x = Af Bx + H + K 1.Subtract H fromeachofthe x -coordinatesofthepointsonthegraphof f .Thisresultsin ahorizontalshifttotheleftif H> 0orrightif H< 0. 2.Dividethe x -coordinatesofthepointsonthegraphobtainedinStep1by B .Thisresults inahorizontalscaling,butmayalsoincludeareectionaboutthe y -axisif B< 0. 3.Multiplythe y -coordinatesofthepointsonthegraphobtainedinStep2by A .Thisresults inaverticalscaling,butmayalsoincludeareectionaboutthe x -axisif A< 0. 4.Add K toeachofthe y -coordinatesofthepointsonthegraphobtainedinStep3.This resultsinaverticalshiftupif K> 0ordownif K< 0. Theorem1.7canbeestablishedbygeneralizingthetechniquesdevelopedinthissection.Suppose a;b isonthegraphof f .Then f a = b ,andtomakegooduseofthisfact,weset Bx + H = a andsolve.Werstsubtractthe H causingthehorizontalshiftandthendivideby B .If B

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92RelationsandFunctions isapositivenumber,thisinducesonlyahorizontalscalingbyafactorof 1 B .If B< 0,then wehaveafactorof )]TJ/F15 10.9091 Tf 8.485 0 Td [(1inplay,anddividingbyitinducesareectionaboutthe y -axis.Sowe have x = a )]TJ/F37 7.9701 Tf 6.586 0 Td [(H B astheinputto g whichcorrespondstotheinput x = a to f .Wenowevaluate g )]TJ/F37 7.9701 Tf 6.195 -4.541 Td [(a )]TJ/F37 7.9701 Tf 6.587 0 Td [(H B = Af )]TJ/F53 10.9091 Tf 5 -8.837 Td [(B a )]TJ/F37 7.9701 Tf 6.586 0 Td [(H B + H + K = Af a + K = Ab + K .Wenoticethattheoutputfrom f is rstmultipliedby A .Aswiththeconstant B ,if A> 0,thisinducesonlyaverticalscaling.If A< 0,thenthe )]TJ/F15 10.9091 Tf 8.485 0 Td [(1inducesareectionacrossthe x -axis.Finally,weadd K totheresult,whichis ourverticalshift.Alessprecise,butmoreintuitivewaytoparaphraseTheorem1.7istothinkof thequantity Bx + H isthe`inside'ofthefunction f .What'shappeninginside f aectstheinputs or x -coordinatesofthepointsonthegraphof f .Tondthe x -coordinatesofthecorresponding pointson g ,weundowhathasbeendoneto x inthesamewaywewouldsolveanequation.What's happeningtotheoutputcanbethoughtofasthingshappening`outside'thefunction, f .Things happeningoutsideaecttheoutputsor y -coordinatesofthepointsonthegraphof f .Here,we followtheusualorderofoperationsagreement:werstmultiplyby A thenadd K tondthe corresponding y -coordinatesonthegraphof g Example 1.8.4 Belowisthecompletegraphof y = f x .Useittograph g x = 4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 f )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x 2 )]TJ/F15 9.9626 Tf 7.749 0 Td [(2 ; 0 ; 0 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(3 )]TJ/F15 9.9626 Tf 7.749 0 Td [(4 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(3 ; 3 x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 Solution. WeuseTheorem1.7totracktheve`keypoints' )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0, ; 3, ; 0and ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(3indicatedonthegraphof f totheirnewlocations.Werstrewrite g x intheform presentedinTheorem1.7, g x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1+2.Weset )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1equaltothe x -coordinatesof thekeypointsandsolve.Forexample,solving )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(4,werstsubtract1toget )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 thendivideby )]TJ/F15 10.9091 Tf 8.485 0 Td [(2toget x = 5 2 .Subtractingthe1isahorizontalshifttotheleft1unit.Dividingby )]TJ/F15 10.9091 Tf 8.485 0 Td [(2canbethoughtofasatwostepprocess:dividingby2whichcompressesthegraphhorizontally byafactorof2followedbydividingmultiplyingby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1whichcausesareectionacrossthe y -axis. Wesummarizetheresultsinthetablebelow.

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1.8Transformations93 a;f a a )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1= a x )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 x = 5 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x = 3 2 ; 3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1=0 x = 1 2 ; 0 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1=2 x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1=4 x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 Next,wetakeeachofthe x valuesandsubstitutetheminto g x = )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(3 2 f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1+2togetthe corresponding y -values.Substituting x = 5 2 ,andusingthefactthat f )]TJ/F15 10.9091 Tf 8.484 0 Td [(4= )]TJ/F15 10.9091 Tf 8.485 0 Td [(3,weget g 5 2 = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(3 2 f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 5 2 +1 +2= )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(3 2 f )]TJ/F15 10.9091 Tf 8.485 0 Td [(4+2= )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3+2= 9 2 +2= 13 2 Weseetheoutputfrom f isrstmultipliedby )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(3 2 .Thinkingofthisasatwostepprocess, multiplyingby 3 2 thenby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,weseewehaveaverticalstretchbyafactorof 3 2 followedbya reectionacrossthe x -axis.Adding2resultsinaverticalshiftup2units.Continuinginthis manner,wegetthetablebelow. x g x x;g x 5 2 13 2 )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(5 2 ; 13 2 3 2 2 )]TJ/F34 7.9701 Tf 6.195 -4.542 Td [(3 2 ; 2 1 2 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(5 2 )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(5 2 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 2 )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 ; 2 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 13 2 )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 ; 13 2 Tograph g ,weploteachofthepointsinthetableaboveandconnecttheminthesameorderand fashionasthepointstowhichtheycorrespond.Plotting f and g side-by-sidegives )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 ; 0 ; 0 ; )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 ; )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 ; 3 x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 5 6 )]TJ/F40 7.9701 Tf 3.882 -6.416 Td [()]TJ/F35 5.9776 Tf 7.782 3.259 Td [(3 2 ; 13 2 )]TJ/F40 7.9701 Tf 3.881 -6.416 Td [()]TJ/F35 5.9776 Tf 7.782 3.258 Td [(1 2 ; 2 )]TJ/F35 5.9776 Tf 5.076 -3.158 Td [(1 2 ; )]TJ/F35 5.9776 Tf 7.782 3.258 Td [(5 2 )]TJ/F35 5.9776 Tf 5.077 -3.158 Td [(3 2 ; 2 )]TJ/F35 5.9776 Tf 5.077 -3.157 Td [(5 2 ; 13 2 x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 3 4 5 6

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94RelationsandFunctions Thereaderisstronglyencouraged 11 tographtheseriesoffunctionswhichshowsthegradualtransformationofthegraphof f intothegraphof g .Wehaveoutlinedthesequenceoftransformations intheaboveexposition;allthatremainsistoplotallveintermediatestages. Ourlastexampleturnsthetablesandasksfortheformulaofafunctiongivenadesiredsequence oftransformations.Ifnothingelse,itisagoodreviewoffunctionnotation. Example 1.8.5 Let f x = x 2 .Findandsimplifytheformulaofthefunction g x whosegraph istheresultof f undergoingthefollowingsequenceoftransformations.Checkyouranswerusing agraphingcalculator. 1.Verticalshiftup2units 2.Reectionacrossthe y -axis 3.Horizontalshiftright1unit 4.Horizontalstretchbyafactorof2 Solution. Webuilduptoaformulafor g x usingintermediatefunctionsaswe'veseeninprevious examples.Welet g 1 takecareofourrststep.Theorem1.2tellsus g 1 x = f x +2= x 2 +2.Next, wereectthegraphof g 1 aboutthe x -axisusingTheorem1.4: g 2 x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(g 1 x = )]TJ/F55 10.9091 Tf 10.303 8.836 Td [()]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 +2 = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 )]TJ/F15 10.9091 Tf 11.51 0 Td [(2.Weshiftthegraphtotheright1unit,accordingtoTheorem1.3,bysetting g 3 x = g 2 x )]TJ/F15 10.9091 Tf 10.889 0 Td [(1= )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.889 0 Td [(1 2 )]TJ/F15 10.9091 Tf 10.889 0 Td [(2= )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 +2 x )]TJ/F15 10.9091 Tf 10.889 0 Td [(3.Finally,weinduceahorizontalstretchbyafactorof2 usingTheorem1.6toget g x = g 3 )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 x = )]TJ/F55 10.9091 Tf 10.303 8.836 Td [()]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 x 2 +2 )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 x )]TJ/F15 10.9091 Tf 10.313 0 Td [(3whichyields g x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 4 x 2 + x )]TJ/F15 10.9091 Tf 10.313 0 Td [(3. Weusethecalculatortographthestagesbelowtoconrmourresult. shiftup2units )454()222()222()222()223()222()222()222()223()222()222()454(! add2toeach y -coordinate y = f x = x 2 y = g 1 x = f x +2= x 2 +2 reectacross x -axis )454()222()222()222()223()222()222()222()223()222()222()454(! multiplyeach y -coordinateby )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 y = g 1 x = x 2 +2 y = g 2 x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(g 1 x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 11 Youreallyshoulddothisonceinyourlife.

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1.8Transformations95 shiftright1unit )454()222()222()222()223()222()222()222()223()222()222()454(! add1toeach x -coordinate y = g 2 x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 y = g 3 x = g 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1= )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 horizontalstretchbyafactorof2 )470()223()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()471(! multiplyeach x -coordinateby2 y = g 3 x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 y = g x = g 3 )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 x = )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 4 x 2 + x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 Wehavekepttheviewingwindowthesameinallofthegraphsabove.Thishadtheundesirable consequenceofmakingthelastgraphlook`incomplete'inthatwecannotseetheoriginalshape of f x = x 2 .Alteringtheviewingwindowresultsinamorecompletegraphofthetransformed functionasseenbelow. y = g x Thisexamplebringsourrstchaptertoaclose.Inthechapterswhichlieahead,beonthelookout fortheconceptsdevelopedheretoresurfaceaswestudydierentfamiliesoffunctions.

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96RelationsandFunctions 1.8.1Exercises 1.Usethegraphof y = S x belowtographthefollowing.Note:Eachgraphbuildsuponthe previousgraph. x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 ; 0 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ; )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 ; 0 ; 3 ; 0 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 Thegraphof y = S x a y = S x +1 b y = S )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +1 c y = 1 2 S )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +1 d y = 1 2 S )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +1+1 2.Thecompletegraphof y = f x isgivenbelow.Useittographthefollowingfunctions. )]TJ/F15 9.9626 Tf 7.749 0 Td [(3 ; 0 ; 3 ; 0 x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 a g x = f x +3 b h x = f x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 c j x = f )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(2 3 d a x = f x +4 e b x = f x +1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 f c x = 3 5 f x g d x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 f x h k x = f )]TJ/F34 7.9701 Tf 6.195 -4.542 Td [(2 3 x i m x = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 4 f x j n x =4 f x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 k p x =4+ f )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x l q x = )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 2 f )]TJ/F37 7.9701 Tf 6.195 -4.541 Td [(x +4 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(3

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1.8Transformations97 3.Thegraphof y = 3 p x isgivenbelowontheleftandthegraphof y = g x isgivenonthe right.Findaformulafor g basedontransformationsofthegraphof f .Checkyouranswer byconrmingthatthepointsshownonthegraphof g satisfytheequation y = g x x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.756 0 Td [(10 )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345678 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 y = 3 p x x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.756 0 Td [(10 )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345678 )]TJ/F35 5.9776 Tf 5.757 0 Td [(5 )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 1 2 3 4 5 y = g x 4.Formanycommonfunctions,thepropertiesofalgebramakeahorizontalscalingthesame asaverticalscalingbypossiblyadierentfactor.Forexample,westatedearlierthat p 9 x =3 p x .Withthehelpofyourclassmates,ndtheequivalentverticalscalingproduced bythehorizontalscalings y = x 3 ;y = j 5 x j ;y = 3 p 27 x and y = )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 x 2 .Whatabout y = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 x 3 ;y = j)]TJ/F15 10.9091 Tf 16.364 0 Td [(5 x j ;y = 3 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(27 x and y = )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 x 2 ? 5.Wementionedearlierinthesectionthat,ingeneral,theorderinwhichtransformationsare appliedmatters,yetinourrstexamplewithtwotransformationstheorderdidnotmatter. Youcouldperformtheshifttotheleftfollowedbytheshiftdownoryoucouldshiftdown andthenlefttoachievethesameresult.Withthehelpofyourclassmates,determinethe situationsinwhichorderdoesmatterandthoseinwhichitdoesnot. 6.Whathappensifyoureectanevenfunctionacrossthe y -axis? 7.Whathappensifyoureectanoddfunctionacrossthe y -axis? 8.Whathappensifyoureectanevenfunctionacrossthe x -axis? 9.Whathappensifyoureectanoddfunctionacrossthe x -axis? 10.Howwouldyoudescribesymmetryabouttheoriginintermsofreections? 11.AswesawinExample1.8.5,theviewingwindowonthegraphingcalculatoraectshowwesee thetransformationsdonetoagraph.Usingtwodierentcalculators,ndviewingwindows sothat f x = x 2 ontheonecalculatorlookslike g x =3 x 2 ontheother.

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98RelationsandFunctions 1.8.2Answers 1.a x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 ; 0 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 ; )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; 0 ; 3 ; 0 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 y = S x +1 b x y ; 0 ; )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 ; 0 ; 3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ; 0 123 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 y = S )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +1 c x y ; 0 )]TJ/F34 7.9701 Tf 3.882 -6.416 Td [(2 ; )]TJ/F35 5.9776 Tf 7.782 3.258 Td [(3 2 ; 0 )]TJ/F34 7.9701 Tf 3.881 -6.416 Td [(0 ; 3 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ; 0 123 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 y = 1 2 S )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +1 d x y ; 1 )]TJ/F34 7.9701 Tf 3.882 -6.416 Td [(2 ; )]TJ/F35 5.9776 Tf 7.782 3.259 Td [(1 2 ; 1 )]TJ/F34 7.9701 Tf 3.881 -6.416 Td [(0 ; 5 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ; 1 )]TJ/F34 7.9701 Tf 6.586 0 Td [(113 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 y = 1 2 S )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +1+1 2.a g x = f x +3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 ; 3 ; 6 ; 3 x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 b h x = f x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 ; )]TJ/F6 4.9813 Tf 6.952 2.345 Td [(1 2 0 ; 5 2 3 ; )]TJ/F6 4.9813 Tf 6.952 2.345 Td [(1 2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3

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1.8Transformations99 c j x = f )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(2 3 )]TJ/F6 4.9813 Tf 6.952 2.346 Td [(7 3 ; 0 2 3 ; 3 11 3 ; 0 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 d a x = f x +4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 ; 0 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 ; 3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 ; 0 x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(7 )]TJ/F35 5.9776 Tf 5.757 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 e b x = f x +1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 ; )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 ; 2 ; )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 f c x = 3 5 f x )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 ; 0 0 ; 9 5 ; 0 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 g d x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 f x )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 ; 0 ; )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 ; 0 x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 h k x = f )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(2 3 x )]TJ/F6 4.9813 Tf 6.952 2.346 Td [(9 2 ; 0 ; 3 9 2 ; 0 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 i m x = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 4 f x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; 0 )]TJ/F34 7.9701 Tf 3.882 -6.416 Td [(0 ; )]TJ/F35 5.9776 Tf 7.782 3.259 Td [(3 4 ; 0 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 j n x =4 f x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 ; )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 ; 6 ; )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 x y 123456 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6

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100RelationsandFunctions k p x =4+ f )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x = f )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 x +1+4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 ; 4 1 2 ; 7 ; 4 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 l q x = )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(1 2 f )]TJ/F10 6.9738 Tf 5.762 -4.148 Td [(x +4 2 )]TJ/F15 9.9626 Tf 9.963 0 Td [(3= )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(1 2 f )]TJ/F7 6.9738 Tf 5.762 -4.148 Td [(1 2 x +2 )]TJ/F15 9.9626 Tf 9.963 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(10 ; )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 ; )]TJ/F6 4.9813 Tf 6.952 2.345 Td [(9 2 ; )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(10 )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 3. g x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 p x +3 )]TJ/F15 10.9091 Tf 10.91 0 Td [(1or g x =2 3 p )]TJ/F53 10.9091 Tf 8.485 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1

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Chapter2 LinearandQuadraticFunctions 2.1LinearFunctions Wenowbeginthestudyoffamiliesoffunctions.Ourrstfamily,linearfunctions,areoldfriendsas weshallsoonsee.RecallfromGeometrythattwodistinctpointsintheplanedetermineaunique linecontainingthosepoints,asindicatedbelow. P x 0 ;y 0 Q x 1 ;y 1 Togiveasenseofthe`steepness'oftheline,werecallwecancomputethe slope ofthelineusing theformulabelow. Equation 2.1 The slope m ofthelinecontainingthepoints P x 0 ;y 0 and Q x 1 ;y 1 is: m = y 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 0 x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 0 ; provided x 1 6 = x 0 AcoupleofnotesaboutEquation2.1areinorder.First,don'taskwhyweusetheletter` m 'to representslope.Therearemanyexplanationsoutthere,butapparentlynoonereallyknowsfor sure. 1 Secondly,thestipulation x 1 6 = x 0 ensuresthatwearen'ttryingtodividebyzero.Thereader isinvitedtopausetothinkaboutwhatishappeninggeometrically;theanxiousreadercanskip alongtothenextexample. Example 2.1.1 Findtheslopeofthelinecontainingthefollowingpairsofpoints,ifitexists.Plot eachpairofpointsandthelinecontainingthem. 1 Seewww.mathforum.org orwww.mathworld.wolfram.com fordiscussionsonthistopic.

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102LinearandQuadraticFunctions 1. P ; 0, Q ; 4 2. P )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 2, Q ; 4 3. P )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3, Q ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 4. P )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 ; 2, Q ; 2 5. P ; 3, Q ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 6. P ; 3, Q : 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 Solution. Ineachoftheseexamples,weapplytheslopeformula,Equation2.1. 1. m = 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 = 4 2 =2 P Q x y 1234 1 2 3 4 2. m = 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 = 2 4 = 1 2 P Q x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 1 2 3 4 3. m = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 4 = )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(3 2 P Q x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 4. m = 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 = 0 7 =0 P Q x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 1 2 3

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2.1LinearFunctions103 5. m = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 ,whichisundened P Q x y 12 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 6. m = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 : 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 : 1 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(40 P Q x y 12 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 AfewcommentsExample2.1.1areinorder.First,forreasonswhichwillbemadeclearsoon,if theslopeispositivethentheresultinglineissaidtobeincreasing.Ifitisnegative,wesaytheline isdecreasing.Aslopeof0resultsinahorizontallinewhichwesayisconstant,andanundened sloperesultsinaverticalline. 2 Second,thelargertheslopeisinabsolutevalue,thesteeperthe line.YoumayrecallfromIntermediateAlgebrathatslopecanbedescribedastheratio` rise run '.For example,inthesecondpartofExample2.1.1,wefoundtheslopetobe 1 2 .Wecaninterpretthis asariseof1unitupwardforevery2unitstotherightwetravelalongtheline,asshownbelow. `over2' `up1' x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 1 2 3 4 Usingmoreformalnotation,givenpoints x 0 ;y 0 and x 1 ;y 1 ,weusetheGreekletterdelta`'to write y = y 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 0 and x = x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 0 .Inmostscienticcircles,thesymbolmeans`changein'. 2 Someauthorsusetheunfortunatemoniker`noslope'whenaslopeisundened.It'seasytoconfusethenotions of`noslope'with`slopeof0'.Forthisreason,wewilldescribeslopesofverticallinesas`undened'.

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104LinearandQuadraticFunctions Hence,wemaywrite m = y x ; whichdescribestheslopeasthe rateofchange of y withrespectto x .Ratesofchangeabound inthe`realworld,'asthenextexampleillustrates. Example 2.1.2 At6AM,itis24 F;at10AM,itis32 F. 1.Findtheslopeofthelinecontainingthepoints ; 24and ; 32. 2.Interpretyouranswertotherstpartintermsoftemperatureandtime. 3.Predictthetemperatureatnoon. Solution. 1.Fortheslope,wehave m = 32 )]TJ/F34 7.9701 Tf 6.586 0 Td [(24 10 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 = 8 4 =2. 2.Sincethevaluesinthenumeratorcorrespondtothetemperaturesin F,andthevaluesin thedenominatorcorrespondtotimeinhours,wecaninterprettheslopeas2= 2 1 = 2 F 1 hour ; or2 Fperhour.Sincetheslopeispositive,weknowthiscorrespondstoanincreasingline. Hence,thetemperatureisincreasingatarateof2 Fperhour. 3.Noonistwohoursafter10AM.Assumingatemperatureincreaseof2 Fperhour,intwo hoursthetemperatureshouldrise4 F.Sincethetemperatureat10AMis32 F,wewould expectthetemperatureatnoontobe32+4=36 F. Nowitmaywellhappenthatinthepreviousscenario,atnoonthetemperatureisonly33 F. Thisdoesn'tmeanourcalculationsareincorrect.Rather,itmeansthatthetemperaturechange throughoutthedayisn'taconstant2 Fperhour.Mathematicsisoftenusedtodescribe,or model realworldphenomena.Mathematicalmodelsarejustthat:models.Thepredictionswegetout ofthemodelsmaybemathematicallyaccurate,butmaynotresemblewhathappensinthereal world.WewilldiscussthismorethoroughlyinSection2.5. InSection1.2,wediscussedtheequationsofverticalandhorizontallines.Usingtheconceptof slope,wecandevelopequationsfortheothervarietiesoflines.Supposealinehasaslopeof m and containsthepoint x 0 ;y 0 .Suppose x;y isanotherpointontheline,asindicatedbelow. x 0 ;y 0 x;y

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2.1LinearFunctions105 Wehave m = y )]TJ/F53 10.9091 Tf 10.91 0 Td [(y 0 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 0 m x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 0 = y )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 0 y )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 0 = m x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 0 y = m x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 0 + y 0 : Wehavejustderivedthe point-slopeform ofaline. 3 Equation 2.2 The point-slopeform ofthelinewithslope m containingthepoint x 0 ;y 0 is theequation y = m x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 0 + y 0 Example 2.1.3 Writetheequationofthelinecontainingthepoints )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 3and ; 1. Solution. InordertouseEquation2.2weneedtondtheslopeofthelineinquestion.Sowe useEquation2.1toget m = y x = 1 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [( )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(2 3 .Wearespoiledforchoiceforapoint x 0 ;y 0 We'lluse )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 3andleaveittothereadertocheckthatusing ; 1resultsinthesameequation. Substitutingintothepoint-slopeformoftheline,weget y = m x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 0 + y 0 y = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(2 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+3 y = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(2 3 x )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(2 3 +3 y = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(2 3 x + 7 3 : Wecancheckouranswerbyshowingthatboth )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 3and ; 1areonthegraphof y = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(2 3 x + 7 3 algebraically,aswedidinSection1.3. Insimplifyingtheequationofthelineinthepreviousexample,weproducedanotherformofa line,the slope-interceptform .Thisisthefamiliar y = mx + b formyouhaveprobablyseenin IntermediateAlgebra.The`intercept'in`slope-intercept'comesfromthefactthatifweset x =0, weget y = b .Inotherwords,the y -interceptoftheline y = mx + b is ;b Equation 2.3 The slope-interceptform ofthelinewithslope m and y -intercept ;b isthe equation y = mx + b: Notethatifwehaveslope m =0,wegettheequation y = b whichmatchesourformulafora horizontallinegiveninSection1.2.TheformulagiveninEquation2.3canbeusedtodescribeall linesexceptverticallines.Alllinesexceptverticallinesarefunctionswhy?andsowehavenally reachedagoodpointtointroduce linearfunctions 3 Wecanalsounderstandthisequationintermsofapplyingtransformationstothefunction I x = x .Seethe exercises.

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106LinearandQuadraticFunctions Definition 2.1 A linearfunction isafunctionoftheform f x = mx + b; where m and b arerealnumberswith m 6 =0.Thedomainofalinearfunctionis ; 1 Forthecase m =0,weget f x = b .Thesearegiventheirownclassication. Definition 2.2 A constantfunction isafunctionoftheform f x = b; where b isrealnumber.Thedomainofaconstantfunctionis ; 1 Recallthattographafunction, f ,wegraphtheequation y = f x .Hence,thegraphofalinear functionisalinewithslope m and y -intercept ;b ;thegraphofaconstantfunctionisahorizontal linewithslope m =0anda y -interceptof ;b .NowthinkbacktoSection1.7.1,specically Denition1.8concerningincreasing,decreasingandconstantfunctions.Alinewithpositiveslope wascalledanincreasinglinebecausealinearfunctionwith m> 0isanincreasingfunction. Similarly,alinewithanegativeslopewascalledadecreasinglinebecausealinearfunctionwith m< 0isadecreasingfunction.Andhorizontallineswerecalledconstantbecause,well,wehope you'vealreadymadetheconnection. Example 2.1.4 Graphthefollowingfunctions.Identifytheslopeand y -intercept. 1. f x =3 2. f x =3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 3. f x = 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 4. f x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 Solution. 1.Tograph f x =3,wegraph y =3.Thisisahorizontalline m =0through ; 3. 2.Thegraphof f x =3 x )]TJ/F15 10.9091 Tf 10.776 0 Td [(1isthegraphoftheline y =3 x )]TJ/F15 10.9091 Tf 10.776 0 Td [(1.Comparisonofthisequation withEquation2.3yields m =3and b = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Hence,ourslopeis3andour y -interceptis ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Togetanotherpointontheline,wecanplot ;f = ; 2. x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 1 2 3 4 f x =3 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 f x =3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1

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2.1LinearFunctions107 3.Atrstglance,thefunction f x = 3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x 4 doesnotttheforminDenition2.1butaftersome rearrangingweget f x = 3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x 4 = 3 4 )]TJ/F34 7.9701 Tf 11.145 4.295 Td [(2 x 4 = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 x + 3 4 .Weidentify m = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 and b = 3 4 .Hence, ourgraphisalinewithaslopeof )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(1 2 anda y -interceptof )]TJ/F15 10.9091 Tf 5 -8.837 Td [(0 ; 3 4 .Plottinganadditional point,wecanchoose ;f toget )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 ; 1 4 4.Ifwesimplifytheexpressionfor f ,weget f x = x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 = x +2 : Ifweweretostate f x = x +2,wewouldbecommittingasinofomission.Remember,to ndthedomainofafunction,wedoso before wesimplify!Inthiscase, f hasbigproblems when x =2,andassuch,thedomainof f is ; 2 [ ; 1 .Toindicatethis,wewrite f x = x +2 ;x 6 =2.So,exceptat x =2,wegraphtheline y = x +2.Theslope m =1 andthe y -interceptis ; 2.Asecondpointonthegraphis ;f = ; 3.Sinceour function f isnotdenedat x =2,weputanopencircleatthepointthatwouldbeonthe line y = x +2when x =2,namely ; 4. x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 1 2 f x = 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(1123 1 2 3 4 f x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +2 Thelasttwofunctionsinthepreviousexampleshowcasesomeofthedicultyindeningalinear functionusingthephrase`oftheform'asinDenition2.1,sincesomealgebraicmanipulations maybeneededtorewriteagivenfunctiontomatch`theform.'Keepinmindthatthedomainsof linearandconstantfunctionsareallrealnumbers, ; 1 ,andsowhile f x = x 2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 simplied toaformula f x = x +2, f isnotconsideredalinearfunctionsinceitsdomainexcludes x =2. However,wewouldconsider f x = 2 x 2 +2 x 2 +1 tobeaconstantfunctionsinceitsdomainisallrealnumberswhy?and f x = 2 x 2 +2 x 2 +1 = 2 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 +1 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 +1 =2 Thefollowingexampleuseslinearfunctionstomodelsomebasiceconomicrelationships.

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108LinearandQuadraticFunctions Example 2.1.5 Thecost, C ,indollars,toproduce x PortaBoy 4 gamesystemsforalocalretailer isgivenby C x =80 x +150for x 0. 1.Findandinterpret C 2.HowmanyPortaBoyscanbeproducedfor$15 ; 000? 3.Explainthesignicanceoftherestrictiononthedomain, x 0. 4.Findandinterpret C 5.Findandinterprettheslopeofthegraphof y = C x Solution. 1.Tond C ,wereplaceeveryoccurrenceof x with10intheformulafor C x toget C =80+150=950.Since x representsthenumberofPortaBoysproduced,and C x representsthecost,indollars, C =950meansitcosts$950toproduce10PortaBoys forthelocalretailer. 2.TondhowmanyPortaBoyscanbeproducedfor$15 ; 000,wesetthecost, C x ,equalto 15000,andsolvefor x C x =15000 80 x +150=15000 80 x =14850 x = 14850 80 =185 : 625 SincewecanonlyproduceawholenumberamountofPortaBoys,wecanproduce185 PortaBoysfor$15 ; 000. 3.Therestriction x 0istheapplieddomain,asdiscussedinSection1.5.Inthiscontext, x representsthenumberofPortaBoysproduced.Itmakesnosensetoproduceanegative quantityofgamesystems. 5 4.Tond C ,wereplaceeveryoccurrenceof x with0intheformulafor C x toget C = 80+150=150.Thismeansitcosts$150toproduce0PortaBoys.The$150isoftencalled the xed or start-up costofthisventure.Whatmightcontributetothiscost? 4 ThesimilarityofthisnametoPortaJohn isdeliberate. 5 Actually,itmakesnosensetoproduceafractionalpartofagamesystem,either,aswesawinthepreviouspart ofthisexample.Thisabsurdity,however,seemsquiteforgivableinsometextbooksbutnottous.

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2.1LinearFunctions109 5.Ifweweretograph y = C x ,wewouldbegraphingtheportionoftheline y =80 x +150 for x 0.Werecognizetheslope, m =80.Likeanyslope,wecaninterpretthisasarateof change.Inthiscase, C x isthecostindollars,while x measuresthenumberofPortaBoys so m = y x = C x =80= 80 1 = $80 1PortaBoy : Inotherwords,thecostisincreasingatarateof$80perPortaBoyproduced.Thisisoften calledthe variablecost forthisventure. Thenextexampleasksustondalinearfunctiontomodelarelatedeconomicproblem. Example 2.1.6 ThelocalretailerinExample2.1.5hasdeterminedthatthenumberofPortaBoy gamesystemssoldinaweek, x ,isrelatedtothepriceofeachsystem, p ,indollars.Whenthe pricewas$220,20gamesystemsweresoldinaweek.Whenthesystemswentonsalethefollowing week,40systemsweresoldat$190apiece. 1.Findalinearfunctionwhichtsthisdata.Usetheweeklysales, x ,astheindependent variableandtheprice p ,asthedependentvariable. 2.Findasuitableapplieddomain. 3.Interprettheslope. 4.Iftheretailerwantstosell150PortaBoysnextweek,whatshouldthepricebe? 5.Whatwouldtheweeklysalesbeifthepriceweresetat$150persystem? Solution. 1.WerecallfromSection1.5themeaningof`independent'and`dependent'variable.Since x istobetheindependentvariable,and p thedependentvariable,wetreat x astheinput variableand p astheoutputvariable.Hence,wearelookingforafunctionoftheform p x = mx + b .Todetermine m and b ,weusethefactthat20PortaBoysweresoldduring theweekthepricewas220dollarsand40unitsweresoldwhenthepricewas190dollars. Usingfunctionnotation,thesetwofactscanbetranslatedas p =220and p =190. Since m representstherateofchangeof p withrespectto x ,wehave m = p x = 190 )]TJ/F15 10.9091 Tf 10.909 0 Td [(220 40 )]TJ/F15 10.9091 Tf 10.909 0 Td [(20 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 20 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 : Wenowhavedetermined p x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x + b .Todetermine b ,wecanuseourgivendataagain. Using p =220,wesubstitute x =20into p x =1 : 5 x + b andsettheresultequalto220: )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5+ b =220.Solving,weget b =250.Hence,weget p x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x +250.Wecan checkourformulabycomputing p and p toseeifweget220and190,respectively. Incidentally,thisequationissometimescalledthe price-demand 6 equationforthisventure. 6 Orsimplythe demand equation

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110LinearandQuadraticFunctions 2.Todeterminetheapplieddomain,welookatthephysicalconstraintsoftheproblem.Certainly,wecan'tsellanegativenumberofPortaBoys,so x 0.However,wealsonotethatthe slopeofthislinearfunctionisnegative,andassuch,thepriceisdecreasingasmoreunitsare sold.Anotherconstraint,then,isthattheprice, p x 0.Solving )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x +250 0results in )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 : 5 x )]TJ/F15 10.9091 Tf 20.111 0 Td [(250or x 500 3 =166 : 6.Since x representsthenumberofPortaBoyssoldina week,werounddownto166.Asaresult,areasonableapplieddomainfor p is[0 ; 166]. 3.Theslope m = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5,onceagain,representstherateofchangeofthepriceofasystemwith respecttoweeklysalesofPortaBoys.Sincetheslopeisnegative,wehavethattheprice isdecreasingatarateof$1 : 50perPortaBoysold.Saiddierently,youcansellonemore PortaBoyforevery$1 : 50dropinprice. 4.Todeterminethepricewhichwillmove150PortaBoys,wend p = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5+250=25. Thatis,thepricewouldhavetobe$25. 5.IfthepriceofaPortaBoyweresetat$150,wehave p x =150,or, )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x +250=150.Solving, weget )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(100or x =66 : 6.Thismeansyouwouldbeabletosell66PortaBoysaweek ifthepricewere$150persystem. Notallreal-worldphenomenacanbemodeledusinglinearfunctions.Nevertheless,itispossibleto usetheconceptofslopetohelpanalyzenon-linearfunctionsusingthefollowing: Definition 2.3 Let f beafunctiondenedontheinterval[ a;b ].The averagerateofchange of f over[ a;b ]isdenedas: f x = f b )]TJ/F53 10.9091 Tf 10.91 0 Td [(f a b )]TJ/F53 10.9091 Tf 10.91 0 Td [(a Geometrically,ifwehavethegraphof y = f x ,theaveragerateofchangeover[ a;b ]istheslopeof thelinewhichconnects a;f a and b;f b .Thisiscalledthe secantline throughthesepoints. Forthatreason,sometextbooksusethenotation m sec fortheaveragerateofchangeofafunction. Notethatforalinearfunction m = m sec ,orinotherwords,itsrateofchangeoveranintervalis thesameasitsaveragerateofchange.

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2.1LinearFunctions111 a;f a b;f b y = f x Thegraphof y = f x anditssecantlinethrough a;f a and b;f b Theinterestedreadermayquestiontheadjective`average'inthephrase`averagerateofchange.' Inthegureabove,wecanseethatthefunctionchangeswildlyon[ a;b ],yettheslopeofthesecant lineonlycapturesasnapshotoftheactionat a and b .Thissituationisentirelyanalogoustothe averagespeedonatrip.Supposeittakesyou2hourstotravel100miles.Youraveragespeedis 100 miles 2 hours =50milesperhour.However,itisentirelypossiblethatatthestartofyourjourney,you traveled25milesperhour,thenspedupto65milesperhour,andsoforth.Theaveragerateof changeisakintoyouraveragespeedonthetrip.Yourspeedometermeasuresyourspeedatany oneinstantalongthetrip,your instantaneousratesofchange ,andthisisoneofthecentral themesofCalculus. 7 Wheninterpretingratesofchange,weinterpretthemthesamewaywedidslopes.Inthecontext offunctions,itmaybehelpfultothinkoftheaveragerateofchangeas: changeinoutputs changeininputs Example 2.1.7 The revenue ofselling x unitsataprice p perunitisgivenbytheformula R = xp SupposeweareinthescenarioofExamples2.1.5and2.1.6. 1.Findandsimplifyanexpressionfortheweeklyrevenue R asafunctionofweeklysales, x 2.Findandinterprettheaveragerateofchangeof R overtheinterval[0 ; 50]. 3.Findandinterprettheaveragerateofchangeof R as x changesfrom50to100andcompare thattoyourresultinpart2. 4.Findandinterprettheaveragerateofchangeofweeklyrevenueasweeklysalesincreasefrom 100PortaBoysto150PortaBoys. Solution. 7 Herewegoagain...

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112LinearandQuadraticFunctions 1.Since R = xp ,wesubstitute p x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x +250fromExample2.1.6toget R x = x )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x +250= )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 : 5 x 2 +250 x 2.UsingDenition2.3,wegettheaveragerateofchangeis R x = R )]TJ/F53 10.9091 Tf 10.909 0 Td [(R 50 )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 = 8750 )]TJ/F15 10.9091 Tf 10.91 0 Td [(0 50 )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 =175 : Interpretingthisslopeaswehaveinsimilarsituations,weconcludethatforeveryadditional PortaBoysoldduringagivenweek,theweeklyrevenueincreases$175. 3.ThewordingofthispartisslightlydierentthanthatinDenition2.3,butitsmeaningisto ndtheaveragerateofchangeof R overtheinterval[50 ; 100].Tondthisrateofchange, wecompute R x = R )]TJ/F53 10.9091 Tf 10.909 0 Td [(R 100 )]TJ/F15 10.9091 Tf 10.909 0 Td [(50 = 10000 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8750 50 =25 : Inotherwords,foreachadditionalPortaBoysold,therevenueincreasesby$25.Notewhile therevenueisstillincreasingbysellingmoregamesystems,wearen'tgettingasmuchofan increaseaswedidinpart2ofthisexample.Canyouthinkofwhythiswouldhappen? 4.TranslatingtheEnglishtothemathematics,wearebeingaskedtondtheaveragerateof changeof R overtheinterval[100 ; 150].Wend R x = R )]TJ/F53 10.9091 Tf 10.909 0 Td [(R 150 )]TJ/F15 10.9091 Tf 10.909 0 Td [(100 = 3750 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10000 50 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(125 : Thismeansthatwearelosing$125dollarsofweeklyrevenueforeachadditionalPortaBoy sold.Canyouthinkwhythisispossible? Weclosethissectionwithanewlookatdierencequotients,rstintroducedinSection1.5.Ifwe wishtocomputetheaveragerateofchangeofafunction f overtheinterval[ x;x + h ],thenwe wouldhave f x = f x + h )]TJ/F53 10.9091 Tf 10.909 0 Td [(f x x + h )]TJ/F53 10.9091 Tf 10.91 0 Td [(x = f x + h )]TJ/F53 10.9091 Tf 10.909 0 Td [(f x h Aswehaveindicated,therateofchangeofafunctionaverageorotherwiseisofgreatimportance inCalculus. 8 8 So,wearenottorturingyouwiththesefornothing.

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2.1LinearFunctions113 2.1.1Exercises 1.Findboththepoint-slopeformandtheslope-interceptformofthelinewiththegivenslope whichpassesthroughthegivenpoint. a m = 1 7 ;P )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 4 b m = )]TJ 8.484 9.025 Td [(p 2 ;P ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 c m = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 ;P p 3 ; 2 p 3 d m =678 ;P )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 2.Findtheslope-interceptformofthelinewhichpassesthroughthegivenpoints. a P ; 0 ;Q )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 ; 5 b P )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ;Q ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 c P ; 0 ;Q ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(8 d P ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 ;Q ; 4 3.Waterfreezesat0 Celsiusand32 Fahrenheitanditboilsat100 Cand212 F. aFindalinearfunction F thatexpressestemperatureintheFahrenheitscaleintermsof degreesCelsius.Usethisfunctiontoconvert20 CintoFahrenheit. bFindalinearfunction C thatexpressestemperatureintheCelsiusscaleintermsof degreesFahrenheit.Usethisfunctiontoconvert110 FintoCelsius. cIsthereatemperature n suchthat F n = C n ? 4.Asalespersonispaid $ 200perweekplus5%commissiononherweeklysalesof x dollars. Findalinearfunctionthatrepresentshertotalweeklypayintermsof x .Whatmusther weeklysalesbeinorderforhertoearn $ 475.00fortheweek? 5.Findallofthepointsontheline y =2 x +1whichare4unitsfromthepoint )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 3. 6.Economicforcesbeyondanyone'scontrolhavechangedthecostfunctionforPortaBoysto C x =105 x +175.ReworkExample2.1.5withthisnewcostfunction. 7.Inresponsetotheeconomicforcesintheexerciseabove,thelocalretailersetstheselling priceofaPortaBoyat $ 250.Remarkably,30unitsweresoldeachweek.Whenthesystems wentonsalefor $ 220,40unitsperweekweresold.ReworkExamples2.1.6and2.1.7with thisnewdata.Whatdicultiesdoyouencounter? 8.LegendhasitthatabullSasquatchinrutwillhowlapproximately9timesperhourwhenitis 40 F outsideandonly5timesperhourifit's70 F .Assumingthatthenumberofhowlsper hour, N ,canberepresentedbyalinearfunctionoftemperatureFahrenheit,ndthenumber ofhowlsperhourhe'llmakewhenit'sonly20 F outside.Whatistheapplieddomainofthis function?Why? 9.ParallelLinesRecallfromIntermediateAlgebrathatparallellineshavethesameslope. Pleasenotethattwoverticallinesarealsoparalleltooneanothereventhoughtheyhave anundenedslope.Intheexercisesbelow,youaregivenalineandapointwhichisnoton thatline.Findthelineparalleltothegivenlinewhichpassesthroughthegivenpoint.

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114LinearandQuadraticFunctions a y =3 x +2 ;P ; 0b y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x +5 ;P ; 2 10.PerpendicularLinesRecallfromIntermediateAlgebrathattwonon-verticallinesareperpendicularifandonlyiftheyhavenegativereciprocalslopes.Thatistosay,ifonelinehas slope m 1 andtheotherhasslope m 2 then m 1 m 2 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1.Youwillbeguidedthroughaproof ofthisresultinthenextexercise.Pleasenotethatahorizontallineisperpendiculartoa verticallineandviceversa,soweassume m 1 6 =0and m 2 6 =0.Intheexercisesbelow,youare givenalineandapointwhichisnotonthatline.Findthelineperpendiculartothegiven linewhichpassesthroughthegivenpoint. a y = 1 3 x +2 ;P ; 0b y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x +5 ;P ; 2 11.Weshallnowprovethat y = m 1 x + b 1 isperpendicularto y = m 2 x + b 2 ifandonlyif m 1 m 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Tomakeourliveseasierweshallassumethat m 1 > 0and m 2 < 0.Wecan alsomove"thelinessothattheirpointofintersectionistheoriginwithoutmessingthings up,sowe'llassume b 1 = b 2 =0 : Takeamomentwithyourclassmatestodiscusswhythisis okay.Graphingthelinesandplottingthepoints O ; 0, P ;m 1 and Q ;m 2 givesus thefollowingsetup. P O Q x y Theline y = m 1 x willbeperpendiculartotheline y = m 2 x ifandonlyif 4 OPQ isaright triangle.Let d 1 bethedistancefrom O to P ,let d 2 bethedistancefrom O to Q andlet d 3 bethedistancefrom P to Q .UsethePythagoreanTheoremtoshowthat 4 OPQ isaright triangleifandonlyif m 1 m 2 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1byshowing d 2 1 + d 2 2 = d 2 3 ifandonlyif m 1 m 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.

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2.1LinearFunctions115 12.Thefunctiondenedby I x = x iscalledtheIdentityFunction. aDiscusswithyourclassmateswhythisnamemakessense. bShowthatthepoint-slopeformofalineEquation2.2canbeobtainedfrom I usinga sequenceofthetransformationsdenedinSection1.8. 13.Computetheaveragerateofchangeofthegivenfunctionoverthespeciedinterval. a f x = x 3 ; [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 2] b f x = 1 x ; [1 ; 5] c f x = p x; [0 ; 16] d f x = x 2 ; [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 3] e f x = x +4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ; [5 ; 7] f f x =3 x 2 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 ; [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 2] 14.Computetheaveragerateofchangeofthegivenfunctionovertheinterval[ x;x + h ].Here weassume[ x;x + h ]isinthedomainofeachfunction. a f x = x 3 b f x = 1 x c f x = x +4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 d f x =3 x 2 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 15.Withthehelpofyourclassmatesndseveralreal-world"examplesofratesofchangethat areusedtodescribenon-linearphenomena.

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116LinearandQuadraticFunctions 2.1.2Answers 1.a y )]TJ/F15 10.9091 Tf 10.909 0 Td [(4= 1 7 x +1 y = 1 7 x + 29 7 b y +3= p 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 y = )]TJ 8.485 9.024 Td [(p 2 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 c y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 p 3= )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x )]TJ 10.909 9.025 Td [(p 3 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x +7 p 3 d y +12=678 x +1 y =678 x +666 2.a y = )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(5 3 x b y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 c y = 8 5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 d y = 9 4 x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(47 4 3.a F C = 9 5 C +32 b C F = 5 9 F )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(160 9 c F )]TJ/F15 10.9091 Tf 8.484 0 Td [(40= )]TJ/F15 10.9091 Tf 8.485 0 Td [(40= C )]TJ/F15 10.9091 Tf 8.485 0 Td [(40. 4. W x =200+ : 05 x; Shemustmake $ 5500inweeklysales. 5. )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(11 5 ; 27 5 8. N T = )]TJ/F34 7.9701 Tf 11.797 4.296 Td [(2 15 T + 43 3 Havinganegativenumberofhowlsmakesnosenseandsince N : 5=0wecanputan upperboundof107 : 5 onthedomain.Thelowerboundistrickierbecausethere'snothing otherthancommonsensetogoon.Asitgetscolder,hehowlsmoreoften.Atsomepoint itwilleitherbesocoldthathefreezestodeathorhe'showlingnon-stop.Sowe'regoingto saythathecanwithstandtemperaturesnolowerthan )]TJ/F15 10.9091 Tf 8.485 0 Td [(60 sothattheapplieddomainis [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(60 ; 107 : 5]. 9.a y =3 x b y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x +20 10.a y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x b y = 1 6 x + 3 2 12.a 2 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 =3 b 1 5 )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 1 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(1 5 c p 16 )]TJ 10.909 9.024 Td [(p 0 16 )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 = 1 4 d 3 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 =0 e 7+4 7 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(5+4 5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 = )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(7 8 f 2 +2 )]TJ/F34 7.9701 Tf 8.469 0 Td [(7 )]TJ/F34 7.9701 Tf 8.468 0 Td [( )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 2 +2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 8.468 0 Td [(7 2 )]TJ/F34 7.9701 Tf 8.468 0 Td [( )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 = )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 13.a3 x 2 +3 xh + h 2 b )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x x + h c6 x +3 h +2 d )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x + h )]TJ/F15 10.9091 Tf 10.909 0 Td [(3

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2.2AbsoluteValueFunctions117 2.2AbsoluteValueFunctions Thereareafewwaystodescribewhatismeantbytheabsolutevalue j x j ofarealnumber x .You mayhavebeentaughtthat j x j isthedistancefromtherealnumber x tothe0onthenumber.So, forexample, j 5 j =5and j)]TJ/F15 10.9091 Tf 16.364 0 Td [(5 j =5,sinceeachis5unitsfrom0onthenumberline. distanceis5unitsdistanceis5units )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1012345 Anotherwaytodeneabsolutevalueisbytheequation j x j = p x 2 .Usingthisdenition,wehave j 5 j = p 2 = p 25=5and j)]TJ/F15 10.9091 Tf 17.281 0 Td [(5 j = p )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 2 = p 25=5.Thelongandshortofbothofthese proceduresisthat j x j takesnegativerealnumbersandassignsthemtotheirpositivecounterparts whileitleavespositivenumbersalone.Thislastdescriptionistheoneweshalladopt,andis summarizedinthefollowingdenition. Definition 2.4 The absolutevalue ofarealnumber x ,denoted j x j ,isgivenby j x j = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x; if x< 0 x; if x 0 InDenition2.4,wedene j x j usingapiecewise-denedfunction.Seepage46inSection1.5.To checkthatthisdenitionagreeswithwhatwepreviouslyunderstoodasabsolutevalue,notethat since5 0,tond j 5 j weusetherule j x j = x ,so j 5 j =5.Similarly,since )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 < 0,weusethe rule j x j = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x ,sothat j)]TJ/F15 10.9091 Tf 16.483 0 Td [(5 j = )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(5=5.Thisisoneofthetimeswhenit'sbesttointerpretthe expression` )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 'as`theoppositeof x 'asopposedto`negative x .'Beforeweembarkonstudying absolutevaluefunctions,weremindourselvesofthepropertiesofabsolutevalue. Theorem 2.1 PropertiesofAbsoluteValue: Let a b ,and x berealnumbersandlet n be aninteger. a Then ProductRule: j ab j = j a jj b j PowerRule: j a n j = j a j n whenever a n isdened QuotientRule: a b = j a j j b j ,provided b 6 =0 j x j =0ifandonlyif x =0. For c> 0, j x j = c ifandonlyif x = c or x = )]TJ/F53 10.9091 Tf 8.484 0 Td [(c For c< 0, j x j = c hasnosolution. a Recallthatthismeans n =0 ; 1 ; 2 ;:::: TheproofoftheProductandQuotientRulesinTheorem2.1boilsdowntocheckingfourcases: whenboth a and b arepositive;whentheyarebothnegative;whenoneispositiveandtheother

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118LinearandQuadraticFunctions isnegative;whenoneorbotharezero.Forexample,supposewewishtoshow j ab j = j a jj b j .We needtoshowthisequationistrueforallrealnumbers a and b .If a and b arebothpositive,then sois ab .Hence, j a j = a j b j = b ,and j ab j = ab .Hence,theequation j ab j = j a jj b j isthesameas ab = ab whichistrue.Ifboth a and b arenegative,then ab ispositive.Hence, j a j = )]TJ/F53 10.9091 Tf 8.485 0 Td [(a j b j = )]TJ/F53 10.9091 Tf 8.485 0 Td [(b and j ab j = ab .Theequation j ab j = j a jj b j becomes ab = )]TJ/F53 10.9091 Tf 8.485 0 Td [(a )]TJ/F53 10.9091 Tf 8.485 0 Td [(b ,whichistrue.Suppose a is positiveand b isnegative.Then ab isnegative,andwehave j ab j = )]TJ/F53 10.9091 Tf 8.485 0 Td [(ab j a j = a and j b j = )]TJ/F53 10.9091 Tf 8.485 0 Td [(b Theequation j ab j = j a jj b j reducesto )]TJ/F53 10.9091 Tf 8.485 0 Td [(ab = a )]TJ/F53 10.9091 Tf 8.485 0 Td [(b whichistrue.Asymmetricargumentshowsthe equation j ab j = j a jj b j holdswhen a isnegativeand b ispositive.Finally,ifeither a or b orboth arezero,thenbothsidesof j ab j = j a jj b j arezero,andsotheequationholdsinthiscase,too.All ofthisrhetorichasshownthattheequation j ab j = j a jj b j holdstrueinallcases.Theproofofthe QuotientRuleisverysimilar,withtheexceptionthat b 6 =0.ThePowerRulecanbeshownby repeatedapplicationoftheProductRule.ThelastthreepropertiescanbeprovedusingDenition 2.4andbylookingatthecaseswhen x 0,inwhichcase j x j = x ,orwhen x< 0,inwhichcase j x j = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x .Forexample,if c> 0,and j x j = c ,thenif x 0,wehave x = j x j = c .If,ontheother hand, x< 0,then )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = j x j = c ,so x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(c .Theremainingpropertiesareprovedsimilarlyandare leftfortheexercises. TographfunctionsinvolvingabsolutevaluewemakeliberaluseofDenition2.4,asthenext exampleillustrates. Example 2.2.1 Grapheachofthefollowingfunctions.Findthezerosofeachfunctionandthe x -and y -interceptsofeachgraph,ifanyexist.Fromthegraph,determinethedomainandrange ofeachfunction,listtheintervalsonwhichthefunctionisincreasing,decreasing,orconstant,and ndtherelativeandabsoluteextrema,iftheyexist. 1. f x = j x j 2. g x = j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 j 3. h x = j x j)]TJ/F15 10.9091 Tf 16.363 0 Td [(3 4. i x =4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 j 3 x +1 j Solution. 1.Tondthezerosof f ,weset f x =0.Weget j x j =0,which,byTheorem2.1givesus x =0. Sincethezerosof f arethe x -coordinatesofthe x -interceptsofthegraphof y = f x ,weget ; 0isouronly x -intercept.Tondthe y -intercept,weset x =0,andnd y = f =0,so that ; 0isour y -interceptaswell. 1 WithSection2.1underourbelts,wecanuseDenition 2.4toget f x = j x j = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x; if x< 0 x; if x 0 Hence,for x< 0,wearegraphingtheline y = )]TJ/F53 10.9091 Tf 8.484 0 Td [(x ;for x 0,wehavetheline y = x ProceedingaswedidinSection1.7,weget 1 Actually,sincefunctionscanhaveatmostone y -interceptwhy?,assoonaswefound ; 0asthe x -intercept, weknewthiswasalsothe y -intercept.

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2.2AbsoluteValueFunctions119 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 1 2 3 4 f x = j x j x< 0 x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1123 1 2 3 4 f x = j x j x 0 Noticewehavean`opencircle'at ; 0inthegraphwhen x< 0.Aswehaveseenbefore, thisisduetothefactthepointson y = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x approach ; 0asthe x -valuesapproach0.Since x isrequiredtobestrictlylessthanzeroonthisstretch,however,theopencircleisdrawn. However,noticethatwhen x 0,wegettollinthepointat ; 0,whicheectively`plugs' theholeindicatedbytheopencircle.Hence,weget x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 1 2 3 4 f x = j x j Byprojectingthegraphtothe x -axis,weseethatthedomainis ; 1 .Projectingto the y -axisgivesustherange[0 ; 1 .Thefunctionisincreasingon[0 ; 1 anddecreasingon ; 0].Therelativeminimumvalueof f isthesameastheabsoluteminimum,namely0 whichoccursat ; 0.Thereisnorelativemaximumvalueof f .Thereisalsonoabsolute maximumvalueof f ,sincethe y valuesonthegraphextendinnitelyupwards. 2.Tondthezerosof g ,weset g x = j x )]TJ/F15 10.9091 Tf 11.542 0 Td [(3 j =0.ByTheorem2.1,weget x )]TJ/F15 10.9091 Tf 11.541 0 Td [(3=0so that x =3.Hence,the x -interceptis ; 0.Tondour y -intercept,weset x =0sothat y = g = j 0 )]TJ/F15 10.9091 Tf 11.501 0 Td [(3 j =3,andso ; 3isour y -intercept.Tograph g x = j x )]TJ/F15 10.9091 Tf 11.501 0 Td [(3 j ,weuse Denition2.4torewriteas g x = j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 j = )]TJ/F15 10.9091 Tf 8.484 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ; if x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 < 0 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ; if x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 0 Simplifying,weget g x = )]TJ/F53 10.9091 Tf 8.484 0 Td [(x +3 ; if x< 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ; if x 3

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120LinearandQuadraticFunctions Asbefore,theopencircleweintroduceat ; 0fromthegraphof y = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +3islledbythe point ; 0fromtheline y = x )]TJ/F15 10.9091 Tf 10.75 0 Td [(3.Wedeterminethedomainas ; 1 andtherangeas [0 ; 1 .Thefunction g isincreasingon[3 ; 1 anddecreasingon ; 3].Therelativeand absoluteminimumvalueof g is0whichoccursat ; 0.Asbefore,thereisnorelativeor absolutemaximumvalueof g 3.Setting h x =0tolookforzerosgives j x j)]TJ/F15 10.9091 Tf 16.537 0 Td [(3=0.Beforewecanuseanyoftheproperties inTheorem2.1,weneedtoisolatetheabsolutevalue.Doingsogives j x j =3sothat x =3 or x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.Asaresult,wehaveapairof x -intercepts: )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 0and ; 0.Setting x =0gives y = h = j 0 j)]TJ/F15 10.9091 Tf 17.298 0 Td [(3= )]TJ/F15 10.9091 Tf 8.485 0 Td [(3,soour y -interceptis ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.Asbefore,werewritetheabsolute valuein h toget h x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ; if x< 0 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ; if x 0 Onceagain,theopencircleat ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3fromonepieceofthegraphof h islledbythepoint ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3fromtheotherpieceof h .Fromthegraph,wedeterminethedomainof h is ; 1 andtherangeis[ )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 ; 1 .On[0 ; 1 h isincreasing;on ; 0]itisdecreasing.Therelative minimumoccursatthepoint ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3onthegraph,andwesee )]TJ/F15 10.9091 Tf 8.485 0 Td [(3isboththerelativeand absoluteminimumvalueof h .Also, h hasnorelativeorabsolutemaximumvalue. x y 12345 1 2 3 4 g x = j x )]TJ/F35 5.9776 Tf 7.454 0 Td [(3 j x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 h x = j x j)]TJ/F35 5.9776 Tf 11.42 0 Td [(3 4.Asbefore,weset i x =0tondthezerosof i toget4 )]TJ/F15 10.9091 Tf 10.444 0 Td [(2 j 3 x +1 j =0.Onceagain,weneed toisolatetheabsolutevalueexpressionbeforeweapplyTheorem2.1.So4 )]TJ/F15 10.9091 Tf 11.179 0 Td [(2 j 3 x +1 j =0 becomes2 j 3 x +1 j =4andhence, j 3 x +1 j =2.ApplyingTheorem2.1,weget3 x +1=2or 3 x +1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,fromwhichweget x = 1 3 or x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1.Our x -interceptsare )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 3 ; 0 and )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0. Substituting x =0gives y = i =4 )]TJ/F15 10.9091 Tf 10.948 0 Td [(2 j 3+1 j =2,fora y -interceptof ; 2.Rewriting theformulafor i x withoutabsolutevaluesgives i x = 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [( x +1 ; if3 x +1 < 0 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 ; if3 x +1 0 = 6 x +6 ; if x< )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x +2 ; if x )]TJ/F34 7.9701 Tf 21.195 4.295 Td [(1 3 Theusualanalysisnearthetroublespot, x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 givesthe`corner'ofthisgraphis )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 ; 4 andwegetthedistinctive` 'shape:

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2.2AbsoluteValueFunctions121 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 i x =4 )]TJ/F35 5.9776 Tf 7.454 0 Td [(2 j 3 x +2 j Thedomainof i is ; 1 whiletherangeis ; 4].Thefunction i isincreasingon )]TJ/F54 10.9091 Tf 5 -8.837 Td [( ; )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 3 anddecreasingon )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 3 ; 1 .Therelativemaximumoccursatthepoint )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 3 ; 4 andtherelativeandabsolutemaximumvalueof i is4.Sincethegraphof i extendsdownwards forevermore,thereisnoabsoluteminimumvalue.Aswecanseefromthegraph,thereisno relativeminimum,either. Notethatallofthefunctionsinthepreviousexamplebearthecharacteristic` 'shapeasthegraph of y = j x j .Infact,wecouldhavegraphedallofthefunctions g h ,and i inExample2.2.1starting withthegraphof f x = j x j andapplyingtransformationsasinSection1.8.Forexample,thefor thefunction g ,wehave g x = j x )]TJ/F15 10.9091 Tf 10.921 0 Td [(3 j = f x )]TJ/F15 10.9091 Tf 10.92 0 Td [(3.Theorem1.3tellsusthiscausesthegraphof f tobeshiftedtotheright3units.Choosingthreerepresentativepointsonthegraphof f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1, ; 0and ; 1,wecangraph g asfollows. x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ; 1 ; 0 ; 1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 1 2 3 4 f x = j x j shiftright3unit )454()222()222()222()223()222()222()222()223()222()222()454(! add3toeach x -coordinate x y ; 1 ; 0 ; 1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11245 1 2 3 4 g x = f x )]TJ/F34 7.9701 Tf 8.468 0 Td [(3= j x )]TJ/F34 7.9701 Tf 8.468 0 Td [(3 j Similarly,thegraphof h inExample2.2.1canbeunderstoodviaTheorem1.2asaverticalshift down3units.Thefunction i canbegraphedusingTheorem1.7byndingthenaldestinations ofthethreepoints )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1, ; 0and ; 1andconnectingtheminthecharacteristic` 'fashion. WhilethemethodsinSection1.8canbeusedtographanentirefamilyofabsolutevaluefunctions, notallfunctionsinvolvingabsolutevaluespossesthecharacteristic` 'shape,asthenextexample illustrates. Example 2.2.2 Grapheachofthefollowingfunctions.Findthezerosofeachfunctionandthe x -and y -interceptsofeachgraph,ifanyexist.Fromthegraph,determinethedomainandrange

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122LinearandQuadraticFunctions ofeachfunction,listtheintervalsonwhichthefunctionisincreasing,decreasing,orconstant,and ndtherelativeandabsoluteextrema,iftheyexist. 1. f x = j x j x 2. g x = j x +2 j)-222(j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 j +1 Solution. 1.Werstnotethat,duetothefractionintheformulaof f x x 6 =0.Thusthedomainis ; 0 [ ; 1 .Tondthezerosof f ,weset f x = j x j x =0.Thislastequationimplies j x j =0,which,fromTheorem2.1,implies x =0.However, x =0isnotinthedomainof f whichmeanswehave,infact,no x -intercepts.Forthesamereason,wehaveno y -intercepts, since f isundened.Re-writingtheabsolutevalueinthefunctiongives f x = 8 < : )]TJ/F53 10.9091 Tf 8.485 0 Td [(x x ; if x< 0 x x ; if x> 0 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; if x< 0 1 ; if x> 0 Tographthisfunction,wegraphtwohorizontallines: y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1for x< 0and y =1for x> 0. Wehaveopencirclesat ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and ; 1Canyouexplainwhy?soweget x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 f x = j x j x Aswefoundearlier,thedomainis ; 0 [ ; 1 .Therangeconsistsofjust2 y values: f)]TJ/F15 10.9091 Tf 13.939 0 Td [(1 ; 1 g 2 Thefunction f isconstanton ; 0and ; 1 .Thelocalminimumvalueof f istheabsoluteminimumvalueof f ,namely )]TJ/F15 10.9091 Tf 8.485 0 Td [(1;thelocalmaximumandabsolutemaximum valuesfor f alsocoincide )]TJ/F15 10.9091 Tf 11.833 0 Td [(theybothare1.Everypointonthegraphof f issimultaneously arelativemaximumandarelativeminimum.CanyouseewhyinlightofDenition1.9? ThiswasexploredintheexercisesinSection1.7.2. 2.Tondthezerosof g ,weset g x =0.Theresultis j x +2 j)-243(j x )]TJ/F15 10.9091 Tf 11.128 0 Td [(3 j +1=0.Attempting toisolatetheabsolutevaluetermiscomplicatedbythefactthatthereare two termswith absolutevalues.Inthiscase,iteasiertoproceedusingcasesbyre-writingthefunction g with twoseparateapplicationsofDenition2.4toremoveeachinstanceoftheabsolutevalues,one atatime.Intherstroundweget g x = )]TJ/F15 10.9091 Tf 8.485 0 Td [( x +2 )-222(j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 j +1 ; if x +2 < 0 x +2 )-222(j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 j +1 ; if x +2 0 = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )-222(j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 j ; if x< )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 x +3 )-222(j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 j ; if x )]TJ/F15 10.9091 Tf 20 0 Td [(2 2 Thesearesetbraces,notparenthesesorbrackets.Weusedthissame`setbuilder'notationinSection1.4.

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2.2AbsoluteValueFunctions123 Giventhat j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 j = )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ; if x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 < 0 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ; if x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 0 = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +3 ; if x< 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ; if x 3 ; weneedtobreakupthedomainagainat x =3.Notethatif x< )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,then x< 3,sowe replace j x )]TJ/F15 10.9091 Tf 11.046 0 Td [(3 j with )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +3forthatpartofthedomain,too.Ourcompletedrevisionofthe formof g yields g x = 8 > < > : )]TJ/F53 10.9091 Tf 8.485 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F53 10.9091 Tf 8.484 0 Td [(x +3 ; if x< )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +3 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F53 10.9091 Tf 8.484 0 Td [(x +3 ; if x )]TJ/F15 10.9091 Tf 20 0 Td [(2and x< 3 x +3 )]TJ/F15 10.9091 Tf 10.909 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ; if x 3 = 8 > < > : )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; if x< )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 x; if )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x< 3 6 ; if x 3 Tosolve g x =0,weseethattheonlypiecewhichcontainsavariableis g x =2 x for )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x< 3. Solving2 x =0gives x =0.Since x =0isintheinterval[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3,wekeepthissolutionandhave ; 0asouronly x -intercept.Accordingly,the y -interceptisalso ; 0.Tograph g ,westartwith x< )]TJ/F15 10.9091 Tf 8.485 0 Td [(2andgraphthehorizontalline y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4withanopencircleat )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4.For )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x< 3, wegraphtheline y =2 x andthepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4patchestheholeleftbythepreviouspiece.An opencircleat ; 6completesthegraphofthispart.Finally,wegraphthehorizontalline y =6 for x 3,andthepoint ; 6llsintheopencircleleftbythepreviouspartofthegraph.The nishedgraphis x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 g x = j x +2 j)-284(j x )]TJ/F35 5.9776 Tf 7.454 0 Td [(3 j +1 Thedomainof g isallrealnumbers, ; 1 ,andtherangeof g isallrealnumbersbetween )]TJ/F15 10.9091 Tf 8.485 0 Td [(4and6inclusive,[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 6].Thefunctionisincreasingon[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3]andconstanton ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2]and [3 ; 1 .Therelativeminimumvalueof f is4whichmatchestheabsoluteminimum.Therelative andabsolutemaximumvaluesalsocoincideat6.Everypointonthegraphof y = g x for x< )]TJ/F15 10.9091 Tf 8.484 0 Td [(2

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124LinearandQuadraticFunctions and x> 3yieldsbotharelativeminimumandrelativemaximum.Thepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4,however, givesonlyarelativeminimumandthepoint ; 6yieldsonlyarelativemaximum.Recallthe exercisesinSection1.7.2whichdealtwithconstantfunctions. Manyoftheapplicationsthattheauthorsareawareofinvolvingabsolutevaluesalsoinvolve absolutevalueinequalities.Forthatreason,wesaveourdiscussionofapplicationsforSection2.4.

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2.2AbsoluteValueFunctions125 2.2.1Exercises 1.Grapheachofthefollowingfunctionsusingtransformationsorthedenitionofabsolute value,asappropriate.Findthezerosofeachfunctionandthe x -and y -interceptsofeach graph,ifanyexist.Fromthegraph,determinethedomainandrangeofeachfunction,list theintervalsonwhichthefunctionisincreasing,decreasing,orconstant,andndtherelative andabsoluteextrema,iftheyexist. a f x = j x +4 j b f x = j x j +4 c f x = j 4 x j d f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 j x j e f x =3 j x +4 j)]TJ/F15 10.9091 Tf 16.363 0 Td [(4 f f x = j x +4 j x +4 g f x = j 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x j 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x h f x = 1 3 j 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 j i f x = j x +4 j + j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 j 2.Withthehelpofyourclassmates,provethesecond,thirdandfthpropertieslistedinTheorem2.1. 3.Withthehelpofyourclassmates,ndafunctioninvolvingabsolutevalueswhosegraphis givenbelow. x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(8 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112345678 1 2 3 4 4.Withthehelpofyourclassmates,provethefollowingtwopropertiesofabsolutevalue. aTheTriangleInequalityForallrealnumbers a and b; j a + b jj a j + j b j bIf j f x j = j g x j theneither f x = g x or f x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(g x 5.UsetheresultfromExercise4babovetosolvethefollowingequations.Interpretyourresults graphically. a j 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 j = j 2 x +7 j b j 3 x +1 j = j 4 x j c j 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x j = j x +1 j d j 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x j =5 j x +1 j

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126LinearandQuadraticFunctions 2.2.2Answers 1.a f x = j x +4 j f )]TJ/F15 10.9091 Tf 8.485 0 Td [(4=0 x -intercept )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 0 y -intercept ; 4 Domain ; 1 Range[0 ; 1 Decreasingon ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4] Increasingon[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 1 Relativeandabsolutemin.at )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 0 Norelativeorabsolutemaximum x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(8 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11 1 2 3 4 b f x = j x j +4 Nozeros No x -intercepts y -intercept ; 4 Domain ; 1 Range[4 ; 1 Decreasingon ; 0] Increasingon[0 ; 1 Relativeandabsoluteminimumat ; 4 Norelativeorabsolutemaximum x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11234 1 2 3 4 5 6 7 8 c f x = j 4 x j f =0 x -intercepts ; 0 y -intercept ; 0 Domain ; 1 Range[0 ; 1 Decreasingon ; 0] Increasingon[0 ; 1 Relativeandabsoluteminimumat ; 0 Norelativeorabsolutemaximum x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112 1 2 3 4 5 6 7 8

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2.2AbsoluteValueFunctions127 d f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 j x j f =0 x -intercepts ; 0 y -intercept ; 0 Domain ; 1 Range ; 0] Increasingon ; 0] Decreasingon[0 ; 1 Relativeandabsolutemaximumat ; 0 Norelativeorabsoluteminimum x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 e f x =3 j x +4 j)]TJ/F15 10.9091 Tf 16.363 0 Td [(4 f )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.681 4.295 Td [(16 3 =0, f )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.681 4.295 Td [(8 3 =0 x -intercepts )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F34 7.9701 Tf 9.68 4.296 Td [(16 3 ; 0 )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F34 7.9701 Tf 9.68 4.296 Td [(8 3 ; 0 y -intercept ; 8 Domain ; 1 Range[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 1 Decreasingon ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4] Increasingon[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 1 Relativeandabsolutemin.at )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 Norelativeorabsolutemaximum x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 1 2 3 4 5 6 7 8 f f x = j x +4 j x +4 Nozeros No x -intercept y -intercept ; 1 Domain ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 1 Range f)]TJ/F15 10.9091 Tf 13.939 0 Td [(1 ; 1 g Constanton ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4] Constanton[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 1 Absoluteminimumateverypoint x; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 where x< )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 Absolutemaximumateverypoint x; 1 where x> )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 RelativemaximumANDminimumateverypointonthegraph x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(8 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 g f x = j 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x j 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x Nozeros No x -intercept y -intercept ; 1 Domain ; 2 [ ; 1 Range f)]TJ/F15 10.9091 Tf 13.939 0 Td [(1 ; 1 g Constanton ; 2] Constanton[2 ; 1 Absoluteminimumateverypoint x; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 where x> 2 Absolutemaximumateverypoint x; 1 where x< 2 RelativemaximumANDminimumateverypointonthegraph x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(112345 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1

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128LinearandQuadraticFunctions h f x = 1 3 j 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 j f )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 =0 x -intercepts )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 ; 0 y -intercept )]TJ/F15 10.9091 Tf 5 -8.837 Td [(0 ; 1 3 Domain ; 1 Range[0 ; 1 Decreasingon )]TJ/F54 10.9091 Tf 5 -8.836 Td [( ; 1 2 Increasingon 1 2 ; 1 Relativeandabsolutemin.at )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 2 ; 0 Norelativeorabsolutemaximum x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 1 2 iRe-write f x = j x +4 j + j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 j as f x = 8 > < > : )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2if x< )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 6if )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 x< 2 2 x +2if x 2 Nozeros No x -intercept y -intercept ; 6 Domain ; 1 Range[6 ; 1 Decreasingon ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4] Constanton[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 2] Increasingon[2 ; 1 Absoluteminimumateverypoint x; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 where )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 x 2 Noabsolutemaximum Relativeminimumateverypoint x; 1 where )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 x 2 Relativemaximumateverypoint x; 1 where )]TJ/F15 10.9091 Tf 8.485 0 Td [(4
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2.3QuadraticFunctions129 2.3QuadraticFunctions YoumayrecallstudyingquadraticequationsinIntermediateAlgebra.Inthissection,wereview thoseequationsinthecontextofournextfamilyoffunctions:thequadraticfunctions. Definition 2.5 A quadraticfunction isafunctionoftheform f x = ax 2 + bx + c; where a b ,and c arerealnumberswith a 6 =0.Thedomainofaquadraticfunctionis ; 1 Example 2.3.1 Grapheachofthefollowingquadraticfunctions.Findthezerosofeachfunction andthe x -and y -interceptsofeachgraph,ifanyexist.Fromthegraph,determinethedomainand rangeofeachfunction,listtheintervalsonwhichthefunctionisincreasing,decreasing,orconstant andndtherelativeandabsoluteextrema,iftheyexist. 1. f x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3.2. g x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 +1. Solution. 1.Tondthezerosof f ,weset f x =0andsolvetheequation x 2 )]TJ/F15 10.9091 Tf 11.279 0 Td [(4 x +3=0.Factoring givesus x )]TJ/F15 10.9091 Tf 10.268 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.268 0 Td [(1=0sothat x =3or x =1.The x -interceptsarethen ; 0and ; 0. Tondthe y -intercept,weset x =0andndthat y = f =3.Hence,the y -interceptis ; 3.Plottingadditionalpoints,weget x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 8 f x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 Fromthegraph,weseethedomainis ; 1 andtherangeis[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 .Thefunction f isincreasingon[2 ; 1 anddecreasingon ; 2].Arelativeminimumoccursatthepoint ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1andthevalue )]TJ/F15 10.9091 Tf 8.485 0 Td [(1isboththerelativeandabsoluteminimumof f .

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130LinearandQuadraticFunctions 2.Notethattheformulafor g x doesn'tmatchtheformgiveninDenition2.5.However,ifwe tookthetimetoexpand g x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.66 0 Td [(3 2 +1,wewouldget g x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 2 +12 x )]TJ/F15 10.9091 Tf 10.66 0 Td [(17which doesmatchwithDenition2.5.Whenwendthezerosof g ,wecanuseeitherformula,since bothareequivalent.Usingtheformulawhichwasgiventous,weget g x =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 +1=0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 = 1 2 divideby )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3= r 1 2 extractsquareroots x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3= p 2 2 rationalizethedenominator x =3 p 2 2 x = 6 p 2 2 getacommondenominator Hence,wehavetwo x -intercepts: 6+ p 2 2 ; 0 and 6 )]TJ 6.587 6.598 Td [(p 2 2 ; 0 .Theinquisitivereadermay wonderwhatwewouldhavedonehadwechosentosettheexpandedformof g x equalto zero.Since )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 2 +12 x )]TJ/F15 10.9091 Tf 11.483 0 Td [(17doesnotfactornicely,wewouldhavehadtoresorttoother methods,whicharereviewedlaterinthissection,tosolve )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 2 +12 x )]TJ/F15 10.9091 Tf 11.378 0 Td [(17=0.Tond the y -intercept,weset x =0andget g = )]TJ/F15 10.9091 Tf 8.485 0 Td [(17.Our y -interceptisthen ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(17.Plotting someadditionalpoints,weget x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 )]TJ/F35 5.9776 Tf 5.757 0 Td [(17 )]TJ/F35 5.9776 Tf 5.757 0 Td [(16 )]TJ/F35 5.9776 Tf 5.757 0 Td [(15 )]TJ/F35 5.9776 Tf 5.757 0 Td [(14 )]TJ/F35 5.9776 Tf 5.757 0 Td [(13 )]TJ/F35 5.9776 Tf 5.757 0 Td [(12 )]TJ/F35 5.9776 Tf 5.757 0 Td [(11 )]TJ/F35 5.9776 Tf 5.757 0 Td [(10 )]TJ/F35 5.9776 Tf 5.757 0 Td [(9 )]TJ/F35 5.9776 Tf 5.757 0 Td [(8 )]TJ/F35 5.9776 Tf 5.757 0 Td [(7 )]TJ/F35 5.9776 Tf 5.757 0 Td [(6 )]TJ/F35 5.9776 Tf 5.757 0 Td [(5 )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 1 g x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 +1

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2.3QuadraticFunctions131 Thedomainof g is ; 1 andtherangeis ; 1].Thefunction g isincreasingon ; 3] anddecreasingon[3 ; 1 .Therelativemaximumoccursatthepoint ; 1with1beingboththe relativeandabsolutemaximumvalueof g Hopefullythepreviousexampleshaveremindedyouofsomeofthebasiccharacteristicsofthe graphsofquadraticequations.Firstandforemost,thegraphof y = ax 2 + bx + c where a b ,and c arerealnumberswith a 6 =0iscalleda parabola .Ifthecoecientof x 2 a ,ispositive,the parabolaopensupwards;if a isnegative,itopensdownwards,asillustratedbelow. 1 vertex a> 0 vertex a< 0 Graphsof y = ax 2 + bx + c Thepointatwhichtherelativeminimumif a> 0orrelativemaximumif a< 0occursiscalled the vertex oftheparabola.Notethateachoftheparabolasaboveissymmetricaboutthedashed verticallinewhichcontainsitsvertex.Thislineiscalledthe axisofsymmetry oftheparabola. Asyoumayrecall,therearetwowaystoquicklyndthevertexofaparabola,dependingonwhich formwearegiven.Theresultsaresummarizedbelow. Equation 2.4 VertexFormulasforQuadraticFunctions :Suppose a b c h ,and k are realnumberswith a 6 =0. If f x = a x )]TJ/F53 10.9091 Tf 10.909 0 Td [(h 2 + k ,thevertexofthegraphof y = f x isthepoint h;k If f x = ax 2 + bx + c ,thevertexofthegraphof y = f x isthepoint )]TJ/F53 10.9091 Tf 12.95 7.38 Td [(b 2 a ;f )]TJ/F53 10.9091 Tf 12.95 7.38 Td [(b 2 a Example 2.3.2 UseEquation2.4tondthevertexofthegraphsinExample2.3.1. Solution. 1.Theformula f x = x 2 )]TJ/F15 10.9091 Tf 9.837 0 Td [(4 x +3isintheform f x = ax 2 + bx + c .Weidentify a =1, b = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4, and c =3,sothat )]TJ/F53 10.9091 Tf 12.95 7.38 Td [(b 2 a = )]TJ 12.408 7.38 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 =2 ; and f )]TJ/F53 10.9091 Tf 12.95 7.381 Td [(b 2 a = f = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; sothevertexis ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1aspreviouslystated. 1 Wewilljustifytheroleof a inthebehavioroftheparabolalaterinthesection.

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132LinearandQuadraticFunctions 2.Weseethattheformula g x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x )]TJ/F15 10.9091 Tf 9.616 0 Td [(3 2 +1isintheform g x = a x )]TJ/F53 10.9091 Tf 9.616 0 Td [(h 2 + k .Weidentify a = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2, x )]TJ/F53 10.9091 Tf 10.909 0 Td [(h as x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3so h =3,and k =1andgetthevertex ; 1,asrequired. Theformula f x = a x )]TJ/F53 10.9091 Tf 10.926 0 Td [(h 2 + k a 6 =0inEquation2.4issometimescalledthe standardform ofaquadraticfunction;theformula f x = ax 2 + bx + c a 6 =0issometimescalledthe general form ofaquadraticfunction. ToseewhytheformulasinEquation2.4producethevertex,letusrstconsideraquadraticfunction instandardform.Ifweconsiderthegraphoftheequation y = a x )]TJ/F53 10.9091 Tf 11.327 0 Td [(h 2 + k weseethatwhen x = h ,weget y = k ,so h;k isonthegraph.If x 6 = h ,then x )]TJ/F53 10.9091 Tf 10.226 0 Td [(h 6 =0andso x )]TJ/F53 10.9091 Tf 10.226 0 Td [(h 2 isapositive number.If a> 0,then a x )]TJ/F53 10.9091 Tf 11.112 0 Td [(h 2 ispositive,andso y = a x )]TJ/F53 10.9091 Tf 11.111 0 Td [(h 2 + k isalwaysanumberlarger than k .Thatmeansthatwhen a> 0, h;k isthelowestpointonthegraphandthustheparabola mustopenupwards,making h;k thevertex.Asimilarargumentshowsthatif a< 0, h;k isthe highestpointonthegraph,sotheparabolaopensdownwards,and h;k isalsothevertexinthis case.Alternatively,wecanapplythemachineryinSection1.8.Thevertexoftheparabola y = x 2 iseasilyseentobetheorigin, ; 0.Weleaveittothereadertoconvinceoneselfthatifweapply anyofthetransformationsinSection1.8shifts,reections,and/orscalingsto y = x 2 ,thevertex oftheresultingparabolawillalwaysbethepointthegraphcorrespondingto ; 0.Toobtainthe formula f x = a x )]TJ/F53 10.9091 Tf 11.014 0 Td [(h 2 + k ,westartwith g x = x 2 andrstdene g 1 x = ag x = ax 2 .This isresultsinaverticalscalingand/orreection. 2 Sincewemultiplytheoutputby a ,wemultiply the y -coordinatesonthegraphof g by a ,sothepoint ; 0remains ; 0andremainsthevertex. Next,wedene g 2 x = g 1 x )]TJ/F53 10.9091 Tf 10.438 0 Td [(h = a x )]TJ/F53 10.9091 Tf 10.438 0 Td [(h 2 .Thisinducesahorizontalshiftrightorleft h units 3 movesthevertex,ineithercase,to h; 0.Finally, f x = g 2 x + k = a x )]TJ/F53 10.9091 Tf 10.993 0 Td [(h 2 + k whicheects averticalshiftupordown k units 4 resultinginthevertexmovingfrom h; 0to h;k Toverifythevertexformulaforaquadraticfunctioningeneralform,wecompletethesquareto convertthegeneralformintothestandardform. 5 f x = ax 2 + bx + c = a x 2 + b a x + c = a x 2 + b a x + b 2 4 a 2 + c )]TJ/F53 10.9091 Tf 10.909 0 Td [(a b 2 4 a 2 completethesquare = a x + b 2 a 2 + 4 ac )]TJ/F53 10.9091 Tf 10.909 0 Td [(b 2 4 a factor;getacommondenominator Comparingthislastexpressionwiththestandardform,weidentify x )]TJ/F53 10.9091 Tf 11.518 0 Td [(h as )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x + b 2 a sothat h = )]TJ/F53 10.9091 Tf 12.95 7.38 Td [(b 2 a .Insteadofmemorizingthevalue k = 4 ac )]TJ/F37 7.9701 Tf 6.587 0 Td [(b 2 4 a ,weseethat f )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F37 7.9701 Tf 12.235 4.296 Td [(b 2 a = 4 ac )]TJ/F37 7.9701 Tf 6.586 0 Td [(b 2 4 a .Assuch, 2 Justascalingif a> 0.If a< 0,thereisareectioninvolved. 3 Rightif h> 0,leftif h< 0. 4 Upif k> 0,downif k< 0 5 Actually,wecouldalsotakethestandardform, f x = a x )]TJ/F64 8.9664 Tf 9.068 0 Td [(h 2 + k ,expandit,andcomparethecoecientsof itandthegeneralformtodeducetheresult.However,wewillhaveanotheruseforthecompletedsquareformofthe generalformofaquadratic,sowe'llproceedwiththeconversion.

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2.3QuadraticFunctions133 wehavederivedthevertexformulaforthegeneralformaswell.Notethatthevalue a playsthe exactsameroleinboththestandardandgeneralequationsofaquadraticfunction )]TJ/F15 10.9091 Tf 13.176 0 Td [(itisthe coecientof x 2 ineach.Nomatterwhattheform,if a> 0,theparabolaopensupwards;if a< 0, theparabolaopensdownwards. Nowthatwehavethecompletedsquareformofthegeneralformofaquadraticfunction,itistime toremindourselvesofthe quadraticformula .Inafunctioncontext,itgivesusameanstond thezerosofaquadraticfunctioningeneralform. Equation 2.5 TheQuadraticFormula: If a b c arerealnumberswith a 6 =0,thenthe solutionsto ax 2 + bx + c =0are x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(b p b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 ac 2 a : AssumingtheconditionsofEquation2.5,thesolutionsto ax 2 + bx + c =0arepreciselythezeros of f x = ax 2 + bx + c .Wehaveshownanequivalentformulafor f is f x = a x + b 2 a 2 + 4 ac )]TJ/F53 10.9091 Tf 10.909 0 Td [(b 2 4 a : Hence,anequationequivalentto ax 2 + bx + c =0is a x + b 2 a 2 + 4 ac )]TJ/F53 10.9091 Tf 10.909 0 Td [(b 2 4 a =0 : Solvinggives a x + b 2 a 2 + 4 ac )]TJ/F53 10.9091 Tf 10.909 0 Td [(b 2 4 a =0 a x + b 2 a 2 = )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(4 ac )]TJ/F53 10.9091 Tf 10.909 0 Td [(b 2 4 a 1 a a x + b 2 a 2 # = 1 a b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 ac 4 a x + b 2 a 2 = b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 ac 4 a 2 x + b 2 a = r b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 ac 4 a 2 extractsquareroots x + b 2 a = p b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 ac 2 a x = )]TJ/F53 10.9091 Tf 12.95 7.38 Td [(b 2 a p b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 ac 2 a x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(b p b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 ac 2 a

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134LinearandQuadraticFunctions Inourdiscussionsofdomain,wewerewarnedagainsthavingnegativenumbersunderneaththe squareroot.Giventhat p b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 ac ispartoftheQuadraticFormula,wewillneedtopayspecial attentiontotheradicand b 2 )]TJ/F15 10.9091 Tf 11.087 0 Td [(4 ac .Itturnsoutthatthequantity b 2 )]TJ/F15 10.9091 Tf 11.087 0 Td [(4 ac playsacriticalrolein determiningthenatureofthesolutionstoaquadraticequation.Itisgivenaspecialnameandis discussedbelow. Definition 2.6 If a b c arerealnumberswith a 6 =0,thenthe discriminant ofthequadratic equation ax 2 + bx + c =0isthequantity b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 ac: Theorem 2.2 DiscriminantTrichotomy: Let a b ,and c berealnumberswith a 6 =0. If b 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(4 ac< 0,theequation ax 2 + bx + c =0hasnorealsolutions. If b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 ac =0,theequation ax 2 + bx + c =0hasexactlyonerealsolution. If b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 ac> 0,theequation ax 2 + bx + c =0hasexactlytworealsolutions. TheproofofTheorem2.2stemsfromthepositionofthediscriminantinthequadraticequation, andisleftasagoodmentalexerciseforthereader.Thenextexampleexploitsthefruitsofallof ourlaborinthissectionthusfar. Example 2.3.3 The prot functionforaproductisdenedbytheequationProt=Revenue )]TJ/F15 10.9091 Tf -459.515 -13.549 Td [(Cost,or P x = R x )]TJ/F53 10.9091 Tf 10.094 0 Td [(C x .RecallfromExample2.1.7thattheweeklyrevenue,indollars,made byselling x PortaBoyGameSystemsisgivenby R x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x 2 +250 x .Thecost,indollars,to produce x PortaBoyGameSystemsisgiveninExample2.1.5as C x =80 x +150, x 0. 1.Determinetheweeklyprotfunction, P x 2.Graph y = P x .Includethe x -and y -interceptsaswellasthevertexandaxisofsymmetry. 3.Interpretthezerosof P 4.Interpretthevertexofthegraphof y = P x 5.Recalltheweeklyprice-demandequationforPortaBoysis: p x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x +250,where p x isthepriceperPortaBoy,indollars,and x istheweeklysales.Whatshouldthepriceper systembeinordertomaximizeprot? Solution. 1.Tondtheprotfunction P x ,wesubtract P x = R x )]TJ/F53 10.9091 Tf 10.909 0 Td [(C x = )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F15 10.9091 Tf 8.484 0 Td [(1 : 5 x 2 +250 x )]TJ/F15 10.9091 Tf 10.909 0 Td [( x +150= )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x 2 +170 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(150 : 2.Tondthe x -intercepts,weset P x =0andsolve )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x 2 +170 x )]TJ/F15 10.9091 Tf 11.522 0 Td [(150=0.Themere thoughtoftryingtofactorthelefthandsideofthisequationcoulddoseriouspsychological damage,soweresorttothequadraticformula,Equation2.5.Identifying a = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5, b =170, and c = )]TJ/F15 10.9091 Tf 8.485 0 Td [(150,weobtain

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2.3QuadraticFunctions135 x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(b p b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 ac 2 a = )]TJ/F15 10.9091 Tf 8.485 0 Td [(170 p 170 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(150 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(170 p 28000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 = 170 20 p 70 3 Wegettwo x -intercepts: 170 )]TJ/F34 7.9701 Tf 6.586 0 Td [(20 p 70 3 ; 0 and 170+20 p 70 3 ; 0 .Tondthe y -intercept,weset x =0andnd y = P = )]TJ/F15 10.9091 Tf 8.485 0 Td [(150fora y -interceptof ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(150.Tondthevertex,weuse thefactthat P x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x 2 +170 x )]TJ/F15 10.9091 Tf 10.893 0 Td [(150isinthegeneralformofaquadraticfunctionand appealtoEquation2.4.Substituting a = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5and b =170,weget x = )]TJ/F34 7.9701 Tf 17.443 4.296 Td [(170 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 : 5 = 170 3 Tondthe y -coordinateofthevertex,wecompute P )]TJ/F34 7.9701 Tf 6.196 -4.542 Td [(170 3 = 14000 3 andndourvertexis )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(170 3 ; 14000 3 .Theaxisofsymmetryistheverticallinepassingthroughthevertexsoitisthe line x = 170 3 .Tosketchareasonablegraph,weapproximatethe x -intercepts, : 89 ; 0and : 44 ; 0,andthevertex, : 67 ; 4666 : 67.Notethatinordertogetthe x -interceptsand thevertextoshowupinthesamepicture,wehadtoscalethe x -axisdierentlythanthe y -axis.Thisresultsintheleft-hand x -interceptandthe y -interceptbeinguncomfortablyclose toeachotherandtotheorigininthepicture. x y 102030405060708090100110120 1000 2000 3000 4000 3.Thezerosof P arethesolutionsto P x =0,whichwehavefoundtobeapproximately 0 : 89and112 : 44.Since P representstheweeklyprot, P x =0meanstheweeklyprot is$0.Sometimes,thesevaluesof x arecalledthe`break-even'pointsoftheprotfunction, sincetheseareplaceswheretherevenueequalsthecost;inotherwordswegavesoldenough producttorecoverthecostspenttomaketheproduct.Moreimportantly,weseefromthe graphthataslongas x isbetween0 : 89and112 : 44,thegraph y = P x isabovethe x -axis, meaning y = P x > 0there.Thismeansthatforthesevaluesof x ,aprotisbeingmade. Since x representstheweeklysalesofPortaBoyGameSystems,weroundthezerostopositive integersandhavethataslongas1,butnomorethan112gamesystemsaresoldweekly,the retailerwillmakeaprot.

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136LinearandQuadraticFunctions 4.Fromthegraph,weseethemaximumvalueof P occursatthevertex,whichisapproximately : 67 ; 4666 : 67.Asabove, x representstheweeklysalesofPortaBoysystems,sowecan't sell56 : 67gamesystems.Comparing P =4666and P =4666 : 5,weconcludewewill makeamaximumprotof$4666 : 50ifwesell57gamesystems. 5.Inthepreviouspart,wefoundweneedtosell57PortaBoysperweektomaximizeprot. TondthepriceperPortaBoy,wesubstitute x =57intotheprice-demandfunctiontoget p = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5+250=164 : 5.Thepriceshouldbesetat$164 : 50. Weconcludethissectionwithamorecomplicatedabsolutevaluefunction. Example 2.3.4 Graph f x = j x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 j Solution. Usingthedenitionofabsolutevalue,Denition2.4,wehave f x = )]TJ/F55 10.9091 Tf 10.303 8.836 Td [()]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.91 0 Td [(6 ; if x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 < 0 x 2 )]TJ/F53 10.9091 Tf 10.91 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 ; if x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 0 Thetroubleisthatwehaveyettodevelopanyanalytictechniquestosolvenonlinearinequalities suchas x 2 )]TJ/F53 10.9091 Tf 10.944 0 Td [(x )]TJ/F15 10.9091 Tf 10.945 0 Td [(6 < 0.Youwon'thavetowaitlong;thisisoneofthemaintopicsofSection2.4. Nevertheless,wecanattackthisproblemgraphically.Tothatend,wegraph y = g x = x 2 )]TJ/F53 10.9091 Tf 10.354 0 Td [(x )]TJ/F15 10.9091 Tf 10.353 0 Td [(6 usingtheinterceptsandthevertex.Tondthe x -intercepts,wesolve x 2 )]TJ/F53 10.9091 Tf 11.156 0 Td [(x )]TJ/F15 10.9091 Tf 11.156 0 Td [(6=0.Factoring gives x )]TJ/F15 10.9091 Tf 11.272 0 Td [(3 x +2=0so x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2or x =3.Hence, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0and ; 0are x -intercepts.The y interceptisfoundbysetting x =0, ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(6.Tondthevertex,wend x = )]TJ/F37 7.9701 Tf 12.235 4.295 Td [(b 2 a = )]TJ/F40 7.9701 Tf 11.797 4.295 Td [()]TJ/F34 7.9701 Tf 6.587 0 Td [(1 2 = 1 2 ,and y = )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 2 )]TJ/F55 10.9091 Tf 9.973 8.837 Td [()]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 )]TJ/F15 10.9091 Tf 9.974 0 Td [(6= )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(25 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 : 25.Plotting,wegettheparabolaseenbelowontheleft.Toobtain pointsonthegraphof y = f x = j x 2 )]TJ/F53 10.9091 Tf 9.957 0 Td [(x )]TJ/F15 10.9091 Tf 9.958 0 Td [(6 j ,wecantakepointsonthegraphof g x = x 2 )]TJ/F53 10.9091 Tf 9.957 0 Td [(x )]TJ/F15 10.9091 Tf 9.957 0 Td [(6 andapplytheabsolutevaluetoeachofthe y valuesontheparabola.Weseefromthegraphof g thatfor x )]TJ/F15 10.9091 Tf 20 0 Td [(2or x 3,the y valuesontheparabolaaregreaterthanorequaltozerosincethe graphisonorabovethe x -axis,sotheabsolutevalueleavesthisportionofthegraphalone.For x between )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and3,however,the y valuesontheparabolaarenegative.Forexample,thepoint ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(6on y = x 2 )]TJ/F53 10.9091 Tf 10.664 0 Td [(x )]TJ/F15 10.9091 Tf 10.664 0 Td [(6wouldresultinthepoint ; j)]TJ/F15 10.9091 Tf 15.874 0 Td [(6 j = ; )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(6= ; 6onthegraphof f x = j x 2 )]TJ/F53 10.9091 Tf 11.066 0 Td [(x )]TJ/F15 10.9091 Tf 11.066 0 Td [(6 j .Proceedinginthismannerforallpointswith x -coordinatesbetween )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and 3resultsinthegraphseenaboveontheright.

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2.3QuadraticFunctions137 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 y = g x = x 2 )]TJ/F37 7.9701 Tf 8.469 0 Td [(x )]TJ/F34 7.9701 Tf 8.469 0 Td [(6 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 y = f x = j x 2 )]TJ/F37 7.9701 Tf 8.469 0 Td [(x )]TJ/F34 7.9701 Tf 8.469 0 Td [(6 j Ifwetakeastepbackandlookatthegraphsof g and f inthelastexample,wenoticethatto obtainthegraphof f fromthegraphof g ,wereecta portion ofthegraphof g aboutthe x -axis. Wecanseethisanalyticallybysubstituting g x = x 2 )]TJ/F53 10.9091 Tf 10.29 0 Td [(x )]TJ/F15 10.9091 Tf 10.29 0 Td [(6intotheformulafor f x andcalling tomindTheorem1.4fromSection1.8. f x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(g x ; if g x < 0 g x ; if g x 0 Thefunction f isdenedsothatwhen g x isnegativei.e.,whenitsgraphisbelowthe x -axis, thegraphof f isitsrefectionacrossthe x -axis.Thisisageneraltemplatetographfunctions oftheform f x = j g x j .Fromthisperspective,thegraphof f x = j x j canbeobtainedby reectiontheportionoftheline g x = x whichisbelowthe x -axisbackabovethe x -axiscreating thecharacteristic` 'shape.

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138LinearandQuadraticFunctions 2.3.1Exercises 1.Grapheachofthefollowingquadraticfunctions.Findthe x -and y -interceptsofeachgraph, ifanyexist.Ifitisgiveninthegeneralform,convertitintostandardform.Findthe domainandrangeofeachfunctionandlisttheintervalsonwhichthefunctionisincreasing ordecreasing.Identifythevertexandtheaxisofsymmetryanddeterminewhetherthevertex yieldsarelativeandabsolutemaximumorminimum. a f x = x 2 +2 b f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [( x +2 2 c f x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 d f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1 2 +4 e f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 2 +4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 f f x = x 2 )]TJ/F15 10.9091 Tf 17.559 7.38 Td [(1 100 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 6 2.Graph f x = j 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 j 3.Findallofthepointsontheline y =1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x whichare2unitsfrom ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4.Withthehelpofyourclassmates,showthatifaquadraticfunction f x = ax 2 + bx + c has tworealzerosthenthe x -coordinateofthevertexisthemidpointofthezeros. 5.AssumingnoairresistanceorforcesotherthantheEarth'sgravity,theheightabovethe groundattime t ofafallingobjectisgivenby s t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 : 9 t 2 + v 0 t + s 0 where h isinmeters, t isinseconds, v 0 istheobject'sinitialvelocityinmeterspersecondand s 0 isitsinitialposition inmeters. aWhatistheapplieddomainofthisfunction? bDiscusswithyourclassmateswhateachof v 0 > 0 ;v 0 =0and v 0 < 0wouldmean. cComeupwithascenarioinwhich s 0 < 0. dLet'ssayaslingshotisusedtoshootamarblestraightupfromtheground s 0 =0with aninitialvelocityof15meterspersecond.Whatisthemarble'smaximumheightabove theground?Atwhattimewillithittheground? eNowshootthemarblefromthetopofatowerwhichis25meterstall.Whendoesithit theground? fWhatwouldtheheightfunctionbeifinsteadofshootingthemarbleupoofthetower, youweretoshootitstraightDOWNfromthetopofthetower? 6.TheInternationalSilverStringsSubmarineBandholdsabakesaleeachyeartofundtheir triptotheNationalSasquatchConvention.Ithasbeendeterminedthatthecostindollars ofbaking x cookiesis C x =0 : 1 x +25andthatthedemandfunctionfortheircookiesis p =10 )]TJ/F53 10.9091 Tf 10.909 0 Td [(: 01 x: Howmanycookiesshouldtheybakeinordertomaximizetheirprot? 6 Wehavealreadyseenthegraphofthisfunction.ItwasusedasanexampleinSection1.7toshowhowthe graphingcalculatorcanbemisleading.

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2.3QuadraticFunctions139 7.Thetwotowersofasuspensionbridgeare400feetapart.Theparaboliccable 7 attachedto thetopsofthetowersis10feetabovethepointonthebridgedeckthatismidwaybetween thetowers.Ifthetowersare100feettall,ndtheheightofthecabledirectlyaboveapoint ofthebridgedeckthatis50feettotherightoftheleft-handtower. 8.Whatisthelargestrectangularareaonecanenclosewith14inchesofstring? 9.Solvethefollowingquadraticequationsfortheindicatedvariable. a x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 y 2 =0for x b y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 y = x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(4for x c x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(mx =1for x d )]TJ/F53 10.9091 Tf 8.485 0 Td [(gt 2 + v 0 t + s 0 =0for t Assume g 6 =0. e y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 y =4 x for y f y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 y = x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(4for y 7 Theweightofthebridgedeckforcesthebridgecableintoaparabolaandafreehangingcablesuchasapower linedoesnotformaparabola.WeshallseeinExercise12inSection6.5whatshapeafreehangingcablemakes.

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140LinearandQuadraticFunctions 2.3.2Answers 1.a f x = x 2 +2 No x -intercepts y -intercept ; 2 Domain: ; 1 Range:[2 ; 1 Decreasingon ; 0] Increasingon[0 ; 1 Vertex ; 2isaminimum Axisofsymmetry x =0 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 1 2 3 4 5 6 7 8 9 10 b f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [( x +2 2 x -intercept )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0 y -intercept ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 Domain: ; 1 Range: ; 0] Increasingon ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2] Decreasingon[ )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 ; 1 Vertex )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0isamaximum Axisofsymmetry x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.757 0 Td [(8 )]TJ/F35 5.9776 Tf 5.757 0 Td [(7 )]TJ/F35 5.9776 Tf 5.757 0 Td [(6 )]TJ/F35 5.9776 Tf 5.757 0 Td [(5 )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 c f x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8= x )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x -intercepts )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0and ; 0 y -intercept ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 Domain: ; 1 Range:[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 ; 1 Decreasingon ; 1] Increasingon[1 ; 1 Vertex ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(9isaminimum Axisofsymmetry x =1 x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.757 0 Td [(9 )]TJ/F35 5.9776 Tf 5.757 0 Td [(8 )]TJ/F35 5.9776 Tf 5.757 0 Td [(7 )]TJ/F35 5.9776 Tf 5.757 0 Td [(6 )]TJ/F35 5.9776 Tf 5.757 0 Td [(5 )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 1 2 d f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1 2 +4 x -intercepts )]TJ/F15 9.9626 Tf 7.749 0 Td [(1 )]TJ 9.963 8.241 Td [(p 2 ; 0and )]TJ/F15 9.9626 Tf 7.749 0 Td [(1+ p 2 ; 0 y -intercept ; 2 Domain: ; 1 Range: ; 4] Increasingon ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1] Decreasingon[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 Vertex )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 4isamaximum Axisofsymmetry x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4

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2.3QuadraticFunctions141 e f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 2 +4 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(7= )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(2 3 2 )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(17 3 No x -intercepts y -intercept ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 Domain: ; 1 Range: ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(17 3 ] Increasingon ; 2 3 ] Decreasingon[ 2 3 ; 1 Vertex 2 3 ; )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(17 3 isamaximum Axisofsymmetry x = 2 3 x y 12 )]TJ/F35 5.9776 Tf 5.756 0 Td [(14 )]TJ/F35 5.9776 Tf 5.756 0 Td [(13 )]TJ/F35 5.9776 Tf 5.756 0 Td [(12 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.756 0 Td [(10 )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 f f x = x 2 )]TJ/F34 7.9701 Tf 16.339 4.295 Td [(1 100 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1= )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x )]TJ/F34 7.9701 Tf 16.339 4.295 Td [(1 200 2 )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(40001 40000 x -intercepts 1+ p 40001 200 and 1 )]TJ 6.587 6.598 Td [(p 40001 200 y -intercept ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 Domain: ; 1 Range: )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(40001 40000 ; 1 Decreasingon )]TJ/F54 10.9091 Tf 5 -8.837 Td [( ; 1 200 Increasingon 1 200 ; 1 Vertex )]TJ/F34 7.9701 Tf 10.429 -4.541 Td [(1 200 ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(40001 40000 isaminimum 8 Axisofsymmetry x = 1 200 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 1 2 3 4 5 6 7 8 2. y = j 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 j x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 1 2 3 4 5 6 7 3. 3 )]TJ 10.91 9.024 Td [(p 7 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+ p 7 2 3+ p 7 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ 10.909 9.024 Td [(p 7 2 5.aTheapplieddomainis[0 ; 1 dTheheightfunctionisthiscaseis s t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 : 9 t 2 +15 t .Thevertexofthisparabola isapproximately : 53 ; 11 : 48sothemaximumheightreachedbythemarbleis11 : 48 meters.Ithitsthegroundagainwhen t 3 : 06seconds. 8 You'llneedtouseyourcalculatortozoominfarenoughtoseethatthevertexisnotthe y -intercept.

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142LinearandQuadraticFunctions eTherevisedheightfunctionis s t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 : 9 t 2 +15 t +25whichhaszerosat t )]TJ/F15 10.9091 Tf 20 0 Td [(1 : 20and t 4 : 26.Weignorethenegativevalueandclaimthatthemarblewillhittheground after4 : 26seconds. fShootingdownmeanstheinitialvelocityisnegativesotheheightfunctionsbecomes s t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 : 9 t 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(15 t +25. 6.500cookies 7.Makethevertexoftheparabola ; 10sothatthepointonthetopoftheleft-handtower wherethecableconnectsis )]TJ/F15 10.9091 Tf 8.485 0 Td [(200 ; 100andthepointonthetopoftheright-handtoweris ; 100.Thentheparabolaisgivenby p x = 9 4000 x 2 +10.Standing50feettotherightof theleft-handtowermeansyou'restandingat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(150and p )]TJ/F15 10.9091 Tf 8.485 0 Td [(150=60 : 625.Sothecable is60.625feetabovethebridgedeckthere. 8.Thelargestrectanglehasarea12 : 25in 2 9.a x = y p 10 b x = y )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 c x = m p m 2 +4 2 d t = v 0 p v 2 0 +4 gs 0 2 g e y = 3 p 16 x +9 2 f y =2 x

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2.4Inequalities 143 2.4Inequalities Inthissection,notonlydowedeveloptechniquesforsolvingvariousclassesofinequalitiesanalytically,wealsolookatthemgraphically.Thenextexamplemotivatesthecoreideas. Example 2.4.1 Let f x =2 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(1and g x =5. 1.Solve f x = g x 2.Solve f x g x 4.Graph y = f x and y = g x onthesamesetofaxesandinterpretyoursolutionstoparts1 through3above. Solution. 1.Tosolve f x = g x ,wereplace f x with2 x )]TJ/F15 10.9091 Tf 10.409 0 Td [(1and g x with5toget2 x )]TJ/F15 10.9091 Tf 10.408 0 Td [(1=5.Solving for x ,weget x =3. 2.Theinequality f x g x ,wesolve2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 > 5.Weget x> 3,or ; 1 4.Tograph y = f x ,wegraph y =2 x )]TJ/F15 10.9091 Tf 11.036 0 Td [(1,whichisalinewitha y -interceptof ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1anda slopeof2.Thegraphof y = g x is y =5whichisahorizontallinethrough ; 5. x y y = f x y = g x 1234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 8 Toseetheconnectionbetweenthegraphandthealgebra,werecalltheFundamentalGraphing PrincipleforFunctionsinSection1.7:thepoint a;b isonthegraphof f ifandonlyif f a = b .Inotherwords,agenericpointonthegraphof y = f x is x;f x ,andageneric

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144LinearandQuadraticFunctions pointonthegraphof y = g x is x;g x .Whenweseeksolutionsto f x = g x ,weare lookingforvalues x whose y valuesonthegraphsof f and g arethesame.Inpart1,wefound x =3isthesolutionto f x = g x .Sureenough, f =5and g =5sothatthepoint ; 5isonbothgraphs.Wesaythegraphsof f and g intersect at ; 5.Inpart2,weset f x g x andfound x> 3.For x> 3,notethatthegraphof f is above thegraphof g ,sincethe y valuesonthegraphof f aregreaterthanthe y valueson thegraphof g forthosevaluesof x y = f x y = g x x y 1234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 8 f x g x Theprecedingexampledemonstratesthefollowing,whichisaconsequenceoftheFundamental GraphingPrincipleforFunctions. GraphicalInterpretationofEquationsandInequalities Suppose f and g arefunctions. Thesolutionsto f x = g x arepreciselythe x valueswherethegraphsof y = f x and y = g x intersect. Thesolutionsto f x g x arepreciselythe x valueswherethegraphof y = f x above thegraphof y = g x Thenextexampleturnsthetablesandfurnishesthegraphsoftwofunctionsandasksforsolutions toequationsandinequalities.

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2.4Inequalities 145 Example 2.4.2 Thegraphsof f and g arebelow.Thegraphof y = f x resemblestheupside down shapeofanabsolutevaluefunctionwhilethegraphof y = g x resemblesaparabola.Use thesegraphstoanswerthefollowingquestions. x y y = f x y = g x ; 2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 1.Solve f x = g x .2.Solve f x g x .Wesolved theformerequationandfound x = 1.Tosolve f x >g x ,welookforwherethegraph of f isabovethegraphof g .Thisappearstohappenbetween x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and x =1,onthe interval )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1.Hence,oursolutionto f x g x is[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1]. x y y = f x y = g x ; 2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(112 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 f x
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146LinearandQuadraticFunctions Wenowturnourattentiontosolvinginequalitiesinvolvingtheabsolutevalue.Wehavethe followingtheoremfromIntermediateAlgebratohelpus. Theorem 2.3 InequalitiesInvolvingtheAbsoluteValue: Let c bearealnumber. For c> 0, j x j c isequivalentto x< )]TJ/F53 10.9091 Tf 8.485 0 Td [(c or x>c For c< 0, j x j >c istrueforallrealnumbers. AswithTheorem2.1inSection2.2,wecouldargueTheorem2.3usingcases.However,inlight ofwhatwehavedevelopedinthissection,wecanunderstandthesestatementsgraphically.For instance,if c> 0,thegraphof y = c isahorizontallinewhichliesabovethe x -axisthrough ;c Tosolve j x j )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3.2 < j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 j 5 4. j x +1 j x +4 2 Solution. 1.Tosolve j x )]TJ/F15 10.9091 Tf 11.525 0 Td [(1 j 3,weseeksolutionsto j x )]TJ/F15 10.9091 Tf 11.525 0 Td [(1 j > 3aswellassolutionsto j x )]TJ/F15 10.9091 Tf 11.525 0 Td [(1 j =3. FromTheorem2.3, j x )]TJ/F15 10.9091 Tf 10.857 0 Td [(1 j > 3isequivalentto x )]TJ/F15 10.9091 Tf 10.856 0 Td [(1 < )]TJ/F15 10.9091 Tf 8.485 0 Td [(3or x )]TJ/F15 10.9091 Tf 10.856 0 Td [(1 > 3.FromTheorem2.1, j x )]TJ/F15 10.9091 Tf 11.26 0 Td [(1 j =3isequivalentto x )]TJ/F15 10.9091 Tf 11.261 0 Td [(1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(3or x )]TJ/F15 10.9091 Tf 11.26 0 Td [(1=3.Combiningtheseequationswiththe inequalities,wesolve x )]TJ/F15 10.9091 Tf 11.258 0 Td [(1 )]TJ/F15 10.9091 Tf 20.873 0 Td [(3or x )]TJ/F15 10.9091 Tf 11.259 0 Td [(1 3.Ouransweris x )]TJ/F15 10.9091 Tf 20.873 0 Td [(2or x 4,which,in intervalnotationis ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2] [ [4 ; 1 .Graphically,wehave

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2.4Inequalities 147 x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 2 3 4 Weseethegraphof y = j x )]TJ/F15 10.9091 Tf 11.262 0 Td [(1 j the isabovethehorizontalline y =3for x< )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x> 4,and,hence,thisiswhere j x )]TJ/F15 10.9091 Tf 11.42 0 Td [(1 j > 3.Thetwographsintersectwhen x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x =4,andsowehavegraphicalconrmationofouranalyticsolution. 2.Tosolve4 )]TJ/F15 10.9091 Tf 11.212 0 Td [(3 j 2 x +1 j > )]TJ/F15 10.9091 Tf 8.485 0 Td [(2analytically,werstisolatetheabsolutevaluebeforeapplying Theorem2.3.Tothatend,weget )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 j 2 x +1 j > )]TJ/F15 10.9091 Tf 8.484 0 Td [(6or j 2 x +1 j < 2.Rewriting,wenowhave )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 < 2 x +1 < 2sothat )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 2andso x )]TJ/F15 10.9091 Tf 11.471 0 Td [(1 < )]TJ/F15 10.9091 Tf 8.485 0 Td [(2or x )]TJ/F15 10.9091 Tf 11.471 0 Td [(1 > 2.Weget x< )]TJ/F15 10.9091 Tf 8.485 0 Td [(1or x> 3.Oursolution totherstinequalityisthen ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ ; 1 .For j x )]TJ/F15 10.9091 Tf 11.679 0 Td [(1 j 5,wecombineresultsin Theorems2.1and2.3toget )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x )]TJ/F15 10.9091 Tf 10.95 0 Td [(1 5sothat )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 x 6,or[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 6].Oursolutionto 2 < j x )]TJ/F15 10.9091 Tf 9.521 0 Td [(1 j 5iscomprisedofvaluesof x whichsatisfybothpartsoftheinequality,andsowe takewhat'scalledthe`settheoreticintersection'of ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ ; 1 with[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 6]toobtain [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ ; 6].Graphically,weseethegraphof y = j x )]TJ/F15 10.9091 Tf 10.393 0 Td [(1 j is`between'thehorizontallines y =2and y =5for x valuesbetween )]TJ/F15 10.9091 Tf 8.485 0 Td [(4and )]TJ/F15 10.9091 Tf 8.485 0 Td [(1aswellasthosebetween3and6.Including the x valueswhere y = j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 j and y =5intersect,weget

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148LinearandQuadraticFunctions x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123456789 2 3 4 5 6 7 8 4.Weneedtoexercisesomespecialcautionwhensolving j x +1 j x +4 2 .Whenvariablesareboth insideandoutsideoftheabsolutevalue,it'susuallybesttorefertothedenitionofabsolute value,Denition2.4,toremovetheabsolutevaluesandproceedfromthere.Tothatend,we have j x +1 j = )]TJ/F15 10.9091 Tf 8.485 0 Td [( x +1if x< )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and j x +1 j = x +1if x )]TJ/F15 10.9091 Tf 20.699 0 Td [(1.Webreaktheinequality intocases,therstcasebeingwhen x< )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Forthesevaluesof x ,ourinequalitybecomes )]TJ/F15 10.9091 Tf 8.485 0 Td [( x +1 x +4 2 .Solving,weget )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x )]TJ/F15 10.9091 Tf 11.157 0 Td [(2 x +4,sothat )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 6,whichmeans x )]TJ/F15 10.9091 Tf 20.618 0 Td [(2. Sinceallofthesesolutionsfallintothecategory x< )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,wekeepthemall.Forthesecond case,weassume x )]TJ/F15 10.9091 Tf 20 0 Td [(1.Ourinequalitybecomes x +1 x +4 2 ,whichgives2 x +2 x +4or x 2.Sinceallofthesevaluesof x aregreaterthanorequalto )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,weacceptallofthese solutionsaswell.Ournalansweris ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2] [ [2 ; 1 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 2 3 4 Wenowturnourattentiontoquadraticinequalities.InthelastexampleofSection2.3,weneeded todeterminethesolutionto x 2 )]TJ/F53 10.9091 Tf 11.114 0 Td [(x )]TJ/F15 10.9091 Tf 11.114 0 Td [(6 < 0.Wewillnowre-visitthisproblemusingsomeofthe techniquesdevelopedinthissectionnotonlytoreinforceoursolutioninSection2.3,buttoalso helpformulateageneralanalyticprocedureforsolvingallquadraticinequalities.Ifweconsider f x = x 2 )]TJ/F53 10.9091 Tf 11.302 0 Td [(x )]TJ/F15 10.9091 Tf 11.302 0 Td [(6and g x =0,thensolving x 2 )]TJ/F53 10.9091 Tf 11.302 0 Td [(x )]TJ/F15 10.9091 Tf 11.303 0 Td [(6 < 0correspondsgraphicallytonding thevaluesof x forwhichthegraphof y = f x = x 2 )]TJ/F53 10.9091 Tf 10.975 0 Td [(x )]TJ/F15 10.9091 Tf 10.974 0 Td [(6theparabolaisbelowthegraphof y = g x =0the x -axis.We'veprovidedthegraphagainforreference.

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2.4Inequalities 149 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 y = x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.91 0 Td [(6 Wecanseethatthegraphof f doesdipbelowthe x -axisbetweenitstwo x -intercepts.Thezeros of f are x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x =3inthiscaseandtheydividethedomainthe x -axisintothreeintervals: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3,and ; 1 .Foreverynumberin ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,thegraphof f isabovethe x -axis; inotherwords, f x > 0forall x in ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Similarly, f x < 0forall x in )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3,and f x > 0 forall x in ; 1 .Wecanschematicallyrepresentthiswiththe signdiagram below. )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 + 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + Here,the+aboveaportionofthenumberlineindicates f x > 0forthosevaluesof x ;the )]TJ/F15 10.9091 Tf 8.485 0 Td [( indicates f x < 0there.Thenumberslabeledonthenumberlinearethezerosof f ,soweplace 0abovethem.Weseeatoncethatthesolutionto f x < 0is )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3. Ournextgoalistoestablishaprocedurebywhichwecangeneratethesigndiagramwithout graphingthefunction.Animportantproperty 1 ofquadraticfunctionsisthatifthefunctionis positiveatonepointandnegativeatanother,thefunctionmusthaveatleastonezeroinbetween. Graphically,thismeansthataparabolacan'tbeabovethe x -axisatonepointandbelowthe x -axis atanotherpointwithoutcrossingthe x -axis.Thisallowsustodeterminethesignof all ofthe functionvaluesonagivenintervalbytestingthefunctionatjust one valueintheinterval.This givesusthefollowing. 1 WewillgivethispropertyanameinChapter3andrevisitthisconceptthen.

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150LinearandQuadraticFunctions StepsforSolvingaQuadraticInequality 1.Rewritetheinequality,ifnecessary,asaquadraticfunction f x ononesideoftheinequality and0ontheother. 2.Findthezerosof f andplacethemonthenumberlinewiththenumber0abovethem. 3.Choosearealnumber,calleda testvalue ,ineachoftheintervalsdeterminedinstep2. 4.Determinethesignof f x foreachtestvalueinstep3,andwritethatsignabovethe correspondinginterval. 5.Choosetheintervalswhichcorrespondtothecorrectsigntosolvetheinequality. Example 2.4.4 Solvethefollowinginequalitiesanalyticallyusingsigndiagrams.Verifyyour answergraphically. 1.2 x 2 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2. x 2 > 2 x +1 3.9 x 2 +4 12 x 4.2 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 j)]TJ/F15 10.9091 Tf 16.364 0 Td [(1 Solution. 1.Tosolve2 x 2 3 )]TJ/F53 10.9091 Tf 10.23 0 Td [(x ,werstget0ononesideoftheinequalitywhichyields2 x 2 + x )]TJ/F15 10.9091 Tf 10.23 0 Td [(3 0. Wendthezerosof f x =2 x 2 + x )]TJ/F15 10.9091 Tf 11.28 0 Td [(3bysolving2 x 2 + x )]TJ/F15 10.9091 Tf 11.28 0 Td [(3=0for x .Factoringgives x +3 x )]TJ/F15 10.9091 Tf 11.151 0 Td [(1=0,so x = )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(3 2 ,or x =1.Weplacethesevaluesonthenumberlinewith0 abovethemandchoosetestvaluesintheintervals )]TJ/F54 10.9091 Tf 5 -8.837 Td [( ; )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(3 2 )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 ; 1 ,and ; 1 .Forthe interval )]TJ/F54 10.9091 Tf 5 -8.836 Td [( ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 ,wechoose 2 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2;for )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.681 4.295 Td [(3 2 ; 1 ,wepick x =0;andfor ; 1 x =2. Evaluatingthefunctionatthethreetestvaluesgivesus f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2=3 > 0soweplace+ above )]TJ/F54 10.9091 Tf 5 -8.836 Td [( ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 ; f = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 < 0so )]TJ/F15 10.9091 Tf 8.485 0 Td [(goesabovetheinterval )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 ; 1 ;and, f =7 whichmeans+isplacedabove ; 1 .Sincewearesolving2 x 2 + x )]TJ/F15 10.9091 Tf 10.904 0 Td [(3 0,welookfor solutionsto2 x 2 + x )]TJ/F15 10.9091 Tf 10.682 0 Td [(3 < 0aswellassolutionsfor2 x 2 + x )]TJ/F15 10.9091 Tf 10.682 0 Td [(3=0.For2 x 2 + x )]TJ/F15 10.9091 Tf 10.682 0 Td [(3 < 0,we needtheintervalswhichwehavea )]TJ/F15 10.9091 Tf 8.485 0 Td [(.Checkingthesigndiagram,weseethisis )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 ; 1 Weknow2 x 2 + x )]TJ/F15 10.9091 Tf 10.111 0 Td [(3=0when x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 and x =1,soornalansweris )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(3 2 ; 1 .Tocheckour solutiongraphically,werefertotheoriginalinequality,2 x 2 3 )]TJ/F53 10.9091 Tf 11.051 0 Td [(x .Welet g x =2 x 2 and h x =3 )]TJ/F53 10.9091 Tf 11.154 0 Td [(x .Wearelookingforthe x valueswherethegraphof g isbelowthatof h the solutionto g x
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2.4Inequalities 151 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(3 2 1 + 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 02 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 2 3 4 5 6 7 2.Onceagain,were-write x 2 > 2 x +1as x 2 )]TJ/F15 10.9091 Tf 10.958 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.959 0 Td [(1 > 0andweidentify f x = x 2 )]TJ/F15 10.9091 Tf 10.959 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.959 0 Td [(1. Whenwegotondthezerosof f ,wend,toourchagrin,thatthequadratic x 2 )]TJ/F15 10.9091 Tf 11.241 0 Td [(2 x )]TJ/F15 10.9091 Tf 11.24 0 Td [(1 doesn'tfactornicely.Hence,weresorttothequadraticformulatosolve x 2 )]TJ/F15 10.9091 Tf 10.348 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.348 0 Td [(1=0,and arriveat x =1 p 2.Asbefore,thesezerosdividethenumberlineintothreepieces.To helpusdecideontestvalues,weapproximate1 )]TJ 11.024 9.024 Td [(p 2 )]TJ/F15 10.9091 Tf 20.289 0 Td [(0 : 4and1+ p 2 2 : 4.Wechoose x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, x =0,and x =3asourtestvaluesandnd f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=2,whichis+; f = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 whichis )]TJ/F15 10.9091 Tf 8.485 0 Td [(;and f =2whichis+again.Oursolutionto x 2 )]TJ/F15 10.9091 Tf 11.403 0 Td [(2 x )]TJ/F15 10.9091 Tf 11.403 0 Td [(1 > 0iswhere wehave+,so,inintervalnotation )]TJ/F54 10.9091 Tf 5 -8.836 Td [( ; 1 )]TJ 10.909 9.024 Td [(p 2 [ )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+ p 2 ; 1 .Tochecktheinequality x 2 > 2 x +1graphically,weset g x = x 2 and h x =2 x +1.Wearelookingforthe x values wherethegraphof g isabovethegraphof h .Asbeforewepresentthegraphsontheright andthesignchartontheleft. 1 )]TJ 10.909 9.024 Td [(p 21+ p 2 + 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 03 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 2 3 4 5 6 7 8 3.Tosolve9 x 2 +4 12 x ,asbefore,wesolve9 x 2 )]TJ/F15 10.9091 Tf 9.246 0 Td [(12 x +4 0.Setting f x =9 x 2 )]TJ/F15 10.9091 Tf 9.245 0 Td [(12 x +4=0, wendtheonlyonezeroof f x = 2 3 .Thisone x valuedividesthenumberlineintotwo intervals,fromwhichwechoose x =0and x =1astestvalues.Wend f =4 > 0and f =1 > 0.Sincewearelookingforsolutionsto9 x 2 )]TJ/F15 10.9091 Tf 11.354 0 Td [(12 x +4 0,wearelookingfor

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152LinearandQuadraticFunctions x valueswhere9 x 2 )]TJ/F15 10.9091 Tf 11.163 0 Td [(12 x +4 < 0aswellaswhere9 x 2 )]TJ/F15 10.9091 Tf 11.163 0 Td [(12 x +4=0.Lookingatoursign diagram,therearenoplaceswhere9 x 2 )]TJ/F15 10.9091 Tf 11.492 0 Td [(12 x +4 < 0thereareno )]TJ/F15 10.9091 Tf 8.485 0 Td [(,sooursolution isonly x = 2 3 where9 x 2 )]TJ/F15 10.9091 Tf 11.746 0 Td [(12 x +4=0.Wewritethisas 2 3 .Graphically,wesolve 9 x 2 +4 12 x bygraphing g x =9 x 2 +4and h x =12 x .Wearelookingforthe x values wherethegraphof g isbelowthegraphof h for9 x 2 +4 < 12 x andwherethetwographs intersect x 2 +4=12 x .Weseethelineandtheparabolatouchat )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(2 3 ; 8 ,buttheparabola isalwaysabovethelineotherwise. 3 2 3 + 0 + 01 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 1 2 3 4 5 6 7 8 9 10 11 12 13 4.Tosolveourlastinequality,2 x )]TJ/F53 10.9091 Tf 10.564 0 Td [(x 2 j x )]TJ/F15 10.9091 Tf 10.564 0 Td [(1 j)]TJ/F15 10.9091 Tf 15.673 0 Td [(1,were-writetheabsolutevalueusingcases. For x< 1, j x )]TJ/F15 10.9091 Tf 11.549 0 Td [(1 j = )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 11.549 0 Td [(1=1 )]TJ/F53 10.9091 Tf 11.549 0 Td [(x ,soweget2 x )]TJ/F53 10.9091 Tf 11.549 0 Td [(x 2 1 )]TJ/F53 10.9091 Tf 11.55 0 Td [(x )]TJ/F15 10.9091 Tf 11.549 0 Td [(1,or x 2 )]TJ/F15 10.9091 Tf 11.549 0 Td [(3 x 0. Findingthezerosof f x = x 2 )]TJ/F15 10.9091 Tf 9.882 0 Td [(3 x ,weget x =0and x =3.However,weareonlyconcerned withtheportionofthenumberlinewhere x< 1,sotheonlyzerothatweconcernourselves withis x =0.Thisdividestheinterval x< 1intotwointervals: ; 0and ; 1.We choose x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and x = 1 2 asourtestvalues.Wend f )]TJ/F15 10.9091 Tf 8.484 0 Td [(1=4and f )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(5 4 .Solving x 2 )]TJ/F15 10.9091 Tf 11.17 0 Td [(3 x 0for x< 1givesus[0 ; 1.Next,weturnourattentiontothecase x 1.Here, j x )]TJ/F15 10.9091 Tf 11.159 0 Td [(1 j = x )]TJ/F15 10.9091 Tf 11.16 0 Td [(1,soouroriginalinequalitybecomes2 x )]TJ/F53 10.9091 Tf 11.159 0 Td [(x 2 x )]TJ/F15 10.9091 Tf 11.159 0 Td [(1 )]TJ/F15 10.9091 Tf 11.159 0 Td [(1,or x 2 )]TJ/F53 10.9091 Tf 11.159 0 Td [(x )]TJ/F15 10.9091 Tf 11.159 0 Td [(2 0. Setting g x = x 2 )]TJ/F53 10.9091 Tf 11.308 0 Td [(x )]TJ/F15 10.9091 Tf 11.308 0 Td [(2,wendthezerosof g tobe x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and x =2.Ofthese,only x =2liesintheregion x 1,soweignore x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Ourtestintervalsarenow[1 ; 2and ; 1 .Wechoose x =1and x =3asourtestvaluesandnd g = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and g =4.To solve g x 0,wehave[1 ; 2. 0 + 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 2 1 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [( 0 + 31 Combiningtheseintoonesigndiagram,wegetoursolutionis[0 ; 2].Graphically,tocheck 2 x )]TJ/F53 10.9091 Tf 10.828 0 Td [(x 2 j x )]TJ/F15 10.9091 Tf 10.828 0 Td [(1 j)]TJ/F15 10.9091 Tf 16.201 0 Td [(1,weset h x =2 x )]TJ/F53 10.9091 Tf 10.828 0 Td [(x 2 and i x = j x )]TJ/F15 10.9091 Tf 10.828 0 Td [(1 j)]TJ/F15 10.9091 Tf 16.202 0 Td [(1andlookforthe x values 3 Inthiscase,wesaytheline y =12 x is tangent to y =9 x 2 +4at )]TJ/F35 5.9776 Tf 5.42 -3.57 Td [(2 3 ; 8 .Findingtangentlinestoarbitrary functionsisthestuoflegends,Imean,Calculus.

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2.4Inequalities 153 wherethegraphof h isabovethethegraphof i thesolutionof h x >i x aswellasthe x -coordinatesoftheintersectionpointsofbothgraphswhere h x = i x .Thecombined signchartisgivenontheleftandthegraphsareontheright. 02 + 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 03 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 Itisquitepossibletoencounterinequalitieswheretheanalyticalmethodsdevelopedsofarwillfail us.Inthiscase,weresorttousingthegraphingcalculatortoapproximatethesolution,asthenext exampleillustrates. Example 2.4.5 Supposetherevenue R ,inthousandsofdollars,fromproducingandselling x hundredLCDTVsisgivenby R x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x 3 +35 x 2 +155 x for x 0,whilethecost,inthousands ofdollars,toproduce x hundredLCDTVsisgivenby C x =200 x +25for x 0.Howmany TVs,tothenearestTV,shouldbeproducedtomakeaprot? Solution. Recallthatprot=revenue )]TJ/F15 10.9091 Tf 11.59 0 Td [(cost.Ifwelet P denotetheprot,inthousandsof dollars,whichresultsfromproducingandselling x hundredTVsthen P x = R x )]TJ/F53 10.9091 Tf 10.909 0 Td [(C x = )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x 3 +35 x 2 +155 x )]TJ/F15 10.9091 Tf 10.909 0 Td [( x +25= )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x 3 +35 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(45 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(25 ; where x 0.Ifwewanttomakeaprot,thenweneedtosolve P x > 0;inotherwords, )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x 3 +35 x 2 )]TJ/F15 10.9091 Tf 11.16 0 Td [(45 x )]TJ/F15 10.9091 Tf 11.159 0 Td [(25 > 0.Wehaveyettodiscusshowtogoaboutndingthezerosof P ,let alonemakingasigndiagramforsuchananimal, 4 assuchweresorttothegraphingcalculator. Afterndingasuitablewindow,weget Wearelookingforthe x valuesforwhich P x > 0,thatis,wherethegraphof P isabovethe x -axis.Wemakeuseofthe`Zero'commandandndtwo x -intercepts. 4 Theprocedure,asweshallseeinChapter3isidenticaltowhatwehavedevelopedhere.

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154LinearandQuadraticFunctions Werememberthat x denotesthenumberofTVsin hundreds ,soifwearetondoursolution usingthecalculator,weneedouranswertotwodecimalplaces.Thezero 5 2 : 414 ::: correspondsto 241 : 4 ::: TVs.Sincewecan'tmakeafractionalpartofaTV,weroundthisupto242TVs. 7 The otherzeroseemsdeadonat5,whichcorrespondsto500TVs.Hencetomakeaprot,weshould produceandsellbetween242and499TVs,inclusive. Ourlastexampleinthesectiondemonstrateshowinequalitiescanbeusedtodescriberegionsin theplane,aswesawearlierinSection1.2. Example 2.4.6 Sketchthefollowingrelations. 1. R = f x;y : y> j x jg 2. S = f x;y : y 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 g 3. T = f x;y : j x j 0soweneedtoroundupto242inordertomakeaprot.

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2.4Inequalities 155 3.Finally,therelation T takesthepointswhose y -coordinatessatisfyboththeconditionsin R and S .Soweshadetheregionbetween y = j x j and y =2 )]TJ/F53 10.9091 Tf 11.619 0 Td [(x 2 ,keepingthosepoints ontheparabola,butnotthepointson y = j x j .Togetanaccurategraph,weneedtond wherethesetwographsintersect,soweset j x j =2 )]TJ/F53 10.9091 Tf 10.876 0 Td [(x 2 .Proceedingasbefore,breakingthis equationintocases,weget x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1.Graphingyields x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(112 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 Thegraphof T

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156LinearandQuadraticFunctions 2.4.1Exercises 1.Solvetheinequality.Expressyouranswerinintervalform. a j 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 j 4 b j 7 x +2 j > 10 c1 < j 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 j 3 d j)]TJ/F15 10.9091 Tf 16.363 0 Td [(2 x +1 j x +5 e j x +3 jj 6 x +9 j f x 2 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 0 g x 2 +1 < 0 h3 x 2 11 x +4 i x>x 2 j2 j x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 j < 9 k x 2 j 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 j l x 2 + x +1 0 2.Provethesecond,thirdandfourthpartsofTheorem2.3. 3.Ifaslingshotisusedtoshootamarblestraightupintotheairfrom2metersabovethe groundwithaninitialvelocityof30meterspersecond,forwhatvaluesoftime t willthe marblebeover35metersabovetheground?RefertoExercise5inSection2.3forassistance ifneeded.Roundyouranswerstotwodecimalplaces. 4.WhattemperaturevaluesindegreesCelsiusareequivalenttothetemperaturerange50 F to 95 F ?RefertoExercise3inSection2.1forassistanceifneeded. 5.Thesurfacearea S ofacubewithedgelength x isgivenby S x =6 x 2 for x> 0.Supposethe cubesyourcompanymanufacturesaresupposedtohaveasurfaceareaofexactly42square centimeters,butthemachinesyouownareoldandcannotalwaysmakeacubewiththe precisesurfaceareadesired.Writeaninequalityusingabsolutevaluethatsaysthesurface areaofagivencubeisnomorethan3squarecentimetersawayhighorlowfromthetarget of42squarecentimeters.Solvetheinequalityandexpressyouranswerinintervalform. 6.Sketchthefollowingrelations. a R = f x;y : y x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 g b R = x;y : y>x 2 +1 c R = f x;y : )]TJ/F15 10.9091 Tf 8.485 0 Td [(1
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2.4Inequalities 157 2.4.2Answers 1.a[ 1 3 ; 3] b ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(12 7 [ 8 7 ; 1 c[3 ; 4 [ ; 6] d ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(4 3 [ ; 1 e[ )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(12 7 ; )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(6 5 ] f ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3] [ [1 ; 1 gNosolution h[ )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 ; 4] i ; 1 j )]TJ 6.587 6.598 Td [(p 18 ; )]TJ 6.586 6.598 Td [(p 11] [ [ )]TJ 6.587 6.598 Td [(p 7 ; 0 [ ; p 7] [ [ p 11 ; p 18 k[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ 10.909 9.025 Td [(p 7 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+ p 7] [ [1 ; 3] l ; 1 3.1 : 44
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158LinearandQuadraticFunctions e x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 f )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123 5 10 15 20

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2.5Regression 159 2.5Regression Inthissection,weusesomebasictoolsfromstatisticalanalysistoquantifylinearandquadratic trendsthatwemayseeindata.Ourgoalistogivethereaderanunderstandingofthebasicprocesses involved,butwearequicktoreferthereadertoamoreadvancedcourse 1 foracompleteexposition ofthismaterial.Supposewecollectedthreedatapoints, f ; 2 ; ; 1 ; ; 3 g .Byplottingthese points,wecanclearlyseetheydonotliealongthesameline.Ifwepickanytwoofthepoints, wecanndalinecontainingbothwhichcompletelymissesthethird,butouraimistondaline whichisinsomesense`close'toallthepoints,eventhoughitmaygothroughnoneofthem.The waywemeasure`closeness'inthiscaseistondthe totalsquarederror betweenthedatapoints andtheline.Considerourthreedatapointsandtheline y = 1 2 x + 1 2 .Foreachofourdatapoints, wendtheverticaldistancebetweeneachdatapointandtheline.Toaccomplishthis,weneedto ndapointonthelinedirectlyaboveorbeloweachdatapoint-inotherwords,pointontheline withthesame x -coordinateasourdatapoint.Forexample,tondthepointonthelinedirectly below ; 2,weplug x =1into y = 1 2 x + 1 2 andwegetthepoint ; 1.Similarly,weget ; 1to correspondto ; 2and )]TJ/F15 10.9091 Tf 5 -8.836 Td [(4 ; 5 2 for ; 3. 1234 1 2 3 4 Wendthetotalsquarederror E bytakingthesumofthesquaresofthedierencesofthe y coordinatesofeachdatapointanditscorrespondingpointontheline.Forthedataandlineabove E = )]TJ/F15 10.9091 Tf 10.481 0 Td [(1 2 + )]TJ/F15 10.9091 Tf 10.481 0 Td [(2 2 + )]TJ/F15 10.9091 Tf 5 -8.836 Td [(3 )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(5 2 2 = 9 4 .Usingadvancedmathematicalmachinery, 2 itispossibleto ndthelinewhichresultsinthelowestvalueof E .Thislineiscalledthe leastsquaresregression line ,orsometimesthe`lineofbestt'.Theformulaforthelineofbesttrequiresnotationwe won'tpresentuntilChapter9.1,sowewillrevisititthen.Thegraphingcalculatorcancometoour assistancehere,sinceithasabuiltinfeaturetocomputetheregressionline.Weenterthedata andperformtheLinearRegressionfeatureandweget 1 andauthorswithmoreexpertiseinthisarea, 2 LikeCalculusandLinearAlgebra

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160LinearandQuadraticFunctions Thecalculatortellsusthatthelineofbesttis y = ax + b wheretheslopeis a 0 : 214andthe y -coordinateofthe y -interceptis b 1 : 428.Wewillsticktousingthreedecimalplacesforour approximations.Usingthisline,wecomputethetotalsquarederrorforourdatatobe E 1 : 786. Thevalue r isthe correlationcoecient andisameasureofhowclosethedataistobeingon thesameline.Thecloser j r j isto1,thebetterthelineart.Since r 0 : 327,thistellsusthatthe lineofbesttdoesn'ttallthatwell-inotherwords,ourdatapointsaren'tclosetobeinglinear. Thevalue r 2 iscalledthe coecientofdetermination andisalsoameasureofthegoodnessof t. 3 Plottingthedatawithitsregressionlineresultsinthepicturebelow. OurrstexamplelooksatenergyconsumptionintheUSoverthepast50years. 4 Year EnergyUsage, inQuads 5 1950 34 : 6 1960 45 : 1 1970 67 : 8 1980 78 : 3 1990 84 : 6 2000 98 : 9 Example 2.5.1 Usingtheenergyconsumptiondatagivenabove, 1.Plotthedatausingagraphingcalculator. 3 WerefertheinterestedreadertoacourseinStatisticstoexplorethesignicanceof r and r 2 4 SeethisDepartmentofEnergy activity 5 Theunit1Quadis1Quadrillion=10 15 BTUs,whichisenoughheattoraiseLakeErieroughly1 F

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2.5Regression 161 2.Findtheleastsquaresregressionlineandcommentonthegoodnessoft. 3.Interprettheslopeofthelineofbestt. 4.UsetheregressionlinetopredicttheannualUSenergyconsumptionintheyear2010. 5.Usetheregressionlinetopredictwhentheannualconsumptionwillreach120Quads. Solution. 1.Enteringthedataintothecalculatorgives Thedatacertainlyappearstobelinearinnature. 2.Performingalinearregressionproduces Wecantellbothfromthecorrelationcoecientaswellasthegraphthattheregressionline isagoodttothedata. 3.Theslopeoftheregressionlineis a 1 : 287.Tointerpretthis,recallthattheslopeisthe rateofchangeofthe y -coordinateswithrespecttothe x -coordinates.Sincethe y -coordinates representtheenergyusageinQuads,andthe x -coordinatesrepresentyears,aslopeofpositive 1 : 287indicatesanincreaseinannualenergyusageattherateof1 : 287Quadsperyear. 4.Topredicttheenergyneedsin2010,wesubstitute x =2010intotheequationofthelineof bestttoget y =1 : 287 )]TJ/F15 10.9091 Tf 11.059 0 Td [(2473 : 890 112 : 980.Thepredictedannualenergyusageof theUSin2010isapproximately112 : 908Quads.

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162LinearandQuadraticFunctions 5.TopredictwhentheannualUSenergyusagewillreach120Quads,wesubstitute y =120 intotheequationofthelineofbestttoget120=1 : 287 x )]TJ/F15 10.9091 Tf 11.084 0 Td [(2473 : 908.Solvingfor x yields x 2015 : 454.Sincetheregressionlineisincreasing,weinterpretthisresultassayingthe annualusagein2015won'tyetbe120Quads,butthatin2016,thedemandwillbemore than120Quads. Ournextexamplegivesusanopportunitytondanonlinearmodeltotthedata.Accordingto theNationalWeatherService,thepredictedhourlytemperaturesforPainesvilleonMarch3,2009 weregivenassummarizedbelow. Time Temperature, F 10AM 17 11AM 19 12PM 21 1PM 23 2PM 24 3PM 24 4PM 23 Toenterthisdataintothecalculator,weneedtoadjustthe x values,sincejustenteringthe numberscouldcauseconfusion.Doyouseewhy?Wehaveafewoptionsavailabletous.Perhaps theeasiestistoconvertthetimesintothe24hourclocktimesothat1PMis13,2PMis14,etc.. Ifweenterthesedataintothegraphingcalculatorandplotthepointsweget Whilethebeginningofthedatalookslinear,thetemperaturebeginstofallintheafternoonhours. Thissortofbehaviorremindsusofparabolas,and,sureenough,itispossibletondaparabolaof besttinthesamewaywefoundalineofbestt.Theprocessiscalled quadraticregression anditsgoalistominimizetheleastsquareerrorofthedatawiththeircorrespondingpointson theparabola.Thecalculatorhasabuiltinfeatureforthisaswell,andweget

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2.5Regression 163 Thecoecientofdetermination r 2 seemsreasonablycloseto1,andthegraphvisuallyseemstobe adecentt.Weusethisinournextexample. Example 2.5.2 Usingthequadraticmodelforthetemperaturedataabove,predictthewarmest temperatureoftheday.Whenwillthisoccur? Solution. Themaximumtemperaturewilloccuratthevertexoftheparabola.Recallingthe VertexFormula,Equation2.4, x = )]TJ/F37 7.9701 Tf 12.236 4.296 Td [(b 2 a )]TJ/F34 7.9701 Tf 30.205 4.296 Td [(9 : 464 2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 321 14 : 741.Thiscorrespondstoroughly2:45 PM.Tondthetemperature,wesubstitute x =14 : 741into y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 321 x 2 +9 : 464 x )]TJ/F15 10.9091 Tf 10.265 0 Td [(45 : 857toget y 23 : 899,or23 : 899 F. Theresultsofthelastexampleshouldremindyouthatregressionmodelsarejustthat,models.Our predictedwarmesttemperaturewasfoundtobe23 : 899 F,butourdatasaysitwillwarmto24 F. It'sallwellandgoodtoobservetrendsandguessatamodel,butamorethoroughinvestigation into why certaindata should belinearorquadraticinnatureisusuallyinorder-andthat,most often,isthebusinessofscientists.

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164LinearandQuadraticFunctions 2.5.1Exercises 1.Usingtheenergyproductiondatagivenbelow Year 1950 1960 1970 1980 1990 2000 Production inQuads 35.6 42.8 63.5 67.2 70.7 71.2 aPlotthedatausingagraphingcalculatorandexplainwhyitdoesnotappeartobe linear. bDiscusswithyourclassmateswhyignoringthersttwodatapointsmaybejustied fromahistoricalperspective. cFindtheleastsquaresregressionlineforthelastfourdatapointsandcommentonthe goodnessoft.Interprettheslopeofthelineofbestt. dUsetheregressionlinetopredicttheannualUSenergyproductionintheyear2010. eUsetheregressionlinetopredictwhentheannualUSenergyproductionwillreach100 Quads. 2.Thechartbelowcontainsaportionofthefuelconsumptioninformationfora2002Toyota EchoIJeusedtoown.Therstrowisthecumulativenumberofgallonsofgasolinethat IhadusedandthesecondrowistheodometerreadingwhenIrelledthegastank.So,for example,thefourthentryisthepoint.25,1051whichsaysthatIhadusedatotalof 28.25gallonsofgasolinewhentheodometerread1051miles. GasolineUsed Gallons 0 9.26 19.03 28.25 36.45 44.64 53.57 62.62 71.93 81.69 90.43 Odometer Miles 41 356 731 1051 1347 1631 1966 2310 2670 3030 3371 Findtheleastsquareslineforthisdata.Isitagoodt?Whatdoestheslopeoftheline represent?Doyouandyourclassmatesbelievethismodelwouldhaveheldfortenyearshad InotcrashedthecarontheTurnpikeafewyearsago?I'mkeepingafuellogformy2006 ScionxAforfutureCollegeAlgebrabookssoIhopenottocrashit,too. 3.OnNewYear'sDay,IJe,againstartedweighingmyselfeverymorninginordertohavean interestingdatasetforthissectionofthebook.Discusswithyourclassmatesifthatmakes meanerdorageek.Also,theprofessionalsintheeldofweightmanagementstrongly discourageweighingyourselfeveryday.Whenyoufocusonthenumberandnotyouroverall health,youtendtolosesightofyourobjectives.Iwasmakinganoblesacriceforscience, butyoushouldnot trythisathome.Thewholechartwouldbetoobigtoputintothebook neatly,soI'vedecidedtogiveonlyasmallportionofthedatatoyou.Thisthenbecomesa Civicslessoninhonesty,asyoushallsoonsee.Therearetwochartsgivenbelow.Onehasmy weightforthersteightThursdaysoftheyearJanuary1,2009wasaThursdayandwe'll countitasDay1.andtheotherhasmyweightfortherst10Saturdaysoftheyear.

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2.5Regression 165 Day# Thursday 1 8 15 22 29 36 43 50 Myweight inpounds 238.2 237.0 235.6 234.4 233.0 233.8 232.8 232.0 Day# Saturday 3 10 17 24 31 38 45 52 59 66 Myweight inpounds 238.4 235.8 235.0 234.2 236.2 236.2 235.2 233.2 236.8 238.2 aFindtheleastsquareslinefortheThursdaydataandcommentonitsgoodnessoft. bFindtheleastsquareslinefortheSaturdaydataandcommentonitsgoodnessoft. cUseQuadraticRegressiontondaparabolawhichmodelstheSaturdaydataandcommentonitsgoodnessoft. dCompareandcontrastthepredictionsthethreemodelsmakeformyweightonJanuary 1,2010Day#366.Cananyofthesemodelsbeusedtomakeapredictionofmyweight 20yearsfromnow?Explainyouranswer. eWhyisthisaCivicslessoninhonesty?Well,comparethetwolinearmodelsyouobtained above.Onewasagoodtandtheotherwasnot,yetbothcamefromcarefulselections ofrealdata.Inpresentingthetablestoyou,Ihavenotliedaboutmyweight,nor haveyouusedanybadmathtofalsifythepredictions.Thewordwe'relookingfor hereis`disingenuous'.Lookitupandthendiscusstheimplicationsthistypeofdata manipulationcouldhaveinalarger,morecomplex,politicallymotivatedsetting.Even Obi-WanpresentedthetruthtoLukeonlyfromacertainpointofview." 4.Datathatisneitherlinearnorquadratic.We'llclosethisexercisesetwithtwodatasetsthat, forreasonspresentedlaterinthebook,cannotbemodeledcorrectlybylinesorparabolas.It isagoodexercise,though,toseewhathappenswhenyouattempttousealinearorquadratic modelwhenit'snotappropriate. aThisrstdatasetcamefromaSummer2003publicationofthePortageCountyAnimal ProtectiveLeaguecalledTattleTails".Theymakethefollowingstatementandthen haveachartofdatathatsupportsit.Itdoesn'ttakelongfortwocatstoturninto80 million.Iftwocatsandtheirsurvivingospringreproducedfortenyears,you'dendup with80,399,780cats."Weassume N =2. Year x 1 2 3 4 5 6 7 8 9 10 Numberof Cats N x 12 66 382 2201 12680 73041 420715 2423316 13968290 80399780 UseQuadraticRegressiontondaparabolawhichmodelsthisdataandcommentonits goodnessoft.SpoilerAlert:Doesanyoneknowwhattypeoffunctionweneedhere?

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166LinearandQuadraticFunctions bThisnextdatasetcomesfromtheU.S.NavalObservatory .Thatsitehasloadsof awesomestuonit,butforthisexerciseIusedthesunrise/sunsettimesinFairbanks, Alaskafor2009togiveyouachartofthenumberofhoursofdaylighttheygetonthe 21 st ofeachmonth.We'lllet x =1representJanuary21,2009, x =2representFebruary 21,2009,andsoon. Month Number 1 2 3 4 5 6 7 8 9 10 11 12 Hoursof Daylight 5.8 9.3 12.4 15.9 19.4 21.8 19.4 15.6 12.4 9.1 5.6 3.3 UseQuadraticRegressiontondaparabolawhichmodelsthisdataandcommentonits goodnessoft.SpoilerAlert:Doesanyoneknowwhattypeoffunctionweneedhere?

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2.5Regression 167 2.5.2Answers 1.c y =0 : 266 x )]TJ/F15 10.9091 Tf 9.296 0 Td [(459 : 86with r =0 : 9607whichindicatesagoodt.Theslope0 : 266indicates thecountry'senergyproductionisincreasingatarateof0 : 266Quadperyear. dAccordingtothemodel,theproductionin2010willbe74 : 8Quad. eAccordingtothemodel,theproductionwillreach100Quadintheyear2105. 2.Thelineis y =36 : 8 x +16 : 39.Wehave r = : 99987and r 2 = : 9997sothisisanexcellentt tothedata.Theslope36 : 8representsmilespergallon. 3.aThelinefortheThursdaydatais y = )]TJ/F53 10.9091 Tf 8.485 0 Td [(: 12 x +237 : 69.Wehave r = )]TJ/F53 10.9091 Tf 8.485 0 Td [(: 9568and r 2 = : 9155sothisisareallygoodt. bThelinefortheSaturdaydatais y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 000693 x +235 : 94.Wehave r = )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 008986and r 2 =0 : 0000807whichishorrible.Thisdataisnotevenclosetolinear. cTheparabolafortheSaturdaydatais y =0 : 003 x 2 )]TJ/F15 10.9091 Tf 8.908 0 Td [(0 : 21 x +238 : 30.Wehave R 2 = : 47497 whichisn'tgood.Thusthedataisn'tmodeledwellbyaquadraticfunction,either. dTheThursdaylinearmodelhadmyweightonJanuary1,2010at193.77pounds.The Saturdaymodelsgive235.69and563.31pounds,respectively.TheThursdaylinehas myweightgoingbelow0poundsinaboutveandahalfyears,sothat'snogood.The quadratichasapositiveleadingcoecientwhichwouldmeanunboundedweightgain fortherestofmylife.TheSaturdayline,whichmathematicallydoesnottthedataat all,yieldsaplausibleweightpredictionintheend.Ithinkthisiswhygrown-upstalk aboutLies,DamnedLiesandStatistics." 4.aThequadraticmodelforthecatsinPortagecountyis y =1917803 : 54 x 2 )]TJ/F15 10.9091 Tf 8.913 0 Td [(16036408 : 29 x + 24094857 : 7.Although R 2 = : 70888thisisnotagoodmodelbecauseit'ssofarofor smallvaluesof x .Caseinpoint,themodelgivesus24,094,858catswhen x =0butwe know N =2. bThequadraticmodelforthehoursofdaylightinFairbanks,Alaskais y = : 51 x 2 +6 : 23 x )]TJ/F53 10.9091 Tf -408.242 -13.549 Td [(: 36.Evenwith R 2 = : 92295weshouldbewaryofmakingpredictionsbeyondthedata. Caseinpoint,themodelgives )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 : 84hoursofdaylightwhen x =13.SoJanuary21, 2010willbeextradark"?Obviouslyaparabolapointingdownisn'ttellingusthewhole story.

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168LinearandQuadraticFunctions

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Chapter3 PolynomialFunctions 3.1GraphsofPolynomials Threeofthefamiliesoffunctionsstudiedthusfar:constant,linearandquadratic,belongtoamuch largergroupoffunctionscalled polynomials .Webeginourformalstudyofgeneralpolynomials withadenitionandsomeexamples. Definition 3.1 A polynomialfunction isafunctionoftheform: f x = a n x n + a n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 x n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 + :::a 2 x 2 + a 1 x + a 0 ; where a 0 a 1 .... a n arerealnumbersand n 1isanaturalnumber. a Thedomainofapolynomial functionis ; 1 a Recallthismeans n isa`countingnumber' n =1 ; 2 ; 3 ;::: ThereareseveralthingsaboutDenition3.1thatmaybeo-puttingordownrightfrightening.The bestthingtodoislookatanexample.Consider f x =4 x 5 )]TJ/F15 10.9091 Tf 10.999 0 Td [(3 x 2 +2 x )]TJ/F15 10.9091 Tf 10.999 0 Td [(5.Isthisapolynomial function?Wecanre-writetheformulafor f as f x =4 x 5 +0 x 4 +0 x 3 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 2 +2 x + )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 : ComparingthiswithDenition3.1,weidentify n =5, a 5 =4, a 4 =0, a 3 =0, a 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3, a 1 =2, and a 0 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5.Inotherwords, a 5 isthecoecientof x 5 a 4 isthecoecientof x 4 ,andsoforth;the subscriptonthe a 'smerelyindicatestowhichpowerof x thecoecientbelongs.Thebusinessof restricting n tobeanaturalnumberletsusfocusonwell-behavedalgebraicanimals. 1 Example 3.1.1 Determineifthefollowingfunctionsarepolynomials.Explainyourreasoning. 1. g x = 4+ x 3 x 2. p x = 4 x + x 3 x 3. q x = 4 x + x 3 x 2 +4 1 Enjoythiswhileitlasts.Beforewe'rethroughwiththebook,you'llhavebeenexposedtothemostterribleof algebraicbeasts.Wewilltamethemall,intime.

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170PolynomialFunctions 4. f x = 3 p x 5. h x = j x j 6. z x =0 Solution. 1.Wenotedirectlythatthedomainof g x = x 3 +4 x is x 6 =0.Bydenition,apolynomialhas allrealnumbersasitsdomain.Hence, g can'tbeapolynomial. 2.Eventhough p x = x 3 +4 x x simpliesto p x = x 2 +4,whichcertainlylooksliketheform giveninDenition3.1,thedomainof p ,which,asyoumayrecall,wedetermine before we simplify,excludes0.Alas, p isnotapolynomialfunctionforthesamereason g isn't. 3.Afterwhathappenedwith p inthepreviouspart,youmaybealittleshyaboutsimplifying q x = x 3 +4 x x 2 +4 to q x = x ,whichcertainlytsDenition3.1.Ifwelookatthedomainof q beforewesimplied,weseethatitis,indeed,allrealnumbers.Afunctionwhichcanbe writtenintheformofDenition3.1whosedomainisallrealnumbersis,infact,apolynomial. 4.Wecanrewrite f x = 3 p x as f x = x 1 3 .Since 1 3 isnotanaturalnumber, f isnota polynomial. 5.Thefunction h x = j x j isn'tapolynomial,sinceitcan'tbewrittenasacombinationof powersof x eventhoughitcanbewrittenasapiecewisefunctioninvolvingpolynomials. Asweshallseeinthissection,graphsofpolynomialspossessaquality 2 thatthegraphof h doesnot. 6.There'snothinginDenition3.1whichpreventsallthecoecients a n ,etc.,frombeing0. Hence, z x =0,isanhonest-to-goodnesspolynomial. Definition 3.2 Suppose f isapolynomialfunction. Given f x = a n x n + a n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 x n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 + :::a 2 x 2 + a 1 x + a 0 with a n 6 =0,wesay { Thenaturalnumber n iscalledthe degree ofthepolynomial f { Theterm a n x n iscalledthe leadingterm ofthepolynomial f { Therealnumber a n iscalledthe leadingcoecient ofthepolynomial f { Therealnumber a 0 iscalledthe constantterm ofthepolynomial f If f x = a 0 ,and a 0 6 =0,wesay f hasdegree0. If f x =0,wesay f hasnodegree. a a Someauthorssay f x =0hasdegree forreasonsnotevenwewillgointo. 2 OnewhichreallyreliesonCalculustoverify.

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3.1GraphsofPolynomials171 Thereadermaywellwonderwhywehavechosentoseparateoconstantfunctionsfromtheother polynomialsinDenition3.2.Whynotjustlumpthemalltogetherand,insteadofforcing n to beanaturalnumber, n =1 ; 2 ;::: ,let n beawholenumber, n =0 ; 1 ; 2 ;::: .Wecouldunifyallthe cases,since,afterall,isn't a 0 x 0 = a 0 ?Theansweris`yes,aslongas x 6 =0.'Thefunction f x =3 and g x =3 x 0 aredierent,becausetheirdomainsaredierent.Thenumber f =3isdened, whereas g =3 0 isnot. 3 Indeed,muchofthetheorywewilldevelopinthischapterdoesn't includetheconstantfunctions,sowemightaswelltreatthemasoutsidersfromthestart.One goodthingthatcomesfromDenition3.2isthatwecannowthinkoflinearfunctionsasdegree 1or`rstdegree'polynomialfunctionsandquadraticfunctionsasdegree2or`seconddegree' polynomialfunctions. Example 3.1.2 Findthedegree,leadingterm,leadingcoecientandconstanttermofthefollowing polynomialfunctions. 1. f x =4 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 2 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 2. g x =12 x + x 3 3. h x = 4 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 5 4. p x = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +2 Solution. 1.Therearenosurpriseswith f x =4 x 5 )]TJ/F15 10.9091 Tf 10.447 0 Td [(3 x 2 +2 x )]TJ/F15 10.9091 Tf 10.448 0 Td [(5.ItiswrittenintheformofDenition 3.2,andweseethedegreeis5,theleadingtermis4 x 5 ,theleadingcoecientis4andthe constanttermis )]TJ/F15 10.9091 Tf 8.485 0 Td [(5. 2.TheformgiveninDenition3.2hasthehighestpowerof x rst.Tothatend,were-write g x =12 x + x 3 = x 3 +12 x ,andseethedegreeof g is3,theleadingtermis x 3 ,theleading coecientis1andtheconstanttermis0. 3.Weneedtorewritetheformulafor h sothatitresemblestheformgiveninDenition3.2: h x = 4 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x 5 = 4 5 )]TJ/F37 7.9701 Tf 12.347 4.295 Td [(x 5 = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 5 x + 4 5 .Weseethedegreeof h is1,theleadingtermis )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 5 x ,the leadingcoecientis )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 5 andtheconstanttermis 4 5 4.Itmayseemthatwehavesomeworkaheadofustoget p intheformofDenition3.2. However,itispossibletogleantheinformationrequestedabout p withoutmultiplyingout theentireexpression x )]TJ/F15 10.9091 Tf 10.838 0 Td [(1 3 x )]TJ/F15 10.9091 Tf 10.838 0 Td [(2 x +2.Theleadingtermof p willbethetermwhich hasthehighestpowerof x .Thewaytogetthistermistomultiplythetermswiththehighest powerof x fromeachfactortogether-inotherwords,theleadingtermof p x istheproductof theleadingtermsofthefactorsof p x .Hence,theleadingtermof p is x 3 x x =24 x 5 Thismeansthedegreeof p is5andtheleadingcoecientis24.Asfortheconstantterm, wecanperformasimilartrick.Theconstanttermisobtainedbymultiplyingtheconstant termsfromeachofthefactors )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2=4. 3 Technically,0 0 isanindeterminantform,whichisaspecialcaseofbeingundened.Theauthorsrealizethisis beyondpedantry,butwewouldn'tmentionitifwedidn'tfeelitwasneccessary.

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172PolynomialFunctions Ournextexampleshowshowpolynomialsofhigherdegreearise`naturally' 4 ineventhemostbasic geometricapplications. Example 3.1.3 Aboxwithnotopistobefashionedfroma10inch 12inchpieceofcardboard bycuttingoutcongruentsquaresfromeachcornerofthecardboardandthenfoldingtheresulting tabs.Let x denotethelengthofthesideofthesquarewhichisremovedfromeachcorner. x x x x x x x x 10in 12in width height depth 1.Findthevolume V oftheboxasafunctionof x .Includeanappropriateapplieddomain. 2.Useagraphingcalculatortograph y = V x onthedomainyoufoundinpart1andapproximatethedimensionsoftheboxwithmaximumvolumetotwodecimalplaces.Whatisthe maximumvolume? Solution. 1.FromGeometry,weknowVolume=width height depth.Thekeyistonowndeachof thesequantitiesintermsof x .Fromthegure,weseetheheightoftheboxis x itself.The cardboardpieceisinitially10incheswide.Removingsquareswithasidelengthof x inches fromeachcornerleaves10 )]TJ/F15 10.9091 Tf 11.438 0 Td [(2 x inchesforthewidth. 5 Asforthedepth,thecardboardis initially12incheslong,soaftercuttingout x inchesfromeachside,wewouldhave12 )]TJ/F15 10.9091 Tf 11.06 0 Td [(2 x inchesremaining.Asafunction 6 of x ,thevolumeis V x = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x =4 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(44 x 2 +120 x Tondasuitableapplieddomain,wenotethattomakeaboxatallweneed x> 0.Alsothe shorterofthetwodimensionsofthecardboardis10inches,andsinceweareremoving2 x inchesfromthisdimension,wealsorequire10 )]TJ/F15 10.9091 Tf 10.489 0 Td [(2 x> 0or x< 5.Hence,ourapplieddomain is0
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3.1GraphsofPolynomials173 InordertosolveExample3.1.3,wemadegooduseofthegraphofthepolynomial y = V x .Sowe oughttoturnourattentiontographsofpolynomialsingeneral.Belowarethegraphsof y = x 2 y = x 4 ,and y = x 6 ,side-by-side.Wehaveomittedtheaxessowecanseethatastheexponent increases,the`bottom'becomes`atter'andthe`sides'become`steeper.'Ifyoutakethethetime tographthesefunctionsbyhand, 7 youwillseewhy. y = x 2 y = x 4 y = x 6 Allofthesefunctionsareeven,Doyourememberhowtoshowthis?anditisexactlybecause theexponentiseven. 8 Oneofthemostimportantfeaturesofthesefunctionswhichwecanbe seengraphicallyistheir endbehavior .Theendbehaviorofafunctionisawaytodescribewhat ishappeningtothefunctionvaluesasthe x valuesapproachthe`ends'ofthe x -axis: 9 thatis, astheybecomesmallwithoutbound 10 written x and,ontheipside,astheybecome largewithoutbound 11 written x !1 .Forexample,given f x = x 2 ,as x ,weimagine substituting x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(100, x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1000,etc.,into f toget f )]TJ/F15 10.9091 Tf 8.485 0 Td [(100=10000, f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1000=1000000,and soon.Thusthefunctionvaluesarebecominglargerandlargerpositivenumberswithoutbound. Todescribethisbehavior,wewrite:as x f x !1 .Ifwestudythebehaviorof f as x !1 ,weseethatinthiscase,too, f x !1 .Thesamecanbesaidforanyfunctionofthe form f x = x n where n isanevennaturalnumber.Ifwegeneralizejustabittoincludevertical scalingsandreectionsacrossthe x -axis, 12 wehave 7 Makesureyouchoosesome x -valuesbetween )]TJ/F63 8.9664 Tf 7.167 0 Td [(1and1. 8 Hereinliesoneofthepossibleoriginsoftheterm`even'whenappliedtofunctions. 9 Ofcourse,therearenoendstothe x -axis. 10 Wethinkof x asbecomingaverylarge negative numberfartotheleftofzero. 11 Wethinkof x asmovingfartotherightofzeroandbecomingaverylarge positive number. 12 SeeTheorems1.4and1.5inSection1.8.

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174PolynomialFunctions EndBehavioroffunctions f x = ax n n even. Suppose f x = ax n where a 6 =0isarealnumberand n isanevennaturalnumber.Theend behaviorofthegraphof y = f x matchesoneofthefollowing: a> 0 a< 0 Wenowturnourattentiontofunctionsoftheform f x = x n where n 3isanoddnatural number. 13 Belowwehavegraphed y = x 3 y = x 5 ,and y = x 7 .The`attening'and`steepening' thatwesawwiththeevenpowerspresentsitselfhereaswell,and,itshouldcomeasnosurprise thatallofthesefunctionsareodd. 14 Theendbehaviorofthesefunctionsisallthesame,with f x as x and f x !1 as x !1 y = x 3 y = x 5 y = x 7 Aswiththeevendegreedfunctionswestudiedearlier,wecangeneralizetheirendbehavior. EndBehavioroffunctions f x = ax n n odd. Suppose f x = ax n where a 6 =0isarealnumberand n 3isanoddnaturalnumber.Theend behaviorofthegraphof y = f x matchesoneofthefollowing: a> 0 a< 0 Despitehavingdierentendbehavior,allfunctionsoftheform f x = ax n fornaturalnumbers n sharetwopropertieswhichhelpdistinguishthemfromotheranimalsinthealgebrazoo:theyare continuous and smooth .WhiletheseconceptsareformallydenedusingCalculus, 15 informally, graphsofcontinuousfunctionshaveno`breaks'or`holes'intheirgraphs,andsmoothfunctionshave no`sharpturns.'Itturnsoutthatthesetraitsarepreservedwhenfunctionsareaddedtogether,so generalpolynomialfunctionsinheritthesequalities.Belowwendthegraphofafunctionwhichis 13 Weignorethecasewhen n =1,sincethegraphof f x = x isalineanddoesn'ttthegeneralpatternof higher-degreeoddpolynomials. 14 Andare,perhaps,theinspirationforthemoniker`oddfunction'. 15 Infact,ifyoutakeCalculus,you'llndthatsmoothfunctionsareautomaticallycontinuous,sothatsaying `polynomialsarecontinuousandsmooth'isredundant.

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3.1GraphsofPolynomials175 neithersmoothnorcontinuous,andtoitsrightwehaveagraphofapolynomial,forcomparison. Thefunctionwhosegraphappearsontheleftfailstobecontinuouswhereithasa`break'or`hole' inthegraph;everywhereelse,thefunctioniscontinuous.Thefunctioniscontinuousatthe`corner' andthe`cusp',butweconsiderthese`sharpturns',sotheseareplaceswherethefunctionfails tobesmooth.Apartfromthesefourplaces,thefunctionissmoothandcontinuous.Polynomial functionsaresmoothandcontinuouseverywhere,asexhibitedingraphontheright. `corner' `break' `cusp' `hole' Pathologiesnotfoundongraphsofpolynomials Thegraphofapolynomial Thenotionofsmoothnessiswhattellsusgraphicallythat,forexample, f x = j x j ,whosegraph isthecharacteric` 'shape,cannotbeapolynomial.Thenotionofcontinuityiswhatallowedus toconstructthesigndiagramforquadraticinequalitiesaswedidinSection2.4.Thislastresultis formalizedinthefollowingtheorem. Theorem 3.1 TheIntermediateValueTheoremPolynomialZeroVersion: If f isa polynomialwhere f a and f b havedierentsigns,then f hasatleastonezerobetween x = a and x = b ;thatis,foratleastonerealnumber c suchthat a
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176PolynomialFunctions StepsforConstructingaSignDiagramforaPolynomialFunction Suppose f isapolynomialfunction. 1.Findthezerosof f andplacethemonthenumberlinewiththenumber0abovethem. 2.Choosearealnumber,calleda testvalue ,ineachoftheintervalsdeterminedinstep1. 3.Determinethesignof f x foreachtestvalueinstep2,andwritethatsignabovethe correspondinginterval. Example 3.1.5 Constructasigndiagramfor f x = x 3 x )]TJ/F15 10.9091 Tf 11.005 0 Td [(3 2 x +2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 +1 .Useittogivea roughsketchofthegraphof y = f x Solution. First,wendthezerosof f bysolving x 3 x )]TJ/F15 10.9091 Tf 11.006 0 Td [(3 2 x +2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 +1 =0.Weget x =0, x =3,and x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Theequation x 2 +1=0producesnorealsolutions.Thesethreepoints dividetherealnumberlineintofourintervals: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0, ; 3and ; 1 .Weselectthe testvalues x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3, x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1, x =1,and x =4.Wend f )]TJ/F15 10.9091 Tf 8.485 0 Td [(3is+, f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1is )]TJ/F15 10.9091 Tf 8.485 0 Td [(and f is+ asis f .Wherever f is+,itsgraphisabovethe x -axis;wherever f is )]TJ/F15 10.9091 Tf 8.485 0 Td [(,itsgraphisbelow the x -axis.The x -interceptsofthegraphof f are )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0, ; 0and ; 0.Knowing f issmooth andcontinuousallowsustosketchitsgraph. )]TJ/F15 10.9091 Tf 8.485 0 Td [(203 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 + 1 0 + 4 x y Asketchof y = f x AcoupleofnotesabouttheExample3.1.5areinorder.First,notethatwepurposefullydidnot labelthe y -axisinthesketchofthegraphof y = f x .Thisisbecausethesigndiagramgivesusthe zerosandtherelativepositionofthegraph-itdoesn'tgiveusanyinformationastohowhighorlow thegraphstraysfromthe x -axis.Furthermore,aswehavementionedearlierinthetext,without Calculus,thevaluesoftherelativemaximumandminimumcanonlybefoundapproximatelyusing acalculator.Ifwetookthetimetondtheleadingtermof f ,wewouldndittobe x 8 .Looking attheendbehaviorof f ,wenoticeitmatchestheendbehaviorof y = x 8 .Thisisnoaccident,as wendoutinthenexttheorem. Theorem 3.2 EndBehaviorforPolynomialFunctions: Theendbehaviorofapolynomial f x = a n x n + a n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 x n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 + :::a 2 x 2 + a 1 x + a 0 with a n 6 =0matchestheendbehaviorof y = a n x n ToseewhyTheorem3.2istrue,let'srstlookataspecicexample.Consider f x =4 x 3 )]TJ/F53 10.9091 Tf 10.442 0 Td [(x +5. Ifwewishtoexamineendbehavior,welooktoseethebehaviorof f as x .Sincewe're concernedwith x 'sfardownthe x -axis,wearefarawayfrom x =0andsocanrewrite f x for thesevaluesof x as

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3.1GraphsofPolynomials177 f x =4 x 3 1 )]TJ/F15 10.9091 Tf 17.588 7.38 Td [(1 4 x 2 + 5 4 x 3 As x becomesunboundedineitherdirection,theterms 1 4 x 2 and 5 4 x 3 becomecloserandcloserto 0,asthetablebelowindicates. x 1 4 x 2 5 4 x 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1000 0 : 00000025 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 00000000125 )]TJ/F15 10.9091 Tf 8.485 0 Td [(100 0 : 000025 )]TJ/F15 10.9091 Tf 8.484 0 Td [(0 : 00000125 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 : 0025 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 00125 10 0 : 0025 0 : 00125 100 0 : 000025 0 : 00000125 1000 0 : 00000025 0 : 00000000125 Inotherwords,as x f x 4 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(0+0=4 x 3 ,whichistheleadingtermof f .The formalproofofTheorem3.2worksinmuchthesameway.Factoringouttheleadingtermleaves f x = a n x n 1+ a n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 a n x + ::: + a 2 a n x n )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 + a 1 a n x n )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 + a 0 a n x n As x ,anytermwithan x inthedenominatorbecomescloserandcloserto0,andwehave f x a n x n .Geometrically,Theorem3.2saysthatifwegraph y = f x ,say,usingagraphing calculator,andcontinueto`zoomout,'thegraphofitanditsleadingtermbecomeindistinguishable. Belowarethegraphsof y =4 x 3 )]TJ/F53 10.9091 Tf 11.212 0 Td [(x +5thethickerlineand y =4 x 3 thethinnerlineintwo dierentwindows. Aview`close'totheorigin.A`zoomedout'view. Let'sreturntothefunctioninExample3.1.5, f x = x 3 x )]TJ/F15 10.9091 Tf 9.354 0 Td [(3 2 x +2 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 +1 ,whosesigndiagram andgrapharereproducedbelowforreference.Theorem3.2tellsusthattheendbehavioristhe sameasthatofitsleadingterm, x 8 .Thistellsusthatthegraphof y = f x startsandendsabove the x -axis.Inotherwords, f x is+as x ,andasaresult,wenolongerneedtoevaluate f atthetestvalues x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3and x =4.Isthereawaytoeliminatetheneedtoevaluate f atthe othertestvalues?Whatwewouldreallyneedtoknowishowthefunctionbehavesnearitszerosdoesitcrossthroughthe x -axisatthesepoints,asitdoesat x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2and x =0,ordoesitsimply touchandreboundlikeitdoesat x =3.Fromthesigndiagram,thegraphof f willcrossthe

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178PolynomialFunctions x -axiswheneverthesignsoneithersideofthezeroswitchliketheydoat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x =0;it willtouchwhenthesignsarethesameoneithersideofthezeroasisthecasewith x =3.What weneedtodetermineisthereasonbehindwhetherornotthesignchangeoccurs. )]TJ/F15 10.9091 Tf 8.485 0 Td [(203 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 + 1 0 + 4 x y Asketchof y = f x Fortunately, f wasgiventousinfactoredform: f x = x 3 x )]TJ/F15 10.9091 Tf 11.178 0 Td [(3 2 x +2.Whenweattemptto determinethesignof f )]TJ/F15 10.9091 Tf 8.485 0 Td [(4,weareattemptingtondthesignofthenumber )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2, whichworksouttobe )]TJ/F15 10.9091 Tf 8.485 0 Td [(+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(whichis+.Ifwemovetotheothersideof x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,andnd thesignof f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,wearedeterminingthesignof )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 +1,whichis )]TJ/F15 10.9091 Tf 8.485 0 Td [(++whichgives usthe )]TJ/F15 10.9091 Tf 8.485 0 Td [(.Noticethatsignsofthersttwofactorsinbothexpressionsarethesamein f )]TJ/F15 10.9091 Tf 8.484 0 Td [(4and f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Theonlyfactorwhichswitchessignisthethirdfactor, x +2,preciselythefactorwhich gaveusthezero x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Ifwemovetotheothersideof0andlookcloselyat f ,wegetthesign pattern+1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 +3or+++andwenotethat,onceagain,goingfrom f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1to f theonlyfactorwhichchangedsignwastherstfactor, x 3 ,whichcorrespondstothezero x =0. Finally,tond f ,wesubstitutetoget+4 3 +2 2 +5whichis+++or+.Thesign didn'tchangeforthemiddlefactor x )]TJ/F15 10.9091 Tf 11.046 0 Td [(3 2 .Eventhoughthisisthefactorwhichcorrespondsto thezero x =3,thefactthatthequantityis squared keptthesignofthemiddlefactorthesame oneithersideof3.Ifwelookbackattheexponentsonthefactors x +2and x 3 ,wenotethey arebothodd-soaswesubstitutevaluestotheleftandrightofthecorrespondingzeros,thesigns ofthecorrespondingfactorschangewhichresultsinthesignofthefunctionvaluechanging.This isthekeytothebehaviorofthefunctionnearthezeros.Weneedadenitionandthenatheorem. Definition 3.3 Suppose f isapolynomialfunctionand m isanaturalnumber.If x )]TJ/F53 10.9091 Tf 10.571 0 Td [(c m isa factorof f x but x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c m +1 isnot,thenwesay x = c isazeroof multiplicity m Hence,rewriting f x = x 3 x )]TJ/F15 10.9091 Tf 10.372 0 Td [(3 2 x +2as f x = x )]TJ/F15 10.9091 Tf 10.372 0 Td [(0 3 x )]TJ/F15 10.9091 Tf 10.372 0 Td [(3 2 x )]TJ/F15 10.9091 Tf 10.372 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 ,weseethat x =0 isazeroofmultiplicity3, x =3isazeroofmultiplicity2,and x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2isazeroofmultiplicity1. Theorem 3.3 TheRoleofMultiplicity: Suppose f isapolynomialfunctionand x = c isa zeroofmultiplicity m If m iseven,thegraphof y = f x touchesandreboundsfromthe x -axisas c; 0. If m isodd,thegraphof y = f x crossesthroughthe x -axisas c; 0.

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3.1GraphsofPolynomials179 Ourlastexampleshowshowendbehaviorandmultiplicityallowustosketchadecentgraphwithout appealingtoasigndiagram. Example 3.1.6 Sketchthegraphof f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x )]TJ/F15 10.9091 Tf 12.032 0 Td [(1 x +1 2 usingendbehaviorandthe multiplicityofitszeros. Solution. Theendbehaviorofthegraphof f willmatchthatofitsleadingterm.Tond theleadingterm,wemultiplybytheleadingtermsofeachfactortoget )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x x 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x 3 Thistellsusthegraphwillstartabovethe x -axis,inQuadrantII,andnishbelowthe x -axis,in QuadrantIV.Next,wendthezerosof f .Fortunatelyforus, f isfactored. 16 Settingeachfactor equaltozerogivesis x = 1 2 and x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1aszeros.Tondthemultiplicityof x = 1 2 wenotethat itcorrespondstothefactor x )]TJ/F15 10.9091 Tf 11.152 0 Td [(1.Thisisn'tstrictlyintheformrequiredinDenition3.3.If wefactoroutthe2,however,weget x )]TJ/F15 10.9091 Tf 11.034 0 Td [(1=2 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(1 2 ,andweseethemultiplicityof x = 1 2 is 1.Since1isanoddnumber,weknowfromTheorem3.3thatthegraphof f willcrossthrough the x -axisat )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 2 ; 0 .Sincethezero x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1correspondstothefactor x +1 2 = x )]TJ/F15 10.9091 Tf 11.18 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 ,we seeitsmultiplicityis2whichisanevennumber.Assuch,thegraphof f willtouchandrebound fromthe x -axisat )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0.Thoughwe'renotaskedto,wecanndthe y -interceptbynding f = )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 11.416 0 Td [(1+1 2 =3.Thus ; 3isanadditionalpointonthegraph.Puttingthis togethergivesusthegraphbelow. x y 16 Obtainingthefactoredformofapolynomialisthemainfocusofthenextfewsections.

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180PolynomialFunctions 3.1.1Exercises 1.Foreachpolynomialgivenbelow,ndthedegree,theleadingterm,theleadingcoecient, theconstanttermandtheendbehavior. a f x = p 3 x 17 +22 : 5 x 10 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 7 + 1 3 b p t = )]TJ/F53 10.9091 Tf 8.485 0 Td [(t 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 t t 2 + t +4 c Z b =42 b )]TJ/F53 10.9091 Tf 10.909 0 Td [(b 3 d s t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 : 9 t 2 + v 0 t + s 0 e P x = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 f q r =1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(16 r 4 2.Foreachpolynomialgivenbelow,nditsrealzerosandtheircorrespondingmultiplicities. Usethisinformationalongwithasigncharttoprovidearoughsketchofthegraphofthe polynomial. a a x = x x +2 2 b F x = x 3 x +2 2 c P x = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 d Z b = b )]TJ/F53 10.9091 Tf 10.909 0 Td [(b 2 e Q x = x +5 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 4 f g x = x x +2 3 3.AccordingtoUSPostalregulations,arectangularshippingboxmustsatisfytheinequality Length+Girth 130inches"forParcelPostandLength+Girth 108inches"forother services. 17 Let'sassumewehaveaclosedrectangularboxwithasquarefaceofsidelength x asdrawnbelow.Thelengthisthelongestsideandisclearlylabeled.Thegirthisthe distancearoundtheboxintheothertwodimensionssoinourcaseitisthesumofthefour sidesofthesquare,4 x aAssumingthatwe'llbemailingtheboxParcelPost,expressthelengthoftheboxin termsof x andthenexpressthevolume, V ,oftheboxintermsof x bFindthedimensionsoftheboxwithmaximumvolumethatcanbeshippedviaParcel Post. cRepeatparts3aand3baboveassumingthatwe'llbeshippingtheboxusingother services". length x x 17 Seehere fordetails.

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3.1GraphsofPolynomials181 4.Usetransformationstosketchthegraphsofthefollowingpolynomials. a f x = x +2 3 +1 b g x = x +2 4 +1 c h x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 d j x =2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 4 5.UsetheIntermediateValueTheoremtondintervalsoflength1whichcontaintherealzeros of f x = x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x +5. 6.Theoriginalfunctionusedtomodelthecostofproducing x PortaBoysGameSystemsgivenin Example2.1.5was C x =80 x +150.Whiledevelopingtheirnewestgame,SasquatchAttack!, themakersofthePortaBoyrevisedtheircostfunctionusingacubicpolynomial.Thenew costofproducing x PortaBoysisgivenby C x = : 03 x 3 )]TJ/F15 10.9091 Tf 11.935 0 Td [(4 : 5 x 2 +225 x +250.Market researchindicatesthatthedemandfunction p x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x +250remainsunchanged.Find theproductionlevel x thatmaximizestheprotmadebyproducingandselling x PortaBoys. 7.WenowrevisitthedatasetfromExercise4binSection2.5.Inthatexercise,youweregiven achartofthenumberofhoursofdaylighttheygetonthe21 st ofeachmonthinFairbanks, Alaskabasedonthe2009sunriseandsunsetdatafoundontheU.S.NavalObservatory website.Welet x =1representJanuary21,2009, x =2representFebruary21,2009,andsoon. Thechartisgivenagainforreference. Month Number 1 2 3 4 5 6 7 8 9 10 11 12 Hoursof Daylight 5.8 9.3 12.4 15.9 19.4 21.8 19.4 15.6 12.4 9.1 5.6 3.3 Findcubicthirddegreeandquarticfourthdegreepolynomialswhichmodelthisdataand commentonthegoodnessoftforeach.Whatcanwesayaboutusingeithermodeltomake predictionsabouttheyear2020?Hint:Thinkabouttheendbehaviorofpolynomials.Use themodelstoseehowmanyhoursofdaylighttheygotonyourbirthdayandthencheckthe websitetoseehowaccuratethemodelsare.KnowingthatSasquatcharelargelynocturnal, whatdaysoftheyearaccordingtoyourmodelsaregoingtoallowforatleast14hoursof darknessforeldresearchontheelusivecreatures? 8.Anelectriccircuitisbuiltwithavariableresistorinstalled.Foreachofthefollowingresistancevaluesmeasuredinkilo-ohms, k ,thecorrespondingpowertotheloadmeasuredin milliwatts, mW isgiveninthetablebelow. 18 Resistance: k 1.012 2.199 3.275 4.676 6.805 9.975 Power: mW 1.063 1.496 1.610 1.613 1.505 1.314 aMakeascatterdiagramofthedatausingtheResistanceastheindependentvariable andPowerasthedependentvariable. 18 TheauthorswishtothankDonAnthanandKenWhiteofLakelandCommunityCollegefordevisingthisproblem andgeneratingtheaccompanyingdataset.

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182PolynomialFunctions bUseyourcalculatortondquadraticnddegree,cubicrddegreeandquarticth degreeregressionmodelsforthedataandjudgethereasonablenessofeach. cForeachofthemodelsfoundabove,ndthepredictedmaximumpowerthatcanbe deliveredtotheload.Whatisthecorrespondingresistancevalue? dDiscusswithyourclassmatesthelimitationsofthesemodels-inparticular,discussthe endbehaviorofeach. 9.Showthattheendbehaviorofalinearfunction f x = mx + b isasitshouldbeaccording totheresultswe'veestablishedinthesectionforpolynomialsofodddegree.Thatis,show thatthegraphofalinearfunctionisupononesideanddownontheother"justlikethe graphof y = a n x n foroddnumbers n 10.Thereisonesubtletyabouttheroleofmultiplicitythatweneedtodiscussfurther;specically weneedtosee`how'thegraphcrossesthe x -axisatazeroofoddmultiplicity.Inthesection, wedeliberatelyexcludedthefunction f x = x fromthediscussionoftheendbehaviorof f x = x n foroddnumbers n andwesaidatthetimethatitwasduetothefactthat f x = x didn'ttthepatternweweretryingtoestablish.Youjustshowedinthepreviousexercise thattheendbehaviorofalinearfunctionbehaveslikeeveryotherpolynomialofodddegree, sowhatdoesn't f x = x dothat g x = x 3 does?It'sthe`attening'forvaluesof x nearzero. Itisthislocalbehaviorthatwilldistinguishbetweenazeroofmultiplicity1andoneofhigher oddmultiplicity.Lookagaincloselyatthegraphsof a x = x x +2 2 and F x = x 3 x +2 2 fromExercise2.Discusswithyourclassmateshowthegraphsarefundamentallydierent attheorigin.Itmighthelptouseagraphingcalculatortozoominontheorigintosee thedierentcrossingbehavior.Alsocomparethebehaviorof a x = x x +2 2 tothatof g x = x x +2 3 nearthepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0.Whatdoyoupredictwillhappenatthezerosof f x = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 5 ? 11.Hereareafewotherquestionsforyoutodiscusswithyourclassmates. aHowmanylocalextremacouldapolynomialofdegree n have?Howfewlocalextrema canithave? bCouldapolynomialhavetwolocalmaximabutnolocalminima? cIfapolynomialhastwolocalmaximaandtwolocalminima,canitbeofodddegree? Canitbeofevendegree? dCanapolynomialhavelocalextremawithouthavinganyrealzeros? eWhymusteverypolynomialofodddegreehaveatleastonerealzero? fCanapolynomialhavetwodistinctrealzerosandnolocalextrema? gCanan x -interceptyieldalocalextrema?Canityieldanabsoluteextrema? hIfthe y -interceptyieldsanabsoluteminimum,whatcanwesayaboutthedegreeofthe polynomialandthesignoftheleadingcoecient?

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3.1GraphsofPolynomials183 3.1.2Answers 1.a f x = p 3 x 17 +22 : 5 x 10 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 7 + 1 3 Degree17 Leadingterm p 3 x 17 Leadingcoecient p 3 Constantterm 1 3 As x ;f x As x !1 ;f x !1 b p t = )]TJ/F53 10.9091 Tf 8.485 0 Td [(t 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 t t 2 + t +4 Degree5 Leadingterm5 t 5 Leadingcoecient5 Constantterm0 As t ;p t As t !1 ;p t !1 c Z b =42 b )]TJ/F53 10.9091 Tf 10.909 0 Td [(b 3 Degree3 Leadingterm )]TJ/F53 10.9091 Tf 8.485 0 Td [(b 3 Leadingcoecient )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 Constantterm0 As b ;Z b !1 As b !1 ;Z b d s t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 : 9 t 2 + v 0 t + s 0 Degree2 Leadingterm )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 : 9 t 2 Leadingcoecient )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 : 9 Constantterm s 0 As t ;s t As t !1 ;s t e P x = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 Degree4 Leadingterm x 4 Leadingcoecient1 Constantterm24 As x ;P x !1 As x !1 ;P x !1 f q r =1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(16 r 4 Degree4 Leadingterm )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 r 4 Leadingcoecient )]TJ/F15 10.9091 Tf 8.484 0 Td [(16 Constantterm1 As r ;q r As r !1 ;q r 2.a a x = x x +2 2 x =0multiplicity1 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2multiplicity2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 b F x = x 3 x +2 2 x =0multiplicity3 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2multiplicity2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1

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184PolynomialFunctions c P x = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x =1multiplicity1 x =2multiplicity1 x =3multiplicity1 x =4multiplicity1 x y 1234 d Z b = b )]TJ/F53 10.9091 Tf 10.909 0 Td [(b 2 b = )]TJ 8.485 9.024 Td [(p 42multiplicity1 b =0multiplicity1 b = p 42multiplicity1 b Z )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123456 e Q x = x +5 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 4 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5multiplicity2 x =3multiplicity4 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 f g x = x x +2 3 x =0multiplicity1 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2multiplicity3 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 3.aOurultimategoalistomaximizethevolume,sowe'llstartwiththemaximumLength +Girthof130 : Thismeansthelengthis130 )]TJ/F15 10.9091 Tf 10.954 0 Td [(4 x .Thevolumeofarectangularboxis alwayslength width heightsoweget V x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 x 3 +130 x 2 bGraphing y = V x on[0 ; 33] [0 ; 21000]showsamaximumat : 67 ; 20342 : 59sothe dimensionsoftheboxwithmaximumvolumeare21 : 67in. 21 : 67in. 43 : 32in.fora volumeof20342 : 59in. 3 cIfwestartwithLength+Girth=108thenthelengthis108 )]TJ/F15 10.9091 Tf 11.698 0 Td [(4 x andthevolume is V x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 x 3 +108 x 2 .Graphing y = V x on[0 ; 27] [0 ; 11700]showsamaximumat : 00 ; 11664 : 00sothedimensionsoftheboxwithmaximumvolumeare 18 : 00in. 18 : 00in. 36in.foravolumeof11664 : 00in. 3 .Calculuswillconrmthatthe measurementswhichmaximizethevolumeareexactly 18in.by18in.by36in.,however, asI'msureyouareawarebynow,wetreatallcalculatorresultsasapproximationsand listthemassuch.

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3.1GraphsofPolynomials185 4.a f x = x +2 3 +1 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(10 )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 8 9 10 11 12 b g x = x +2 4 +1 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 c h x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.756 0 Td [(10 )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 8 9 10 d j x =2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 4 x y 12 )]TJ/F35 5.9776 Tf 5.756 0 Td [(13 )]TJ/F35 5.9776 Tf 5.756 0 Td [(12 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.756 0 Td [(10 )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 5.Wehave f )]TJ/F15 10.9091 Tf 8.485 0 Td [(4= )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 ;f )]TJ/F15 10.9091 Tf 8.485 0 Td [(3=5 ;f =5 ;f = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ;f = )]TJ/F15 10.9091 Tf 8.484 0 Td [(5and f =5sothe IntermediateValueTheoremtellsusthat f x = x 3 )]TJ/F15 10.9091 Tf 11.111 0 Td [(9 x +5hasrealzerosintheintervals [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3] ; [0 ; 1]and[2 ; 3]. 6.Makingandselling71PortaBoysyieldsamaximizedprotof $ 5910.67. 7.Thecubicregressionmodelis p 3 x =0 : 0226 x 3 )]TJ/F15 10.9091 Tf 11.428 0 Td [(0 : 9508 x 2 +8 : 615 x )]TJ/F15 10.9091 Tf 11.429 0 Td [(3 : 446.Ithas R 2 = 0 : 93765whichisn'tbad.Thegraphof y = p 3 x intheviewingwindow[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 13] [0 ; 24] alongwiththescatterplotisshownbelowontheleft.Noticethat p 3 hitsthe x -axisatabout x =12 : 45makingthisabadmodelforfuturepredictions.Tousethemodeltoapproximate thenumberofhoursofsunlightonyourbirthday,you'llhavetogureoutwhatdecimalvalue

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186PolynomialFunctions of x iscloseenoughtoyourbirthdayandthenplugitintothemodel.MyJe'sbirthday isJuly31whichis10daysafterJuly21 x =7.Assuming30daysinamonth,Ithink x =7 : 33shouldworkformybirthdayand p 3 : 33 17 : 5.Thewebsitesaystherewillbe about18 : 25hoursofdaylightthatday.Tohave14hoursofdarknessweneed10hoursof daylight.Weseethat p 3 : 96 10and p 3 : 05 10soitseemsreasonabletosaythat we'llhaveatleast14hoursofdarknessfromDecember21,2008 x =0toFebruary21,2009 x =2andthenagainfromOctober21,2009 x =10toDecember21,2009 x =12. Thequarticregressionmodelis p 4 x =0 : 0144 x 4 )]TJ/F15 10.9091 Tf 9.573 0 Td [(0 : 3507 x 3 +2 : 259 x 2 )]TJ/F15 10.9091 Tf 9.573 0 Td [(1 : 571 x +5 : 513.Ithas R 2 =0 : 98594whichisgood.Thegraphof y = p 4 x intheviewingwindow[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 15] [0 ; 35] alongwiththescatterplotisshownbelowontheright.Noticethat p 4 isabove24making thisabadmodelaswellforfuturepredictions.However, p 4 : 33 18 : 71makingitmuch betteratpredictingthehoursofdaylightonJuly31mybirthday.Thismodelsayswe'll haveatleast14hoursofdarknessfromDecember21,2008 x =0toaboutMarch1,2009 x =2 : 30andthenagainfromOctober10,2009 x =9 : 667toDecember21,2009 x =12. y = p 3 x y = p 4 x 8.aThescatterplotisshownbelowwitheachofthethreeregressionmodels. bThequadraticmodelis P 2 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 02 x 2 +0 : 241 x +0 : 956with R 2 =0 : 77708. Thecubicmodelis P 3 x =0 : 005 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 103 x 2 +0 : 602 x +0 : 573with R 2 =0 : 98153. Thequarticmodelis P 4 x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(0 : 000969 x 4 +0 : 0253 x 3 )]TJ/F15 10.9091 Tf 10.974 0 Td [(0 : 240 x 2 +0 : 944 x +0 : 330with R 2 =0 : 99929. cThemaximumspredictedbythethreemodelsare P 2 : 737 1 : 648, P 3 : 232 1 : 657 and P 4 : 784 1 : 630,respectively. y = P 2 x y = P 3 x y = P 4 x

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3.2TheFactorTheoremandTheRemainderTheorem187 3.2TheFactorTheoremandTheRemainderTheorem Supposewewishtondthezerosof f x = x 3 +4 x 2 )]TJ/F15 10.9091 Tf 11.217 0 Td [(5 x )]TJ/F15 10.9091 Tf 11.217 0 Td [(14.Setting f x =0resultsinthe polynomialequation x 3 +4 x 2 )]TJ/F15 10.9091 Tf 11.22 0 Td [(5 x )]TJ/F15 10.9091 Tf 11.22 0 Td [(14=0.Despiteallofthefactoringtechniqueswelearned 1 inIntermediateAlgebra,thisequationfoils 2 usateveryturn.Ifwegraph f usingthegraphing calculator,weget Thegraphsuggeststhat x =2isazero,andwecanverify f =0.Theothertwozerosseemto belessfriendly,and,eventhoughwecouldusethe`Zero'commandtonddecimalapproximations forthese,weseekamethodtondtheremainingzerosexactly.Basedonourexperience,if x =2 isazero,itseemsthatthereshouldbeafactorof x )]TJ/F15 10.9091 Tf 11.285 0 Td [(2lurkingaroundinthefactorizationof f x .Inotherwords,itseemsreasonabletoexpectthat x 3 +4 x 2 )]TJ/F15 10.9091 Tf 10.942 0 Td [(5 x )]TJ/F15 10.9091 Tf 10.942 0 Td [(14= x )]TJ/F15 10.9091 Tf 10.942 0 Td [(2 q x ,where q x issomeotherpolynomial.Howcouldwendsucha q x ,ifitevenexists?Theanswercomes fromouroldfriend,polynomialdivision.Dividing x 3 +4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(14by x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2gives x 2 +6 x +7 x )]TJ/F15 10.9091 Tf 9.091 0 Td [(2 x 3 +4 x 2 )]TJ/F15 10.9091 Tf 14.131 0 Td [(5 x )]TJ/F15 10.9091 Tf 11.403 0 Td [(14 )]TJ/F55 10.9091 Tf 8.685 8.837 Td [()]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 3 )]TJ/F15 10.9091 Tf 9.879 0 Td [(2 x 2 6 x 2 )]TJ/F15 10.9091 Tf 14.131 0 Td [(5 x )]TJ/F55 10.9091 Tf 8.684 8.836 Td [()]TJ/F15 10.9091 Tf 5 -8.836 Td [(6 x 2 )]TJ/F15 10.9091 Tf 9.88 0 Td [(12 x 7 x )]TJ/F15 10.9091 Tf 11.403 0 Td [(14 )]TJ/F15 10.9091 Tf 11.412 0 Td [( x )]TJ/F15 10.9091 Tf 9.879 0 Td [(14 0 Asyoumayrecall,thismeans x 3 +4 x 2 )]TJ/F15 10.9091 Tf 10.71 0 Td [(5 x )]TJ/F15 10.9091 Tf 10.71 0 Td [(14= x )]TJ/F15 10.9091 Tf 10.709 0 Td [(2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 +6 x +7 ,andsotondthezeros of f ,wenowcansolve x )]TJ/F15 10.9091 Tf 11.346 0 Td [(2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 +6 x +7 =0.Weget x )]TJ/F15 10.9091 Tf 11.346 0 Td [(2=0whichgivesusourknown zero, x =2aswellas x 2 +6 x +7=0.Thelatterdoesn'tfactornicely,soweapplytheQuadratic Formulatoget x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 p 2.Thepointofthissectionistogeneralizethetechniqueappliedhere. Firstupisafriendlyreminderofwhatwecanexpectwhenwedividepolynomials. Theorem 3.4 PolynomialDivision: Suppose d x and p x arenonzeropolynomialswhere thedegreeof p isgreaterthanorequaltothedegreeof d .Thereexisttwouniquepolynomials, q x and r x ,suchthat p x = d x q x + r x ; whereeither r x =0orthedegreeof r is strictlylessthanthedegreeof d 1 andprobablyforgot 2 punintended

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188PolynomialFunctions Asyoumayrecall,allofthepolynomialsinTheorem3.4havespecialnames.Thepolynomial p iscalledthe dividend ; d isthe divisor ; q isthe quotient ; r isthe remainder .If r x =0then d iscalleda factor of p .TheproofofTheorem3.4isusuallyrelegatedtoacourseinAbstract Algebra, 3 butwewillusetheresulttoestablishtwoimportantfactswhicharethebasisoftherest ofthechapter. Theorem 3.5 TheRemainderTheorem: Suppose p isapolynomialofdegreeatleast1and c isarealnumber.When p x isdividedby x )]TJ/F53 10.9091 Tf 10.91 0 Td [(c theremainderis p c TheproofofTheorem3.5isadirectconsequenceofTheorem3.4.Whenapolynomialisdivided by x )]TJ/F53 10.9091 Tf 10.451 0 Td [(c ,theremainderiseither0orhasdegreelessthanthedegreeof x )]TJ/F53 10.9091 Tf 10.452 0 Td [(c .Since x )]TJ/F53 10.9091 Tf 10.451 0 Td [(c isdegree 1,thismeansthedegreeoftheremaindermustbe0,whichmeanstheremainderisaconstant. Hence,ineithercase, p x = x )]TJ/F53 10.9091 Tf 11.202 0 Td [(c q x + r ,where r ,theremainder,isarealnumber,possibly 0.Itfollowsthat p c = c )]TJ/F53 10.9091 Tf 11.306 0 Td [(c q c + r =0 q c + r = r ,andsoweget r = p c ,asrequired. Thereisonelast`lowhangingfruit' 4 tocollect-itisanimmediateconsequenceofTheRemainder Theorem. Theorem 3.6 TheFactorTheorem: Suppose p isanonzeropolynomial.Therealnumber c isazeroof p ifandonlyif x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c isafactorof p x TheproofofTheFactorTheoremisaconsequenceofwhatwealreadyknow.If x )]TJ/F53 10.9091 Tf 10.757 0 Td [(c isafactor of p x ,thismeans p x = x )]TJ/F53 10.9091 Tf 9.805 0 Td [(c q x forsomepolynomial q .Hence, p c = c )]TJ/F53 10.9091 Tf 9.805 0 Td [(c q c =0,andso c isazeroof p .Conversely,if c isazeroof p ,then p c =0.Inthiscase,TheRemainderTheorem tellsustheremainderwhen p x isdividedby x )]TJ/F53 10.9091 Tf 10.922 0 Td [(c ,namely p c ,is0,whichmeans x )]TJ/F53 10.9091 Tf 10.922 0 Td [(c isa factorof p .Whatwehaveestablishedisthefundamentalconnectionbetweenzerosofpolynomials andfactorsofpolynomials. OfthethingsTheFactorTheoremtellsus,themostpragmaticisthatwehadbetterndamore ecientwaytodividepolynomialsbyquantitiesoftheform x )]TJ/F53 10.9091 Tf 10.507 0 Td [(c .Fortunately,peoplelikeRuni andHorner havealreadyblazedthistrail.Let'stakeacloserlookatthelongdivisionweperformed atthebeginningofthesectionandtrytostreamlineit.Firsto,let'schangeallofthesubtractions intoadditionsbydistributingthroughthe )]TJ/F15 10.9091 Tf 8.485 0 Td [(1s. x 2 +6 x +7 x )]TJ/F15 10.9091 Tf 8.684 0 Td [(2 x 3 +4 x 2 )]TJ/F15 10.9091 Tf 12.926 0 Td [(5 x )]TJ/F15 10.9091 Tf 8.684 0 Td [(14 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 3 +2 x 2 6 x 2 )]TJ/F15 10.9091 Tf 12.926 0 Td [(5 x )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x 2 +12 x 7 x )]TJ/F15 10.9091 Tf 8.684 0 Td [(14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 x +14 0 Next,observethattheterms )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 x 2 ,and )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 x aretheexactoppositeofthetermsabovethem. Thealgorithmweuseensuresthisisalwaysthecase,sowecanomitthemwithoutlosingany 3 Yes,Virginia,therearealgebracoursesmoreabstractthanthisone. 4 JehatesthisexpressionandCarlincludeditjusttoannoyhim.

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3.2TheFactorTheoremandTheRemainderTheorem189 information.Alsonotethatthetermswe`bringdown'namelythe )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x and )]TJ/F15 10.9091 Tf 8.485 0 Td [(14aren'treally necessarytorecopy,andsoweomitthem,too. x 2 +6 x +7 x )]TJ/F15 10.9091 Tf 8.684 0 Td [(2 x 3 +4 x 2 )]TJ/F15 10.9091 Tf 11.411 0 Td [(5 x )]TJ/F15 10.9091 Tf 8.684 0 Td [(14 2 x 2 6 x 2 12 x 7 x 14 0 Now,let'smovethingsupabitand,forreasonswhichwillbecomeclearinamoment,copythe x 3 intothelastrow. x 2 +6 x +7 x )]TJ/F15 10.9091 Tf 8.684 0 Td [(2 x 3 +4 x 2 )]TJ/F15 10.9091 Tf 11.411 0 Td [(5 x )]TJ/F15 10.9091 Tf 8.684 0 Td [(14 2 x 2 12 x 14 x 3 6 x 2 7 x 0 Notethatbyarrangingthingsinthismanner,eachterminthelastrowisobtainedbyaddingthe twotermsaboveit.Noticealsothatthequotientpolynomialcanbeobtainedbydividingeachof therstthreetermsinthelastrowby x andaddingtheresults.Ifyoutakethetimetoworkback throughtheoriginaldivisionproblem,youwillndthatthisisexactlythewaywedeterminedthe quotientpolynomial.Thismeansthatwenolongerneedtowritethequotientpolynomialdown, northe x inthedivisor,todetermineouranswer. )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 3 +4 x 2 )]TJ/F15 10.9091 Tf 11.411 0 Td [(5 x )]TJ/F15 10.9091 Tf 8.684 0 Td [(14 2 x 2 12 x 14 x 3 6 x 2 7 x 0 We'vestreamlinedthingsquiteabitsofar,butwecanstilldomore.Let'stakeamomentto remindourselveswherethe2 x 2 ,12 x ,and14camefrominthesecondrow.Eachoftheseterms wasobtainedbymultiplyingthetermsinthequotient, x 2 ,6 x and7,respectively,bythe )]TJ/F15 10.9091 Tf 8.484 0 Td [(2in x )]TJ/F15 10.9091 Tf 11.213 0 Td [(2,thenby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1whenwechangedthesubtractiontoaddition.Multiplyingby )]TJ/F15 10.9091 Tf 8.485 0 Td [(2thenby )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 isthesameasmultiplyingby2,andsowereplacethe )]TJ/F15 10.9091 Tf 8.484 0 Td [(2inthedivisorby2.Furthermore,the coecientsofthequotientpolynomialmatchthecoecientsoftherstthreetermsinthelastrow, sowenowtaketheplungeandwriteonlythecoecientsofthetermstoget 2 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 21214 1670

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190PolynomialFunctions Wehaveconstructedisthe syntheticdivisiontableau forthispolynomialdivisionproblem. Let'sre-workourdivisionproblemusingthistableautoseehowitgreatlystreamlinesthedivision process.Todivide x 3 +4 x 2 )]TJ/F15 10.9091 Tf 11.509 0 Td [(5 x )]TJ/F15 10.9091 Tf 11.509 0 Td [(14by x )]TJ/F15 10.9091 Tf 11.509 0 Td [(2,wewrite2intheplaceofthedivisorandthe coecientsof x 3 +4 x 2 )]TJ/F15 10.9091 Tf 10.759 0 Td [(5 x )]TJ/F15 10.9091 Tf 10.759 0 Td [(14inforthedividend.Then`bringdown'therstcoecientofthe dividend. 2 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 2 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 # 1 Next,takethe2fromthedivisorandmultiplybythe1thatwas`broughtdown'toget2.Write thisunderneaththe4,thenaddtoget6. 2 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 # 2 1 2 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 # 2 16 Nowtakethe2fromthedivisortimesthe6toget12,andaddittothe )]TJ/F15 10.9091 Tf 8.485 0 Td [(5toget7. 2 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 # 212 16 2 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 # 212 167 Finally,takethe2inthedivisortimesthe7toget14,andaddittothe )]TJ/F15 10.9091 Tf 8.485 0 Td [(14toget0. 2 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 # 21214 167 2 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 # 21214 167 0 Therstthreenumbersinthelastrowofourtableauarethecoecientsofthequotientpolynomial. Remember,westartedwithathirddegreepolynomialanddividedbyarstdegreepolynomial,so thequotientisaseconddegreepolynomial.Hencethequotientis x 2 +6 x +7.Thenumberinthe boxistheremainder.Syntheticdivisionisourtoolofchoicefordividingpolynomialsbydivisors oftheform x )]TJ/F53 10.9091 Tf 11.438 0 Td [(c .Itisimportanttonotethatitworks only forthesekindsofdivisors. 5 Also takenotethatwhenapolynomialofdegreeatleast1isdividedby x )]TJ/F53 10.9091 Tf 11.365 0 Td [(c ,theresultwillbea polynomialofexactlyonelessdegree.Finally,itisworththetimetotraceeachstepinsynthetic divisionbacktoitscorrespondingstepinlongdivision.Whiletheauthorshavedonetheirbestto indicatewherethealgorithmcomesfrom,thereisnosubstituteforworkingthroughityourself. Example 3.2.1 Usesyntheticdivisiontoperformthefollowingpolynomialdivisions.Findthe quotientandtheremainderpolynomials,thenwritethedividend,quotientandremainderinthe formgiveninTheorem3.4. 5 You'llneedtousegoodold-fashionedpolynomiallongdivisionfordivisorsofdegreelargerthan1.

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3.2TheFactorTheoremandTheRemainderTheorem191 1. )]TJ/F15 10.9091 Tf 5 -8.836 Td [(5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 +1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2. )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 3 +8 x +2 3. 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 2 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 Solution. 1.Whensettingupthesyntheticdivisiontableau,weneedtoenter0forthecoecientof x in thedividend.Doingsogives 3 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(201 # 1539117 51339 118 Sincethedividendwasathirddegreepolynomial,thequotientisaquadraticpolynomial withcoecients5,13and39.Ourquotientis q x =5 x 2 +13 x +39andtheremainderis r x =118.AccordingtoTheorem3.4,wehave5 x 3 )]TJ/F15 10.9091 Tf 9.12 0 Td [(2 x 2 +1= x )]TJ/F15 10.9091 Tf 9.12 0 Td [(3 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(5 x 2 +13 x +39 +118. 2.Forthisdivision,werewrite x +2as x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(2andproceedasbefore )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1008 #)]TJ/F15 10.9091 Tf 23.902 0 Td [(24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 0 Wegetthequotient q x = x 2 )]TJ/F15 10.9091 Tf 11.002 0 Td [(2 x +4andtheremainder r x =0.Relatingthedividend, quotientandremaindergives x 3 +8= x +2 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +4 3.Todivide4 )]TJ/F15 10.9091 Tf 11.291 0 Td [(8 x )]TJ/F15 10.9091 Tf 11.291 0 Td [(12 x 2 by2 x )]TJ/F15 10.9091 Tf 11.291 0 Td [(3,twothingsmustbedone.First,wewritethedividend indescendingpowersof x as )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 x 2 )]TJ/F15 10.9091 Tf 11.452 0 Td [(8 x +4.Second,sincesyntheticdivisionworksonly forfactorsoftheform x )]TJ/F53 10.9091 Tf 11.359 0 Td [(c ,wefactor2 x )]TJ/F15 10.9091 Tf 11.358 0 Td [(3as2 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(3 2 .Ourstrategyistorstdivide )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 x 2 )]TJ/F15 10.9091 Tf 10.114 0 Td [(8 x +4by2,toget )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.114 0 Td [(4 x +2.Next,wedivideby )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(3 2 .Thetableaubecomes 3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(42 #)]TJ/F15 10.9091 Tf 29.356 0 Td [(9 )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(39 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(35 2 Fromthis,weget )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x 2 )]TJ/F15 10.9091 Tf 11.59 0 Td [(4 x +2= )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x )]TJ/F15 10.9091 Tf 11.591 0 Td [(13 )]TJ/F34 7.9701 Tf 12.787 4.295 Td [(35 2 .Multiplyingbothsidesby2and distributinggives )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 x 2 )]TJ/F15 10.9091 Tf 11.736 0 Td [(8 x +4= x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x )]TJ/F15 10.9091 Tf 11.736 0 Td [(13 )]TJ/F15 10.9091 Tf 11.737 0 Td [(35.Atthisstage,wehavewritten )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 x 2 )]TJ/F15 10.9091 Tf 10.114 0 Td [(8 x +4inthe form x )]TJ/F15 10.9091 Tf 10.114 0 Td [(3 q x + r x ,buthowcanwebesurethequotientpolynomialis )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x )]TJ/F15 10.9091 Tf 9.76 0 Td [(13andtheremainderis )]TJ/F15 10.9091 Tf 8.485 0 Td [(35?Theansweristheword`unique'inTheorem3.4.Thetheorem statesthatthereisonlyonewaytodecompose )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 x 2 )]TJ/F15 10.9091 Tf 11.071 0 Td [(8 x +4intoamultipleof x )]TJ/F15 10.9091 Tf 11.071 0 Td [(3plusa constantterm.Sincewehavefoundsuchaway,wecanbesureitistheonlyway. Thenextexamplepullstogetheralloftheconceptsdiscussedinthissection.

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192PolynomialFunctions Example 3.2.2 Let p x =2 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x +3. 1.Find p )]TJ/F15 10.9091 Tf 8.485 0 Td [(2usingTheRemainderTheorem.Checkyouranswerbysubstitution. 2.Usethefactthat x =1isazeroof p tofactor p x andndalloftherealzerosof p Solution. 1.TheRemainderTheoremstates p )]TJ/F15 10.9091 Tf 8.485 0 Td [(2istheremainderwhen p x isdividedby x )]TJ/F15 10.9091 Tf 11.421 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(2. Wesetupoursyntheticdivisiontableaubelow.Wearecarefultorecordthecoecientof x 2 as0,andproceedasabove. )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(53 #)]TJ/F15 10.9091 Tf 23.902 0 Td [(48 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(43 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 AccordingtotheRemainderTheorem, p )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.Wecancheckthisbydirectsubstitution intotheformulafor p x : p )]TJ/F15 10.9091 Tf 8.485 0 Td [(2=2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+3= )]TJ/F15 10.9091 Tf 8.485 0 Td [(16+10+3= )]TJ/F15 10.9091 Tf 8.485 0 Td [(3. 2.TheFactorTheoremtellsusthatsince x =1isazeroof p x )]TJ/F15 10.9091 Tf 9.951 0 Td [(1isafactorof p x .Tofactor p x ,wedivide 1 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(53 # 22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 Wegetaremainderof0whichveriesthat,indeed, p =0.Ourquotientpolynomialisa seconddegreepolynomialwithcoecients2,2,and )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.So q x =2 x 2 +2 x )]TJ/F15 10.9091 Tf 11.145 0 Td [(3.Theorem 3.4tellsus p x = x )]TJ/F15 10.9091 Tf 11.096 0 Td [(1 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 x 2 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 .Tondtheremainingrealzerosof p ,weneedto solve2 x 2 +2 x )]TJ/F15 10.9091 Tf 10.418 0 Td [(3=0for x .Sincethisdoesn'tfactornicely,weusethequadraticformulato ndthattheremainingzerosare x = )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 p 7 2 InSection3.1,wediscussedthenotionofthemultiplicityofazero.Roughlyspeaking,azerowith multiplicity2canbedividedtwiceintoapolynomial;multiplicity3,threetimesandsoon.This isillustratedinthenextexample. Example 3.2.3 Let p x =4 x 4 )]TJ/F15 10.9091 Tf 10.612 0 Td [(4 x 3 )]TJ/F15 10.9091 Tf 10.612 0 Td [(11 x 2 +12 x )]TJ/F15 10.9091 Tf 10.612 0 Td [(3.Giventhat x = 1 2 isazeroofmultiplicity 2,ndalloftherealzerosof p Solution. Wesetupforsyntheticdivision.Sincewearetoldthemultiplicityof 1 2 istwo,we continueourtableauanddivide 1 2 intothequotientpolynomial 1 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1112 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 # 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(63 1 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(126 0 # 20 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 40 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 0

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3.2TheFactorTheoremandTheRemainderTheorem193 Fromtherstdivision,weget4 x 4 )]TJ/F15 10.9091 Tf 11.098 0 Td [(4 x 3 )]TJ/F15 10.9091 Tf 11.098 0 Td [(11 x 2 +12 x )]TJ/F15 10.9091 Tf 11.098 0 Td [(3= )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 )]TJ/F15 10.9091 Tf 11.818 -8.836 Td [(4 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x +6 .The seconddivisiontellsus4 x 3 )]TJ/F15 10.9091 Tf 11.487 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 11.487 0 Td [(12 x +6= )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 2 )]TJ/F15 10.9091 Tf 11.818 -8.836 Td [(4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 .Combiningtheseresults,we have4 x 4 )]TJ/F15 10.9091 Tf 11.144 0 Td [(4 x 3 )]TJ/F15 10.9091 Tf 11.144 0 Td [(11 x 2 +12 x )]TJ/F15 10.9091 Tf 11.144 0 Td [(3= )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 .Tondtheremainingzerosof p ,weset 4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12=0andget x = p 3. Acoupleofthingsaboutthelastexampleareworthmentioning.First,theextensionofthe syntheticdivisiontableauforrepeateddivisionswillbeacommonsiteinthesectionstocome. Typically,wewillstartwithahigherorderpolynomialandpeeloonezeroatatimeuntilweare leftwithaquadratic,whoserootscanalwaysbefoundusingtheQuadraticFormula.Secondly,we found x = p 3arezerosof p .TheFactorTheoremguarantees )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x )]TJ 10.909 9.025 Td [(p 3 and )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x )]TJ/F55 10.9091 Tf 10.909 8.837 Td [()]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ 8.485 9.025 Td [(p 3 are bothfactorsof p .WecancertainlyputtheFactorTheoremtothetestandcontinuethesynthetic divisiontableaufromabovetoseewhathappens. 1 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1112 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 # 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(63 1 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(126 0 # 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 p 3 40 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 0 # 4 p 312 )]TJ 8.485 9.025 Td [(p 3 44 p 3 0 #)]TJ/F15 10.9091 Tf 23.903 0 Td [(4 p 3 4 0 Thisgivesus4 x 4 )]TJ/F15 10.9091 Tf 11.085 0 Td [(4 x 3 )]TJ/F15 10.9091 Tf 11.085 0 Td [(11 x 2 +12 x )]TJ/F15 10.9091 Tf 11.086 0 Td [(3= )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 2 2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x )]TJ 10.909 9.024 Td [(p 3 )]TJ/F53 10.9091 Tf 11.818 -8.836 Td [(x )]TJ/F55 10.9091 Tf 10.909 8.836 Td [()]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ 8.485 9.024 Td [(p 3 ,or,whenwritten withtheconstantinfront p x =4 x )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 2 x )]TJ 10.909 9.556 Td [(p 3 x )]TJ/F55 10.9091 Tf 10.909 12.109 Td [( )]TJ 8.485 9.556 Td [(p 3 Wehaveshownthat p isaproductofitsleadingtermtimeslinearfactorsoftheform x )]TJ/F53 10.9091 Tf 10.001 0 Td [(c where c arezerosof p .Itmaysurpriseanddelightthereaderthat,intheory,allpolynomialscanbe reducedtothiskindoffactorization.WeleavethatdiscussiontoSection3.4,becausethezeros maynotberealnumbers.Ournaltheoreminthesectiongivesusanupperboundonthenumber ofrealzeros. Theorem 3.7 Suppose f isapolynomialofdegree n n 1.Then f hasatmost n realzeros, countingmultiplicities. Theorem3.7isaconsequenceoftheFactorTheoremandpolynomialmultiplication.Everyzero c of f givesusafactoroftheform x )]TJ/F53 10.9091 Tf 10.762 0 Td [(c for f x .Since f hasdegree n ,therecanbeatmost n of thesefactors.Thenextsectionprovidesussometoolswhichnotonlyhelpusdeterminewherethe realzerosaretobefound,butwhichrealnumberstheymaybe. Weclosethissectionwithasummaryofseveralconceptspreviouslypresented.Youshouldtake thetimetolookbackthroughthetexttoseewhereeachconceptwasrstintroducedandwhere eachconnectiontotheotherconceptswasmade.

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194PolynomialFunctions ConnectionsBetweenZeros,FactorsandGraphsofPolynomialFunctions Suppose p isapolynomialfunctionofdegree n 1.Thefollowingstatementsareequivalent: Therealnumber c isazeroof p p c =0 x = c isasolutiontothepolynomialequation p x =0 x )]TJ/F53 10.9091 Tf 10.91 0 Td [(c isafactorof p x Thepoint c; 0isan x -interceptofthegraphof y = p x

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3.2TheFactorTheoremandTheRemainderTheorem195 3.2.1Exercises 1.AnIntermediateAlgebrareviewexerciseUsepolynomiallongdivisiontoperformtheindicateddivision.Writethepolynomialintheform p x = d x q x + r x a x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 +2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 +4 b )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 5 +7 x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 +1 c x 3 +5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 d x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(23 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2.UsesyntheticdivisionandtheRemainderTheoremtotestwhetherornotthegivennumber isazeroofthepolynomial p x =15 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(121 x 4 +17 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(73 x 2 +2 x +48. a c = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 b c =8 c c = 1 2 d c = 2 3 e c =0 f c = )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(3 5 3.Foreachpolynomialgivenbelow,youaregivenoneofitszeros.Usethetechniquesinthis sectiontondtherestoftherealzerosandfactorthepolynomial. a x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +11 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 ;c =1 b x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(24 x 2 +192 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(512 ;c =8 c4 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(28 x 3 +61 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(42 x +9 ;c = 1 2 d3 x 3 +4 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ;c = 2 3 e x 4 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 ;c =0 f x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ;c =1 )]TJ 10.91 9.024 Td [(p 3 g125 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(275 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2265 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3213 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1728 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(324 ;c = )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(3 5 4.Createapolynomial p withthefollowingattributes. As x ;p x !1 Thepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0yieldsalocalmaximum. Thedegreeof p is5. Thepoint ; 0isoneofthe x -interceptsofthegraphof p 5.Findaquadraticpolynomialwithinteger coecientswhichhas x = 3 5 p 29 5 asitsrealzeros.

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196PolynomialFunctions 3.2.2Answers 1.a5 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 +2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1= x 2 +4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(18+ x +71 b )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 5 +7 x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x = x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 +1 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x +6+ x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 c9 x 3 +5= x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 9 2 x 2 + 27 4 x + 81 8 + 283 8 d4 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(23= x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ )]TJ/F53 10.9091 Tf 8.485 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(19 2.a p )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(100 b p =0 c p 1 2 = 825 32 d p 2 3 =0 e p =48 f p )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 5 =0 3.a x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +11 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6= x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 b x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(24 x 2 +192 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(512= x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 3 c4 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(28 x 3 +61 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(42 x +9=4 x )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 2 2 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 2 d3 x 3 +4 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2=3 x )]TJ/F34 7.9701 Tf 12.104 4.296 Td [(2 3 x +1 2 e x 4 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = x 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x +1 f x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2= x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ 10.909 9.024 Td [(p 3 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(+ p 3 g125 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(275 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2265 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3213 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1728 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(324=125 x + 3 5 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x +2 4.Somethinglike p x = )]TJ/F15 10.9091 Tf 8.485 0 Td [( x +2 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x +3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4willwork. 5. q x =5 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4

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3.3RealZerosofPolynomials197 3.3RealZerosofPolynomials InSection3.2,wefoundthatwecanusesyntheticdivisiontodetermineifagivenrealnumberis azeroofapolynomialfunction.Thissectionpresentsresultswhichwillhelpusdeterminegood candidatestotestusingsyntheticdivision.Therearetwoapproachestothetopicofndingthe realzerosofapolynomial.Therstapproachwhichisgainingpopularityistousealittlebitof mathematicsfollowedbyagooduseoftechnologylikegraphingcalculators.Thesecondapproach forpuristsmakesgooduseofmathematicalmachinerytheoremsonly.Forcompleteness,we includethetwoapproachesbutinseparatesubsections. 1 Bothapproachesbenetfromthefollowing twotheorems,therstofwhichisduetothefamousmathematicianAugustinCauchy .Itgivesus anintervalonwhichalloftherealzerosofapolynomialcanbefound. Theorem 3.8 Cauchy'sBound: Suppose f x = a n x n + a n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 x n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 + ::: + a 1 x + a 0 isapolynomial ofdegree n with n 1.Let M bethelargestofthenumbers: j a 0 j j a n j j a 1 j j a n j ,..., j a n )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 j j a n j .Thenallthe realzerosof f lieinintheinterval[ )]TJ/F15 10.9091 Tf 8.485 0 Td [( M +1 ;M +1]. Theproofofthisfactisnoteasilyexplainedwithintheconnesofthistext.Thispaper contains theresultandgivesreferencestoitsproof.Likemanyoftheresultsinthissection,Cauchy'sBound isbestunderstoodwithanexample. Example 3.3.1 Let f x =2 x 4 +4 x 3 )]TJ/F53 10.9091 Tf 10.987 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.987 0 Td [(6 x )]TJ/F15 10.9091 Tf 10.988 0 Td [(3.Determineanintervalwhichcontainsallof therealzerosof f Solution. Cauchy'sBoundsaystotaketheabsolutevalueofeachofthenon-leadingcoecients of f ,namely,4,1,6and3,anddividethembytheabsolutevalueoftheleadingcoecient,2. Doingsoproducesthelistofnumbers2, 1 2 ,3,and 3 2 .Next,wetakethelargestofthesevalues,3, asourvalue M inthetheoremandaddonetoittoget4.Therealzerosof f areguaranteedtolie intheinterval[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 4]. Whereasthepreviousresulttellsuswherewecanndtherealzerosofapolynomial,thenext theoremgivesusalistofpossiblerealzeros. Theorem 3.9 RationalZerosTheorem: Suppose f x = a n x n + a n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 x n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 + ::: + a 1 x + a 0 isapolynomialofdegree n with n 1,and a 0 a 1 ,... a n areintegers.If r isarationalzeroof f ,then r isoftheform p q ,where p isafactoroftheconstantterm a 0 ,and q isafactorofthe leadingcoecient a n Therationalzerostheoremgivesusalistofnumberstotryinoursyntheticdivisionandthat isalotnicerthansimplyguessing.Ifnoneofthenumbersinthelistarezeros,theneitherthe polynomialhasnorealzerosatall,oralloftherealzerosareirrationalnumbers.Toseewhythe RationalZerosTheoremworks,suppose c isazeroof f and c = p q inlowestterms.Thismeans p and q havenocommonfactors.Since f c =0,wehave a n p q n + a n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 p q n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 + ::: + a 1 p q + a 0 =0 : 1 Carlisthepuristandisresponsibleforallofthetheoremsinthissection.Je,ontheotherhand,hasspenttoo muchtimeinschoolpoliticsandhasbeenpollutedwithnotionsof`compromise.'Youcanblametheslowdeclineof civilizationonhimandthoselikehimwhominglemathematicswithtechnology.

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198PolynomialFunctions Multiplyingbothsidesofthisequationby q n ,weclearthedenominatorstoget a n p n + a n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 p n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 q + ::: + a 1 pq n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 + a 0 q n =0 Rearrangingthisequation,weget a n p n = )]TJ/F53 10.9091 Tf 8.485 0 Td [(a n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 p n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 q )]TJ/F53 10.9091 Tf 10.909 0 Td [(::: )]TJ/F53 10.9091 Tf 10.909 0 Td [(a 1 pq n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(a 0 q n Now,thelefthandsideisanintegermultipleof p ,andtherighthandsideisanintegermultipleof q .Canyouseewhy?Thismeans a n p n isbothamultipleof p andamultipleof q .Since p and q havenocommonfactors, a n mustbeamultipleof q .Ifwerearrangetheequation a n p n + a n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 p n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 q + ::: + a 1 pq n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 + a 0 q n =0 as a 0 q n = )]TJ/F53 10.9091 Tf 8.485 0 Td [(a n p n )]TJ/F53 10.9091 Tf 10.909 0 Td [(a n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 p n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 q )]TJ/F53 10.9091 Tf 10.909 0 Td [(::: )]TJ/F53 10.9091 Tf 10.909 0 Td [(a 1 pq n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 wecanplaythesamegameandconclude a 0 isamultipleof p ,andwehavetheresult. Example 3.3.2 Let f x =2 x 4 +4 x 3 )]TJ/F53 10.9091 Tf 11.084 0 Td [(x 2 )]TJ/F15 10.9091 Tf 11.085 0 Td [(6 x )]TJ/F15 10.9091 Tf 11.085 0 Td [(3.UsetheRationalZerosTheoremtolistall possiblerationalzerosof f Solution. Togenerateacompletelistofrationalzeros,weneedtotakeeachofthefactorsof constantterm, a 0 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3,anddividethembyeachofthefactorsoftheleadingcoecient a 4 =2. Thefactorsof )]TJ/F15 10.9091 Tf 8.485 0 Td [(3are 1and 3.SincetheRationalZerosTheoremtacksona anyway,for themoment,weconsideronlythepositivefactors1and3.Thefactorsof2are1and2,sothe RationalZerosTheoremgivesthelist 1 1 ; 1 2 ; 3 1 ; 3 2 or 1 2 ; 1 ; 3 2 ; 3 Ourdiscussionnowdivergesbetweenthosewhowishtousetechnologyandthosewhodonot. 3.3.1ForThoseWishingtouseaGraphingCalculator Atthisstage,weknownotonlytheintervalinwhichallofthezerosof f x =2 x 4 +4 x 3 )]TJ/F53 10.9091 Tf 9.258 0 Td [(x 2 )]TJ/F15 10.9091 Tf 9.259 0 Td [(6 x )]TJ/F15 10.9091 Tf 9.258 0 Td [(3 arelocated,butwealsoknowsomepotentialcandidates.Wecannowuseourcalculatortohelp usdeterminealloftherealzerosof f ,asillustratedinthenextexample. Example 3.3.3 Let f x =2 x 4 +4 x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3. 1.Graph y = f x onthecalculatorusingtheintervalobtainedinExample3.3.1asaguide. 2.Usethegraphtohelpnarrowdownthelistofcandidatesforrationalzerosyouobtainedin Example3.3.2. 3.Usesyntheticdivisiontondtherealzerosof f ,andstatetheirmultiplicities. Solution. 1.InExample3.3.1,wedeterminedalloftherealzerosof f lieintheinterval[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 4].Weset ourwindowaccordinglyandget

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3.3RealZerosofPolynomials199 2.InExample3.3.2,welearnedthatanyrationalzeroof f mustbeinthelist 1 2 ; 1 ; 3 2 ; 3 Fromthegraph,itlooksasifwecanruleoutanyofthepositiverationalzeros,sincethe graphseemstocrossthe x -axisatavaluejustalittlegreaterthan1.Onthenegativeside, )]TJ/F15 10.9091 Tf 8.485 0 Td [(1looksgood,sowetrythatforoursyntheticdivision. )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 #)]TJ/F15 10.9091 Tf 23.902 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(233 22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 Wehaveawinner!Rememberingthat f wasafourthdegreepolynomial,wenowknowour quotientisathirddegreepolynomial.Ifwecandoonemoresuccessfuldivision,wewillhave knockedthequotientdowntoaquadratic,and,ifallelsefails,wecanusethequadratic formulatondthelasttwozeros.Sincethereseemstobenootherrationalzerostotry,we continuewith )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Also,theshapeofthecrossingat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1leadsustowonderifthezero x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1hasmultiplicity3. )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 #)]TJ/F15 10.9091 Tf 23.902 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(233 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 #)]TJ/F15 10.9091 Tf 23.902 0 Td [(203 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 Success!Ourquotientpolynomialisnow2 x 2 )]TJ/F15 10.9091 Tf 11.063 0 Td [(3.Settingthistozerogives2 x 2 )]TJ/F15 10.9091 Tf 11.063 0 Td [(3=0,or x 2 = 3 2 ,whichgivesus x = p 6 2 .Concerningmultiplicities,basedonourdivision,wehave that )]TJ/F15 10.9091 Tf 8.485 0 Td [(1hasamultiplicityofatleast2.TheFactorTheoremtellsusourremainingzeros, p 6 2 ,eachhavemultiplicityatleast1.However,Theorem3.7tellsus f canhaveatmost4 realzeros,countingmultiplicity,andsoweconcludethat )]TJ/F15 10.9091 Tf 8.485 0 Td [(1isofmultiplicityexactly2and p 6 2 eachhasmultiplicity1.Thus,wewerewrong tothinkthat )]TJ/F15 10.9091 Tf 8.484 0 Td [(1hadmultiplicity3. Itisinterestingtonotethatwecouldgreatlyimproveonthegraphof y = f x intheprevious examplegiventousbythecalculator.Forinstance,fromourdeterminationofthezerosof f and theirmultiplicities,weknowthegraphcrossesat x = )]TJ/F40 7.9701 Tf 9.681 10.993 Td [(p 6 2 )]TJ/F15 10.9091 Tf 20.785 0 Td [(1 : 22thenturnsbackupwardsto touchthe x )]TJ/F15 10.9091 Tf 8.485 0 Td [(axisat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Thistellsusthat,despitewhatthecalculatorshowedusthersttime, thereisarelativemaximumoccurringat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1andnota`attenedcrossing'asweoriginally

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200PolynomialFunctions believed.Afterresizingthewindow,weseenotonlytherelativemaximumbutalsoarelative minimumjusttotheleftof x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1whichshowsus,onceagain,thatMathematicsenhancesthe technology,insteadofvice-versa. Ournextexampleshowshowevenamild-manneredpolynomialcancauseproblems. Example 3.3.4 Let f x = x 4 + x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12. 1.UseCauchy'sBoundtodetermineanintervalinwhichalloftherealzerosof f lie. 2.UsetheRationalZerosTheoremtodeterminealistofpossiblerationalzerosof f 3.Graph y = f x usingyourgraphingcalculator. 4.Findalloftherealzerosof f andtheirmultiplicities. Solution. 1.ApplyingCauchy'sBound,wend M =12,soalloftherealzeroslieintheinterval[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 ; 13]. 2.ApplyingtheRationalZerosTheoremwithconstantterm a 0 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(12andleadingcoecient a 4 =1,wegetthelist f 1, 2, 3, 4, 6, 12 g 3.Graphing y = f x ontheinterval[ )]TJ/F15 10.9091 Tf 8.484 0 Td [(13 ; 13]producesthegraphbelowontheleft.Zooming inabitgivesthegraphbelowontheright.Basedonthegraph,noneofourrationalzeros willwork.Doyouseewhynot?

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3.3RealZerosofPolynomials201 4.Fromthegraph,weknow f hastworealzeros,onepositive,andonenegative.Ouronlyhope atthispointistotryandndthezerosof f bysetting f x =0andsolving.Doingsoresults intheequation x 4 + x 2 )]TJ/F15 10.9091 Tf 10.606 0 Td [(12=0.Ifwestareatthisequationlongenough,wemayrecognize itasa`quadraticindisguise'. 2 Inotherwords,wehavethreeterms: x 4 x 2 ,and12,andthe exponentontherstterm, x 4 ,isexactlytwicethatofthesecondterm, x 2 .Wemayrewrite thisas )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 2 + )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F15 10.9091 Tf 9.722 0 Td [(12=0.Tobetterseetheforestforthetrees,letusmomentarilyreplace x 2 withthevariable u .Intermsof u ,ourequationbecomes u 2 + u )]TJ/F15 10.9091 Tf 10.608 0 Td [(12=0.Wecanreadily factorthelefthandsideofthisequationas u +4 u )]TJ/F15 10.9091 Tf 10.247 0 Td [(3=0,whichmeanswehavefactored thelefthandsideof x 4 + x 2 )]TJ/F15 10.9091 Tf 10.176 0 Td [(12=0as )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F53 10.9091 Tf 11.818 -8.837 Td [(x 2 +4 =0.Weget x 2 =3,whichgivesus x = p 3,or x 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4,whichadmitsnorealsolutions.Since p 3 1 : 73,thetwozerosmatch whatweexpectedfromthegraph.Intermsofmultiplicity,theFactorTheoremguarantees )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x )]TJ 10.909 9.025 Td [(p 3 and )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x + p 3 arefactorsof f x .Wenotethatourworkforndingthezerosof f shows f x canbefactoredas f x = )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F53 10.9091 Tf 11.818 -8.837 Td [(x 2 +4 .Since x 2 +4hasnorealzeros,the quantities )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x )]TJ 10.909 9.024 Td [(p 3 and )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x + p 3 mustbothbefactorsof x 2 )]TJ/F15 10.9091 Tf 9.95 0 Td [(3.AccordingtoTheorem3.7, x 2 )]TJ/F15 10.9091 Tf 11.213 0 Td [(3canhaveatmost2zeros,countingmultiplicity,henceeachof p 3isazeroof f of multiplicity1. Thetechniqueusedtofactor f x inExample3.3.4iscalled u -substitution .Weshallseemoreof thistechniqueinSection5.3.Ingeneral,substitutioncanhelpusidentifya`quadraticindisguise' providedthatthereareexactlythreetermsandtheexponentofthersttermisexactlytwicethat ofthesecond.Itisentirelypossiblethatapolynomialhasnorealrootsatall,orworse,ithas realrootsbutnoneofthetechniquesdiscussedinthissectioncanhelpusndthemexactly.In thelattercase,weareforcedtoapproximate,whichinthissubsectionmeansweusethe`Zero' commandonthegraphingcalculator.Wenowpresentothertheoremsanddiscusshowtondzeros ofpolynomialswhenwedonothaveaccesstoagraphingcalculator. 3.3.2ForThoseWishingNOTtouseaGraphingCalculator Supposewewishtondthezerosof f x =2 x 4 +4 x 3 )]TJ/F53 10.9091 Tf 10.848 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.848 0 Td [(6 x )]TJ/F15 10.9091 Tf 10.848 0 Td [(3withoutusingthecalculator. Inthissubsection,wepresentsomemoremathematicaltoolstohelpus.Ourrstresultisdueto ReneDescartes andgivesusanestimateofhowmanypositiveandhowmanynegativerealzeros aretobefound.Thetheoremrequiresustodiscusswhatismeantbythe variationsinsign of apolynomialfunction.Forexample,consider f x =2 x 4 +4 x 3 )]TJ/F53 10.9091 Tf 10.863 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.863 0 Td [(6 x )]TJ/F15 10.9091 Tf 10.863 0 Td [(3.Ifwefocusononly thesignsofthecoecients,westartwitha+,followedbyanother+,thenswitchto )]TJ/F15 10.9091 Tf 8.485 0 Td [(,and stay )]TJ/F15 10.9091 Tf 8.485 0 Td [(fortheremainingtwocoecients.Sincethesignsofthecoecientsswitchedonce,wesay f x hasonevariationinsign.Whenwespeakofthevariationsinsignofapolynomialfunction, f ,weassumetheformulafor f x iswrittenwithdescendingpowersof x ,asinDenition3.1,and concernourselvesonlywiththenonzerocoecients. 2 Moreappropriately,thisequationis`quadraticinform.'Carllikestocallita`quadraticindisguise'becauseit remindshimofTheTransformers.

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202PolynomialFunctions Theorem 3.10 Descartes'RuleofSigns: Suppose f x istheformulaforapolynomial functionwrittenwithdescendingpowersof x If P denotesthenumberofvariationsofsignintheformulafor f x ,thenthenumberof positiverealzeroscountingmultiplicityisoneofthenumbers f P P )]TJ/F15 10.9091 Tf 10.909 0 Td [(2, P )]TJ/F15 10.9091 Tf 10.909 0 Td [(4,... g If N denotesthenumberofvariationsofsignintheformulafor f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x ,thenthenumberof negativerealzeroscountingmultiplicityisoneofthenumbers f N N )]TJ/F15 10.9091 Tf 10.909 0 Td [(2, N )]TJ/F15 10.9091 Tf 10.909 0 Td [(4,... g AcoupleofremarksaboutDescartes'RuleofSignsareinorder.First,Descartes'RuleofSigns givesusanestimatetothe number ofrealzeros,nottheactual value ofthezeros.Second, Descartes'RuleofSignscountsmultiplicities.Thismeansthat,forexample,ifoneofthezeroshas multiplicity2,Descsartes'RuleofSignswouldcountthisas two zeros.Lastly,notethatthenumber ofpositiveornegativerealzerosalwaysstartswiththenumberofsignchangesanddecreasesby anevennumber.Forexample,if f x has7changesinsign,then,countingmultplicities, f has either7,5,3,or1positiverealzero.Thisimpliesthatthegraphof y = f x crossesthepositive x -axisatleastonce.If f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x resultsin4signchanges,then,countingmultiplicities, f has4,2,or 0negativerealzeros;hence,thegraphof y = f x maynotcrossthenegative x -axisatall.The proofofDescartes'RuleofSignsisabittechnical,andcanbefoundhere Example 3.3.5 Let f x =2 x 4 +4 x 3 )]TJ/F53 10.9091 Tf 11.166 0 Td [(x 2 )]TJ/F15 10.9091 Tf 11.166 0 Td [(6 x )]TJ/F15 10.9091 Tf 11.166 0 Td [(3.UseDescartes'RuleofSignstodetermine thepossiblenumberandlocationoftherealzerosof f Solution: Asnotedabove,thevariationsofsignof f x is1.Thismeans,countingmultiplicities, f hasexactly1positiverealzero.Tondthepossiblenumberofnegativerealzeros,weconsider f )]TJ/F53 10.9091 Tf 8.484 0 Td [(x =2 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 4 +4 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 3 )]TJ/F15 10.9091 Tf 10.99 0 Td [( )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.99 0 Td [(6 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x )]TJ/F15 10.9091 Tf 10.99 0 Td [(3whichsimpliesto2 x 4 )]TJ/F15 10.9091 Tf 10.991 0 Td [(4 x 3 )]TJ/F53 10.9091 Tf 10.99 0 Td [(x 2 +6 x )]TJ/F15 10.9091 Tf 10.99 0 Td [(3.This has3variationsinsign,hence f haseither3negativerealzerosor1negativerealzero,counting multiplicities. Cauchy'sBoundgivesusageneralboundonthezerosofapolynomialfunction.Ournextresult helpsusdetermineboundsontherealzerosofapolynomialaswesyntheticallydividewhichare oftensharper 3 boundsthanCauchy'sBound. Theorem 3.11 UpperandLowerBounds: Suppose f isapolynomialofdegree n with n 1. If c> 0issyntheticallydividedinto f andallofthenumbersinthenallineofthedivision tableauhavethesamesigns,then c isanupperboundfortherealzerosof f .Thatis,there arenorealzerosgreaterthan c If c< 0issyntheticallydividedinto f andthenumbersinthenallineofthedivision tableaualternatesigns,then c isalowerboundfortherealzerosof f .Thatis,thereare norealzeroslessthan c NOTE: Ifthenumber0occursinthenallineofthedivisiontableauineitheroftheabove cases,itcanbetreatedas+or )]TJ/F15 10.9091 Tf 8.485 0 Td [(asneeded. 3 Thatis,better,ormoreaccurate.

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3.3RealZerosofPolynomials203 TheUpperandLowerBoundsTheoremworksbecauseofTheorem3.4.Fortheupperboundpartof thetheorem,suppose c> 0isdividedinto f andtheresultinglineinthedivisiontableaucontains, forexample,allnonnegativenumbers.Thismeans f x = x )]TJ/F53 10.9091 Tf 11.181 0 Td [(c q x + r ,wherethecoecients ofthequotientpolynomialandtheremainderarenonnegative.Notetheleadingcoecientof q isthesameas f andso q x isnotthezeropolynomial.If b>c ,then f b = b )]TJ/F53 10.9091 Tf 11.399 0 Td [(c q b + r where b )]TJ/F53 10.9091 Tf 11.345 0 Td [(c and q b arebothpositiveand r 0.Hence f b > 0whichshows b cannotbea zeroof f .Thusnorealnumber b>c canbeazeroof f ,asrequired.Asimilarargumentproves f b < 0ifallthenumbersinthenallineofthesyntheticdivisiontableauarenon-positive.To provethelowerboundpartofthetheorem,wenotethatalowerboundforthenegativerealzeros of f x isanupperboundforthepositiverealzerosof f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x .Applyingtheupperboundportion to f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x givestheresult.Doyouseewherethealternatingsignscomein?Withtheadditional mathematicalmachineryofDescartes'RuleofSignsandtheUpperandLowerBoundsTheorem, wecanndtherealzerosof f x =2 x 4 +4 x 3 )]TJ/F53 10.9091 Tf 9.685 0 Td [(x 2 )]TJ/F15 10.9091 Tf 9.684 0 Td [(6 x )]TJ/F15 10.9091 Tf 9.685 0 Td [(3withouttheuseofagraphingcalculator. Example 3.3.6 Let f x =2 x 4 +4 x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3. 1.Findallrealzerosof f andtheirmultiplicities. 2.Sketchthegraphof y = f x Solution. 1.WeknowfromCauchy'sBoundthatalloftherealzeroslieintheinterval[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 4]andthat ourpossiblerationalzerosare 1 2 1,, 3 2 ,and 3.Descartes'RuleofSignsguarantees usatleastonenegativerealzeroandexactlyonepositiverealzero,countingmultiplicity.We tryourpositiverationalzeros,startingwiththesmallest, 1 2 .Sincetheremainderisn'tzero, weknow 1 2 isn'tazero.Sadly,thenallineinthedivisiontableauhasbothpositiveand negativenumbers,so 1 2 isnotanupperbound.Theonlyinformationwegetfromthisdivision iscourtesyoftheRemainderTheoremwhichtellsus f )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(45 8 sothepoint )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(45 8 is onthegraphof f .Wecontinuetoournextpossiblezero,1.Asbefore,theonlyinformation wecangleanfromthisisthat ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4isonthegraphof f .Whenwetryournextpossible zero, 3 2 ,wegetthatitisnotazero,andwealsoseethatitisanupperboundonthezerosof f ,sinceallofthenumbersinthenallineofthedivisiontableauarepositive.Thismeans thereisnopointtryingourlastpossiblerationalzero,3.Descartes'RuleofSignsguaranteed usapositiverealzero,andatthispointwehaveshownthiszeroisirrational.Furthermore, theIntermediateValueTheorem,Theorem3.1,tellsusthezeroliesbetween1and 3 2 ,since f < 0and f )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(3 2 > 0. 1 2 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 # 1 5 2 3 4 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(21 8 25 3 2 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(21 4 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(45 8 1 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 # 265 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 265 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 2 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 # 3 21 2 57 4 99 8 27 19 2 33 4 75 8 Wenowturnourattentiontonegativerealzeros.Wetrythelargestpossiblezero, )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 Syntheticdivisionshowsusitisnotazero,norisitalowerboundsincethenumbersin

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204PolynomialFunctions thenallineofthedivisiontableaudonotalternate,soweproceedto )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Thisdivision shows )]TJ/F15 10.9091 Tf 8.485 0 Td [(1isazero.Descartes'RuleofSignstoldusthatwemayhaveuptothreenegative realzeros,countingmultiplicity,sowetry )]TJ/F15 10.9091 Tf 8.485 0 Td [(1again,anditworksoncemore.Atthispoint, wehavetaken f ,afourthdegreepolynomial,andperformedtwosuccessfuldivisions.Our quotientpolynomialisquadratic,sowelookatittondtheremainingzeros. )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 2 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 #)]TJ/F15 10.9091 Tf 23.902 0 Td [(1 )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(3 2 5 4 19 8 23 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(5 2 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(19 4 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(5 8 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 #)]TJ/F15 10.9091 Tf 23.903 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(233 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 #)]TJ/F15 10.9091 Tf 23.902 0 Td [(203 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 Settingthequotientpolynomialequaltozeroyields2 x 2 )]TJ/F15 10.9091 Tf 9.734 0 Td [(3=0,sothat x 2 = 3 2 ,or x = p 6 2 Descartes'RuleofSignstellsusthatthepositiverealzerowefound, p 6 2 hasmultiplicity1. Descartesalsotellsusthetotalmultiplicityofnegativerealzerosis3,whichforces )]TJ/F15 10.9091 Tf 8.485 0 Td [(1tobe azeroofmultiplicity2and )]TJ/F40 7.9701 Tf 9.68 10.993 Td [(p 6 2 tohavemultiplicity1. 2.Weknowtheendbehaviorof y = f x resemblesitsleadingterm, y =2 x 4 .Thismeans thegraphentersthesceneinQuadrantIIandexitsinQuadrantI.Since p 6 2 arezerosof oddmultiplicity,weknowthegraphcrossesthroughthe x -axisatthepoints )]TJ/F40 7.9701 Tf 9.68 10.993 Td [(p 6 2 ; 0 and p 6 2 ; 0 .Since )]TJ/F15 10.9091 Tf 8.485 0 Td [(1isazeroofmultiplicity2,weknowthegraphof y = f x touchesand reboundsothe x -axisat )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0.Puttingthistogether,weget x y Youcanseewhythe`nocalculator'approachisnotverypopularthesedays.Itrequiresmore computationandmoretheoremsthanthealternative. 4 Ingeneral,nomatterhowmanytheorems 4 Thisisapparentlyabadthing.

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3.3RealZerosofPolynomials205 youthrowatapolynomial,itmaywellbeimpossible 5 tondtheirzerosexactly.Thepolynomial f x = x 5 )]TJ/F53 10.9091 Tf 11.371 0 Td [(x )]TJ/F15 10.9091 Tf 11.37 0 Td [(1isonesuchbeast. 6 AccordingtoDescartes'RuleofSigns, f hasexactlyone positiverealzero,anditcouldhavetwonegativerealzeros,ornoneatall.TheRationalZeros Testgivesus 1asrationalzerostotrybutneitheroftheseworksince f = f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= )]TJ/F15 10.9091 Tf 8.484 0 Td [(1.If wetrythesubstitutiontechniqueweusedinExample3.3.4,wend f x hasthreeterms,butthe exponentonthe x 5 isn'texactlytwicetheexponenton x .Howcouldwegoaboutapproximating thiszerowithoutresortingtothe`Zero'commandofagraphingcalculator?Weusethe Bisection Method .TherststepintheBisectionMethodistondanintervalonwhich f changessign. Weknow f = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1andwend f =29.BytheIntermediateValueTheorem,weknowthe zeroof f liesintheinterval[1 ; 2].Next,we`bisect'thisintervalandndthemidpoint,1 : 5.We nd f : 5 5 : 09.Thismeansourzeroisbetween1and1 : 5,since f changessignonthisinterval. Now,we`bisect'theinterval[1 ; 1 : 5]andnd f : 25 0 : 08,sonowwehavethezerobetween1and 1 : 25.Bisecting[1 ; 1 : 25],wend f : 125 )]TJ/F15 10.9091 Tf 20 0 Td [(0 : 32,whichmeansthezeroof f isbetween1 : 125and 1 : 25.Wecontinueinthisfashionuntilwehave`sandwiched'thezerobetweentwonumberswhich dierbynomorethanadesiredaccuracy.YoucanthinkoftheBisectionMethodasreversing thesigndiagramprocess:insteadofndingthezerosandcheckingthesignof f usingtestvalues, weareusingtestvaluestodeterminewherethesignsswitchtondthezeros.Itisaslowand tedious,yetfool-proof,methodfordetermininganapproximationofarealzero.Ourlastexample remindsusthatndingthezerosofpolynomialsisacriticalstepinsolvingpolynomialequations andinequalities. Example 3.3.7 1.Findalloftherealsolutionstotheequation2 x 5 +6 x 3 +3=3 x 4 +8 x 2 2.Solvetheinequality2 x 5 +6 x 3 +3 3 x 4 +8 x 2 3.Interpretyouranswertopart2graphically,andverifyusingagraphingcalculator. Solution. 1.Findingtherealsolutionsto2 x 5 +6 x 3 +3=3 x 4 +8 x 2 isthesameasndingthereal solutionsto2 x 5 )]TJ/F15 10.9091 Tf 10.911 0 Td [(3 x 4 +6 x 3 )]TJ/F15 10.9091 Tf 10.911 0 Td [(8 x 2 +3=0.Inotherwords,wearelookingfortherealzeros of p x =2 x 5 )]TJ/F15 10.9091 Tf 10.069 0 Td [(3 x 4 +6 x 3 )]TJ/F15 10.9091 Tf 10.069 0 Td [(8 x 2 +3.Usingthetechniquesdevelopedinthissection,wedivide asfollows. 5 Wedon'tusethiswordlightly;itcanbeproventhatthezerosofsomepolynomialscannotbeexpressedusing theusualalgebraicsymbols. 6 Seethispage .

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206PolynomialFunctions 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 )]TJ/F15 10.9091 Tf 8.485 0 Td [(803 # 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 # 2163 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 2163 0 #)]TJ/F15 10.9091 Tf 23.902 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 206 0 Thequotientpolynomialis2 x 2 +6whichhasnorealzerossoweget x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 and x =1. 2.Tosolvethisnonlinearinequality,wefollowthesameguidelinessetforthinSection2.4:weget 0ononesideoftheinequalityandconstructasigndiagram.Ouroriginalinequalitycanbe rewrittenas2 x 5 )]TJ/F15 10.9091 Tf 8.898 0 Td [(3 x 4 +6 x 3 )]TJ/F15 10.9091 Tf 8.898 0 Td [(8 x 2 +3 0.Wefoundthezerosof p x =2 x 5 )]TJ/F15 10.9091 Tf 8.898 0 Td [(3 x 4 +6 x 3 )]TJ/F15 10.9091 Tf 8.898 0 Td [(8 x 2 +3 inpart1tobe x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 and x =1.Weconstructoursigndiagramasbefore. )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + 0 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 02 Thesolutionto p x < 0is )]TJ/F54 10.9091 Tf 5 -8.836 Td [( ; )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 ,andweknow p x =0at x = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 and x =1.Hence, thesolutionto p x 0is )]TJ/F54 10.9091 Tf 5 -8.836 Td [( ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 [f 1 g 3.Tointerpretthissolutiongraphically,weset f x =2 x 5 +6 x 3 +3and g x =3 x 4 +8 x 2 .We recallthatthesolutionto f x g x isthesetof x valuesforwhichthegraphof f isbelow thegraphof g where f x
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3.3RealZerosofPolynomials207 3.3.3Exercises 1.Findtherealzerosofthepolynomialusingthetechniquesspeciedbyyourinstructor.State themultiplicityofeachrealzero. a p x = x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x +6 b p x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 x 3 +19 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(49 x +20 c p x = x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +12 d p x = x 3 +4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(11 x +6 e p x =3 x 3 +3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(11 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 f p x = x 4 +2 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(40 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(32 g p x =6 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x 2 h p x =36 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(11 x 2 +2 x +1 i p x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(17 x 3 +5 x 2 +34 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 j p x =25 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(105 x 4 +174 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(142 x 2 +57 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 k p x = x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(60 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(80 x 2 +960 x +2304 l p x = x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 2 + x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 m p x =90 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(399 x 3 +622 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(399 x +90 n p x =9 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2.Findtherealzerosof f x = x 3 )]TJ/F34 7.9701 Tf 14.585 4.296 Td [(1 12 x 2 )]TJ/F34 7.9701 Tf 14.585 4.296 Td [(7 12 x + 1 72 byrstndingapolynomial q x with integercoecientssuchthat q x = N f x forsomeinteger N .RecallthattheRational ZerosTheoremrequiredthepolynomialinquestiontohaveintegercoecients.Showthat f and q havethesamerealzeros. 3.Solvethepolynomialinequalityandgiveyouranswerinintervalform. a )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 3 +19 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(49 x +20 > 0 b x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x 2 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 c x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 4 d 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x 3 +35 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(45 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(25 > 0 e4 x 3 3 x +1 f x 3 +2 x 2 2
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208PolynomialFunctions 3.3.4Answers 1.a p x = x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x +6 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2mult.1, x =1mult.1, x =3mult.1 b p x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 x 3 +19 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(49 x +20 x = 1 2 mult.1, x =4mult.1, x =5mult.1 c p x = x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +12 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2mult.2, x =1mult.1, x =3mult.1 d p x = x 3 +4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(11 x +6 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6mult.1, x =1mult.2 e p x =3 x 3 +3 x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(11 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2mult.1, x = 3 p 69 6 eachhasmult.1 f p x = x 4 +2 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(40 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(32 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2mult.3, x =4mult.1 g p x =6 x 4 )]TJ/F15 10.9091 Tf 10.91 0 Td [(5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x 2 x =0mult.2, x = 5 p 241 12 eachhasmult.1 h p x =36 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(11 x 2 +2 x +1 x = 1 2 mult.2, x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 mult.2 i p x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(17 x 3 +5 x 2 +34 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 x = 5 17 mult.1, x = p 2eachmult.1 j p x =25 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(105 x 4 +174 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(142 x 2 +57 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x = 3 5 mult.2, x =1mult.3 k p x = x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(60 x 3 )]TJ/F15 10.9091 Tf 10.91 0 Td [(80 x 2 +960 x +2304 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4mult.3, x =6mult.2 l p x = x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 2 + x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x =7mult.1 m p x =90 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(399 x 3 +622 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(399 x +90 x = 2 3 mult.1, x = 3 2 mult.1, x = 5 3 mult.1, x = 3 5 mult.1 n p x =9 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x x =0mult.1, x = 5 p 61 18 eachhasmult.1 2.Wechoose q x =72 x 3 )]TJ/F15 10.9091 Tf 11.067 0 Td [(6 x 2 )]TJ/F15 10.9091 Tf 11.067 0 Td [(7 x +1=72 f x .Clearly f x =0ifandonlyif q x =0 sotheyhavethesamerealzeros.Inthiscase, x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 ;x = 1 6 and x = 1 4 aretherealzeros ofboth f and q 3.a ; 1 2 [ ; 5 b f)]TJ/F15 10.9091 Tf 13.939 0 Td [(2 g[ [1 ; 3] c ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1] [ [3 ; 1 d )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 [ ; 1 e )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(1 2 [ [1 ; 1 f ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 [ )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ 8.485 9.025 Td [(p 2 ; p 2 g[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 2] h )]TJ/F54 10.9091 Tf 5 -8.836 Td [( ; )]TJ 8.485 9.024 Td [(p 3 [ )]TJ/F54 10.9091 Tf 5 0.188 Td [(p 3 ; 1

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3.4ComplexZerosandtheFundamentalTheoremofAlgebra209 3.4ComplexZerosandtheFundamentalTheoremofAlgebra InSection3.3,wewerefocusedonndingtherealzerosofapolynomialfunction.Inthissection,we expandourhorizonsandlookforthenon-realzerosaswell.Considerthepolynomial p x = x 2 +1. Thezerosof p arethesolutionsto x 2 +1=0,or x 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Thisequationhasnorealsolutions,but youmayrecallfromIntermediateAlgebrathatwecanformallyextractthesquarerootsofboth sidestoget x = p )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Thequantity p )]TJ/F15 10.9091 Tf 8.485 0 Td [(1isusuallyre-labeled i ,theso-called imaginaryunit 1 Thenumber i ,whilenotarealnumber,playsalongwellwithrealnumbers,andactsverymuch likeanyotherradicalexpression.Forinstance,3 i =6 i ,7 i )]TJ/F15 10.9091 Tf 10.443 0 Td [(3 i =4 i )]TJ/F15 10.9091 Tf 10.443 0 Td [(7 i ++4 i =5 )]TJ/F15 10.9091 Tf 10.443 0 Td [(3 i andsoforth.Thekeypropertieswhichdistinguish i fromtherealnumbersarelistedbelow. Definition 3.4 Theimaginaryunit i satisesthetwofollowingproperties 1. i 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2.If c isarealnumberwith c 0then p )]TJ/F53 10.9091 Tf 8.485 0 Td [(c = i p c Property1inDenition3.4establishesthat i doesactasasquareroot 2 of )]TJ/F15 10.9091 Tf 8.484 0 Td [(1,andproperty2 establisheswhatwemeanbythe`principalsquareroot'ofanegativerealnumber.Inproperty 2,itisimportanttoremembertherestrictionon c .Forexample,itisperfectlyacceptabletosay p )]TJ/F15 10.9091 Tf 8.485 0 Td [(4= i p 4= i =2 i .However, p )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 6 = i p )]TJ/F15 10.9091 Tf 8.484 0 Td [(4,otherwise,we'dget 2= p 4= p )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(4= i p )]TJ/F15 10.9091 Tf 8.485 0 Td [(4= i i =2 i 2 =2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; whichisunacceptable. 3 Wearenowinthepositiontodenethe complexnumbers Definition 3.5 A complexnumber isanumberoftheform a + bi ,where a and b arereal numbersand i istheimaginaryunit. Complexnumbersincludethingsyou'dnormallyexpect,like3+2 i and 2 5 )]TJ/F53 10.9091 Tf 11.062 0 Td [(i p 3.However,don't forgetthat a or b couldbezero,whichmeansnumberslike3 i and6arealsocomplexnumbers.In otherwords,don'tforgetthatthecomplexnumbers include therealnumbers,so0and )]TJ 11.015 9.024 Td [(p 21 arebothconsideredcomplexnumbers.Thearithmeticofcomplexnumbersisasyouwouldexpect. TheonlythingyouneedtorememberarethetwopropertiesinDenition3.4.Thenextexample shouldhelprecallhowtheseanimalsbehave. Example 3.4.1 Performtheindicatedoperationsandsimplify.Writeyournalanswerinthe form 4 a + bi 1 Sometechnicalmathematicstextbookslabelit` j '. 2 Notetheuseoftheindenitearticle`a'.Whateverbeastischosentobe i )]TJ/F64 8.9664 Tf 7.167 0 Td [(i istheothersquarerootof )]TJ/F63 8.9664 Tf 7.168 0 Td [(1. 3 Wewanttoenlargethenumbersystemsowecansolvethingslike x 2 = )]TJ/F63 8.9664 Tf 7.167 0 Td [(1,butnotatthecostoftheestablished rulesalreadysetinplace.Forthatreason,thegeneralpropertiesofradicalssimplydonotapplyforevenrootsof negativequantities. 4 We'llacceptananswerofsay3 )]TJ/F63 8.9664 Tf 9.306 0 Td [(2 i ,although,technically,weshouldwritethisas3+ )]TJ/F63 8.9664 Tf 7.168 0 Td [(2 i .Evenwepedants haveourlimits.

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210PolynomialFunctions 1. )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i )]TJ/F15 10.9091 Tf 10.909 0 Td [(+4 i 2. )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i +4 i 3. 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 i 4. p )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 5. p )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 6. x )]TJ/F15 10.9091 Tf 10.909 0 Td [([1+2 i ] x )]TJ/F15 10.9091 Tf 10.909 0 Td [([1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i ] Solution. 1.Asmentionedearlier,wetreatexpressionsinvolving i aswewouldanyotherradical.We combineliketermstoget )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 i )]TJ/F15 10.9091 Tf 10.909 0 Td [(+4 i =1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 i 2.Usingthedistributiveproperty,weget )]TJ/F15 10.9091 Tf 9.858 0 Td [(2 i +4 i =+ i )]TJ/F15 10.9091 Tf 9.859 0 Td [( i )]TJ/F15 10.9091 Tf 9.858 0 Td [( i i = 3+4 i )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 i )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 i 2 .Recalling i 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,weget3+4 i )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 i )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 i 2 =3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(8=11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i 3.Howintheworldarewesupposedtosimplify 1 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 i 3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 i ?Well,wedealwiththedenominator 3 )]TJ/F15 10.9091 Tf 10.198 0 Td [(4 i aswewouldanyotherdenominatorcontainingaradical,andmultiplybothnumerator anddenominatorby3+4 i theconjugateof3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 i 5 Doingsoproduces 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 i 3+4 i 3+4 i = )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i +4 i )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 i +4 i = 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i 25 = 11 25 )]TJ/F15 10.9091 Tf 14.832 7.38 Td [(2 25 i 4.Weuseproperty2ofDenition3.4rst,thenapplytherulesofradicalsapplicabletoreal radicalstoget p )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(12= )]TJ/F53 10.9091 Tf 5 -8.837 Td [(i p 3 )]TJ/F53 10.9091 Tf 11.819 -8.837 Td [(i p 12 = i 2 p 3 12= )]TJ 8.484 9.024 Td [(p 36= )]TJ/F15 10.9091 Tf 8.485 0 Td [(6. 5.Weadheretotheorderofoperationshereandperformthemultiplicationbeforetheradical toget p )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12= p 36=6. 6.Wecanbruteforcemultiplyusingthedistributivepropertyandseethat x )]TJ/F15 10.9091 Tf 10.909 0 Td [([1+2 i ] x )]TJ/F15 10.9091 Tf 10.91 0 Td [([1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i ]= x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x [1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i ] )]TJ/F53 10.9091 Tf 10.909 0 Td [(x [1+2 i ]+[1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i ][1+2 i ] = x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x +2 ix )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ix +1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i +2 i )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 i 2 = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +5 Acoupleofremarksaboutthelastexampleareinorder.First,the conjugate ofacomplexnumber a + bi isthenumber a )]TJ/F53 10.9091 Tf 10.127 0 Td [(bi .Thenotationcommonlyusedforconjugationisa`bar': a + bi = a )]TJ/F53 10.9091 Tf 10.128 0 Td [(bi Forexample, 3+2 i =3 )]TJ/F15 10.9091 Tf 9.624 0 Td [(2 i 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i =3+2 i 6=6, 4 i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 i ,and 3+ p 5=3+ p 5.Theproperties oftheconjugatearesummarizedinthefollowingtheorem. 5 Wewilltalkmoreaboutthisinamoment.

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3.4ComplexZerosandtheFundamentalTheoremofAlgebra211 Theorem 3.12 Suppose z and w arecomplexnumbers. z = z z + w = z + w z w = zw z n = z n ,foranynaturalnumber n =1 ; 2 ; 3 ;::: z isarealnumberifandonlyif z = z Essentially,Theorem3.12saysthatcomplexconjugationworkswellwithaddition,multiplication, andpowers.Theproofofthesepropertiescanbestbeachievedbywritingout z = a + bi and w = c + di forrealnumbers a b c ,and d .Next,wecomputetheleftandrighthandside ofeachequationandchecktoseethattheyarethesame.Theproofoftherstpropertyis averyquickexercise. 6 Toprovethesecondproperty,wecompare z + w and z + w .Wehave z + w = a + bi + c + di = a )]TJ/F53 10.9091 Tf 10.909 0 Td [(bi + c )]TJ/F53 10.9091 Tf 10.909 0 Td [(di .Tond z + w ,werstcompute z + w = a + bi + c + di = a + c + b + d i so z + w = a + c + b + d i = a + c )]TJ/F15 10.9091 Tf 10.91 0 Td [( b + d i = a )]TJ/F53 10.9091 Tf 10.909 0 Td [(bi + c )]TJ/F53 10.9091 Tf 10.91 0 Td [(di Assuch,wehaveestablished z + w = z + w .Theproofformultiplicationworkssimilarly.Theproof thattheconjugateworkswellwithpowerscanbeviewedasarepeatedapplicationoftheproduct rule,andisbestprovedusingatechniquecalledMathematicalInduction. 7 Thelastpropertyisa characterizationofrealnumbers.If z isreal,then z = a +0 i ,so z = a )]TJ/F15 10.9091 Tf 10.82 0 Td [(0 i = a = z .Ontheother hand,if z = z ,then a + bi = a )]TJ/F53 10.9091 Tf 10.104 0 Td [(bi whichmeans b = )]TJ/F53 10.9091 Tf 8.485 0 Td [(b so b =0.Hence, z = a +0 i = a andisreal. Wenowreturntothebusinessofzeros.Supposewewishtondthezerosof f x = x 2 )]TJ/F15 10.9091 Tf 10.999 0 Td [(2 x +5. Tosolvetheequation x 2 )]TJ/F15 10.9091 Tf 10.894 0 Td [(2 x +5=0,wenotethequadraticdoesn'tfactornicely,soweresortto theQuadraticFormula,Equation2.5andobtain x = )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 2 = 2 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 2 = 2 4 i 2 =1 2 i: Twothingsareimportanttonote.First,thezeros,1+2 i and1 )]TJ/F15 10.9091 Tf 11.666 0 Td [(2 i arecomplexconjugates. Ifeverweobtainnon-realzerostoaquadraticfunctionwithreal coecients,thezeroswillbea complexconjugatepair.Doyouseewhy?Next,wenotethatinExample3.4.1,part6,wefound x )]TJ/F15 10.9091 Tf 10.431 0 Td [([1+2 i ] x )]TJ/F15 10.9091 Tf 10.431 0 Td [([1 )]TJ/F15 10.9091 Tf 10.431 0 Td [(2 i ]= x 2 )]TJ/F15 10.9091 Tf 10.431 0 Td [(2 x +5.Thisdemonstratesthatthefactortheoremholdsevenfor non-realzeros,i.e, x =1+2 i isazeroof f ,and,sureenough, x )]TJ/F15 10.9091 Tf 10.948 0 Td [([1+2 i ]isafactorof f x .It turnsoutthatpolynomialdivisionworksthesamewayforallcomplexnumbers,realandnon-real alike,andsotheFactorandRemainderTheoremsholdaswell.Buthowdoweknowifageneral polynomialhasanycomplexzerosatall?Wehavemanyexamplesofpolynomialswithnoreal 6 Trustusonthis. 7 SeeSection9.3.

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212PolynomialFunctions zeros.Cantherebepolynomialswithnozeroswhatsoever?Theanswertothatlastquestionis No."andthetheoremwhichprovidesthatanswerisTheFundamentalTheoremofAlgebra. Theorem 3.13 TheFundamentalTheoremofAlgebra: Suppose f isapolynomialfunction withcomplexnumbercoecientsofdegree n 1,then f hasleastonecomplexzero. TheFundamentalTheoremofAlgebraisanexampleofan`existence'theoreminmathematics.Like theIntermediateValueTheorem,Theorem3.1,theFundamentalTheoremofAlgebraguarantees theexistenceofatleastonezero,butgivesusnoalgorithmtouseinndingit.Infact,aswe mentionedinSection3.3,therearepolynomialswhoserealzeros,thoughtheyexist,cannotbe expressedusingthe`usual'combinationsofarithmeticsymbols,andmustbeapproximated.The authorsarefullyawarethatthefullimpactandprofoundnatureoftheFundamentalTheorem ofAlgebraislostonmoststudentsthislevel,andthat'sne.Ittookmathematiciansliterally hundredsofyearstoprovethetheoreminitsfullgenerality,andsomeofthathistoryisrecorded here .NotethattheFundamentalTheoremofAlgebraappliestopolynomialfunctionswithnot onlyrealcoecients,but,thosewithcomplexnumbercoecientsaswell. Suppose f isapolynomialofdegree n with n 1.TheFundamentalTheoremofAlgebraguarantees usatleastonecomplexzero, z 1 ,and,assuch,theFactorTheoremguaranteesthat f x factors as f x = x )]TJ/F53 10.9091 Tf 10.909 0 Td [(z 1 q 1 x forapolynomialfunction q 1 ,ofdegreeexactly n )]TJ/F15 10.9091 Tf 11.175 0 Td [(1.If n )]TJ/F15 10.9091 Tf 11.175 0 Td [(1 1,then theFundamentalTheoremofAlgebraguaranteesacomplexzeroof q 1 aswell,say z 2 ,andsothe FactorTheoremgivesus q 1 x = x )]TJ/F53 10.9091 Tf 10.909 0 Td [(z 2 q 2 x ,andhence f x = x )]TJ/F53 10.9091 Tf 10.91 0 Td [(z 1 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(z 2 q 2 x .Wecan continuethisprocessexactly n times,atwhichpointourquotientpolynomial q n hasdegree0so it'saconstant.Thisargumentgivesusthefollowingfactorizationtheorem. Theorem 3.14 ComplexFactorizationTheorem: Suppose f isapolynomialfunctionwith complexnumbercoecients.Ifthedegreeof f is n and n 1,then f hasexactly n complex zeros,countingmultiplicity.If z 1 z 2 ,..., z k arethedistinctzerosof f ,withmultiplicities m 1 m 2 ,..., m k ,respectively,then f x = a x )]TJ/F53 10.9091 Tf 10.91 0 Td [(z 1 m 1 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(z 2 m 2 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(z k m k Notethatthevalue a inTheorem3.14istheleadingcoecientof f x Canyouseewhy?andas such,weseethatapolynomialiscompletelydeterminedbyitszeros,theirmultiplicities,andits leadingcoecient.Weputthistheoremtogooduseinthenextexample. Example 3.4.2 Let f x =12 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(20 x 4 +19 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1. 1.Findallcomplexzerosof f andstatetheirmultiplicities. 2.Factor f x usingTheorem3.14 Solution. 1.Since f isafthdegreepolynomial,weknowweneedtoperformatleastthreesuccessful divisionstogetthequotientdowntoaquadraticfunction.Atthatpoint,wecanndthe remainingzerosusingtheQuadraticFormula,ifnecessary.Usingthetechniquesdeveloped inSection3.3,weget

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3.4ComplexZerosandtheFundamentalTheoremofAlgebra213 1 2 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2019 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 # 6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(760 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 2 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14120 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 # 6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(442 )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 3 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(884 0 #)]TJ/F15 10.9091 Tf 29.356 0 Td [(44 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1212 0 Ourquotientis12 x 2 )]TJ/F15 10.9091 Tf 11.149 0 Td [(12 x +12,whosezeroswendtobe 1 i p 3 2 .FromTheorem3.14,we know f hasexactly5zeros,countingmultiplicities,andassuchwehavethezero 1 2 with multiplicity2,andthezeros )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 3 1+ i p 3 2 and 1 )]TJ/F37 7.9701 Tf 6.586 0 Td [(i p 3 2 ,eachofmultiplicity1. 2.ApplyingTheorem3.14,weareguaranteedthat f factorsas f x =12 x )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 2 x + 1 3 x )]TJ/F55 10.9091 Tf 10.909 18.655 Td [(" 1+ i p 3 2 #! x )]TJ/F55 10.9091 Tf 10.909 18.655 Td [(" 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(i p 3 2 #! AtruetestofTheorem3.14andastudent'smettle!wouldbetotakethefactoredformof f x inthepreviousexampleandmultiplyitout 8 toseethatitreallydoesreducetotheformula f x = 12 x 5 )]TJ/F15 10.9091 Tf 10.606 0 Td [(20 x 4 +19 x 3 )]TJ/F15 10.9091 Tf 10.606 0 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.606 0 Td [(2 x +1.WhenfactoringapolynomialusingTheorem3.14,wesaythat itis factoredcompletelyoverthecomplexnumbers ,meaningthatitisimpossibletofactor thepolynomialanyfurtherusingcomplexnumbers.Ifwewantedtocompletelyfactor f x over the realnumbers thenwewouldhavestoppedshortofndingthenonrealzerosof f andfactored f usingourworkfromthesyntheticdivisiontowrite f x = )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x + 1 3 )]TJ/F15 10.9091 Tf 11.818 -8.836 Td [(12 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x +12 or f x =12 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(1 2 2 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x + 1 3 )]TJ/F53 10.9091 Tf 11.818 -8.837 Td [(x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x +1 .Sincethezerosof x 2 )]TJ/F53 10.9091 Tf 11.847 0 Td [(x +1arenonreal,wecall x 2 )]TJ/F53 10.9091 Tf 10.681 0 Td [(x +1an irreduciblequadratic meaningitisimpossibletobreakitdownanyfurtherusing real numbers.Thelasttworesultsofthesectionshowusthat,atleastintheory,ifwehavea polynomialfunctionwithrealcoecients,wecanalwaysfactoritdownenoughsothatanynonreal zeroscomefromirreduciblequadratics. Theorem 3.15 ConjugatePairsTheorem: If f isapolynomialfunctionwithrealnumber coecientsand z isazeroof f ,thensois z Toprovethetheorem,suppose f isapolynomialwithrealnumbercoecients.Specically,let f x = a n x n + a n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 x n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 + :::a 2 x 2 + a 1 x + a 0 .If z isazeroof f ,then f z =0,whichmeans a n z n + a n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 z n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 + :::a 2 z 2 + a 1 z + a 0 =0.Next,weconsider f z andapplyTheorem3.12below. 8 Youreallyshoulddothisonceinyourlifetoconvinceyourselfthatallofthetheoryactuallydoeswork!

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214PolynomialFunctions f z = a n z n + a n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 z n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 + :::a 2 z 2 + a 1 z + a 0 = a n z n + a n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 z n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 + :::a 2 z 2 + a 1 z + a 0 since z n = z n = a n z n + a n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 z n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 + ::: a 2 z 2 + a 1 z + a 0 sincethecoecientsarereal = a n z n + a n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 z n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 + ::: a 2 z 2 + a 1 z + a 0 since z w = zw = a n z n + a n )]TJ/F35 5.9776 Tf 6.586 0 Td [(1 z n )]TJ/F35 5.9776 Tf 6.587 0 Td [(1 + :::a 2 z 2 + a 1 z + a 0 since z + w = z + w = f z = 0 =0 Thisshowsthat z isazeroof f .So,if f isapolynomialfunctionwithrealnumbercoecients, Theorem3.15tellsusif a + bi isanonrealzeroof f ,thensois a )]TJ/F53 10.9091 Tf 11.372 0 Td [(bi .Inotherwords,nonreal zerosof f comeinconjugatepairs.TheFactorTheoremkicksintogiveusboth x )]TJ/F15 10.9091 Tf 10.604 0 Td [([ a + bi ]and x )]TJ/F15 10.9091 Tf 11.052 0 Td [([ a )]TJ/F53 10.9091 Tf 11.052 0 Td [(bi ]asfactorsof f x whichmeans x )]TJ/F15 10.9091 Tf 11.052 0 Td [([ a + bi ] x )]TJ/F15 10.9091 Tf 11.052 0 Td [([ a )]TJ/F53 10.9091 Tf 11.052 0 Td [(bi ]= x 2 +2 ax + )]TJ/F53 10.9091 Tf 5 -8.837 Td [(a 2 + b 2 is anirreduciblequadraticfactorof f .Asaresult,wehaveourlastresultofthesection. Theorem 3.16 RealFactorizationTheorem: Suppose f isapolynomialfunctionwithreal numbercoecients.Then f x canbefactoredintoaproductoflinearfactorscorrespondingto therealzerosof f andirreduciblequadraticfactorswhichgivethenonrealzerosof f Wenowpresentanexamplewhichpullstogetherallofthemajorideasofthissection. Example 3.4.3 Let f x = x 4 +64. 1.Usesyntheticdivisiontoshow x =2+2 i isazeroof f 2.Findtheremainingcomplexzerosof f 3.Completelyfactor f x overthecomplexnumbers. 4.Completelyfactor f x overtherealnumbers. Solution. 1.Rememberingtoinsertthe0'sinthesyntheticdivisiontableauwehave 2+2 i 100064 # 2+2 i 8 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(16+16 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(64 12+2 i 8 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(16+16 i 0 2.Since f isafourthdegreepolynomial,weneedtomaketwosuccessfuldivisionstogeta quadraticquotient.Since2+2 i isazero,weknowfromTheorem3.15that2 )]TJ/F15 10.9091 Tf 11.079 0 Td [(2 i isalsoa zero.Wecontinueoursyntheticdivisiontableau.

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3.4ComplexZerosandtheFundamentalTheoremofAlgebra215 2+2 i 100064 # 2+2 i 8 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(16+16 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(64 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i 12+2 i 8 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(16+16 i 0 # 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i 8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 i 16 )]TJ/F15 10.9091 Tf 10.91 0 Td [(16 i 148 0 Ourquotientpolynomialis x 2 +4 x +8.Usingthequadraticformula,weobtaintheremaining zeros )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+2 i and )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i 3.UsingTheorem3.14,weget f x = x )]TJ/F15 10.9091 Tf 10.878 0 Td [([2 )]TJ/F15 10.9091 Tf 10.878 0 Td [(2 i ] x )]TJ/F15 10.9091 Tf 10.878 0 Td [([2+2 i ] x )]TJ/F15 10.9091 Tf 10.878 0 Td [([ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+2 i ] x )]TJ/F15 10.9091 Tf 10.878 0 Td [([ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.878 0 Td [(2 i ]. 4.Wemultiplythelinearfactorsof f x whichcorrespondtocomplexconjugatepairs.Wend x )]TJ/F15 10.9091 Tf 11.063 0 Td [([2 )]TJ/F15 10.9091 Tf 11.062 0 Td [(2 i ] x )]TJ/F15 10.9091 Tf 11.063 0 Td [([2+2 i ]= x 2 )]TJ/F15 10.9091 Tf 11.063 0 Td [(4 x +8,and x )]TJ/F15 10.9091 Tf 11.063 0 Td [([ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+2 i ] x )]TJ/F15 10.9091 Tf 11.063 0 Td [([ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 11.063 0 Td [(2 i ]= x 2 +4 x +8. Ournalanswer f x = )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +8 )]TJ/F53 10.9091 Tf 11.818 -8.837 Td [(x 2 +4 x +8 Ourlastexampleturnsthetablesandasksustomanufactureapolynomialwithcertainproperties ofitsgraphandzeros. Example 3.4.4 Findapolynomial p oflowestdegreethathasintegercoecientsandsatisesall ofthefollowingcriteria: thegraphof y = p x touchesthe x -axisat )]TJ/F34 7.9701 Tf 6.196 -4.542 Td [(1 3 ; 0 x =3 i isazeroof p as x p x as x !1 p x Solution. Tosolvethisproblem,wewillneedagoodunderstandingoftherelationshipbetween the x -interceptsofthegraphofafunctionandthezerosofafunction,theFactorTheorem,the roleofmultiplicity,complexconjugates,theComplexFactorizationTheorem,andendbehaviorof polynomialfunctions.Inshort,you'llneedmostofthemajorconceptsofthischapter.Sincethe graphof p touchesthe x -axisat )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 3 ; 0 ,weknow x = 1 3 isazeroofevenmultiplicity.Sincewe areafterapolynomialoflowestdegree,weneed x = 1 3 tohavemultiplicityexactly2.TheFactor Theoremnowtellsus )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 3 2 isafactorof p x .Since x =3 i isazeroandournalansweristo haveintegerrealcoecients, x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 i isalsoazero.TheFactorTheoremkicksinagaintogiveus x )]TJ/F15 10.9091 Tf 9.251 0 Td [(3 i and x +3 i asfactorsof p x .Wearegivennofurtherinformationaboutzerosorintercepts soweconclude,bytheComplexFactorizationTheoremthat p x = a )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 3 2 x )]TJ/F15 10.9091 Tf 10.762 0 Td [(3 i x +3 i for somerealnumber a .Expandingthis,weget p x = ax 4 )]TJ/F34 7.9701 Tf 10.771 4.295 Td [(2 a 3 x 3 + 82 a 9 x 2 )]TJ/F15 10.9091 Tf 9.575 0 Td [(6 ax + a .Inordertoobtain integercoecients,weknow a mustbeanintegermultipleof9.Ourlastconcernisendbehavior. Sincetheleadingtermof p x is ax 4 ,weneed a< 0toget p x as x .Hence,ifwe choose x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(9,weget p x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 x 4 +6 x 3 )]TJ/F15 10.9091 Tf 11.161 0 Td [(82 x 2 +54 x )]TJ/F15 10.9091 Tf 11.161 0 Td [(9.Wecanverifyourhandiworkusing thetechniquesdevelopedinthischapter.

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216PolynomialFunctions Thisexampleconcludesourstudyofpolynomialfunctions. 9 Thelastfewsectionshavecontained whatisconsideredbymanytobe`heavy'mathematics.Likeaheavymeal,heavymathematics takestimetodigest.Don'tbeoverlyconcernedifitdoesn'tseemtosinkinallatonce,andpace yourselfontheexercisesoryou'reliabletogetmentalcramps.Butbeforewegettotheexercises, we'dliketooerabitofanepilogue. Ourmaingoalinpresentingthematerialonthecomplexzerosofapolynomialwastogivethe chapterasenseofcompleteness.Giventhatitcanbeshownthatsomepolynomialshaverealzeros whichcannotbeexpressedusingtheusualalgebraicoperations,andstillothershavenorealzeros atall,itwasnicetodiscoverthateverypolynomialofdegree n 1has n complexzeros.Solike wesaid,itgivesusasenseofclosure.Buttheobservantreaderwillnotethatwedidnotgiveany examplesofapplicationswhichinvolvecomplexnumbers.Studentsoftenwonderwhencomplex numberswillbeusedin`real-world'applications.Afterall,didn'twecall i theimaginary unit? Howcanimaginarythingsbeusedinreality?Itturnsoutthatcomplexnumbersareveryusefulin manyappliedeldssuchasuiddynamics,electromagnetismandquantummechanics,butmost oftheapplicationsrequireMathematicswellbeyondCollegeAlgebratofullyunderstandthem. Thatdoesnotmeanyou'llneverbebeabletounderstandthem;infact,itistheauthors'sincere hopethatallofyouwillreachapointinyourstudieswhentheglory,aweandsplendorofcomplex numbersarerevealedtoyou.Fornow,however,thereallygoodstuisbeyondthescopeofthis text.Weinviteyouandyourclassmatestondafewexamplesofcomplexnumberapplications andseewhatyoucanmakeofthem.AsimpleInternetsearchwiththephrase`complexnumbersin reallife'shouldgetyoustarted.Basicelectronicsclassesareanotherplacetolook,butremember, theymightusetheletter j wherewehaveused i Fortheremainderofthetext,wewillrestrictourattentiontorealnumbers.Wedothisprimarily becausetherstCalculussequenceyouwilltake,ostensiblytheonethatthistextispreparingyou for,studiesonlyfunctionsofrealvariables.Also,lotsofreallycoolscienticthingsdon'trequireany deepunderstandingofcomplexnumberstostudythem,buttheydoneedmoreMathematicslike exponential,logarithmicandtrigonometricfunctions.Webelieveitmakesmoresensepedagogically foryoutolearnaboutthosefunctionsnowandthen,afteryou'vecompletedtheCalculussequence, takeacourseinComplexFunctionTheoryinyourjuniororsenioryear.Itisinthatcoursethat thetruepowerofthecomplexnumbersisreleased.Butfornow,inordertofullyprepareyoufor lifeimmediatelyafterCollegeAlgebra,wewillsaythatfunctionslike f x = 1 x 2 +1 haveadomain ofallrealnumbers,eventhoughweknow x 2 +1=0hastwocomplexsolutions,namely x = i Because x 2 +1 > 0forall real numbers x ,thefraction 1 x 2 +1 isneverundenedinourrealvariable setting. 9 Withtheexceptionoftheexercisesonthenextpage,ofcourse.

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3.4ComplexZerosandtheFundamentalTheoremofAlgebra217 3.4.1Exercises 1.Weknow i 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1whichmeans i 3 = i 2 i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i = )]TJ/F53 10.9091 Tf 8.485 0 Td [(i and i 4 = i 2 i 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=1. Usethisinformationtosimplifythefollowing. a i 5 b i 304 c i 3 d )]TJ/F53 10.9091 Tf 8.485 0 Td [(i 23 2.Let z =3+4 i and w =2 )]TJ/F53 10.9091 Tf 10.39 0 Td [(i .Computethefollowingandexpressyouranswerin a + bi form. a z + w b w )]TJ/F53 10.9091 Tf 10.909 0 Td [(z c z w d z w e w z f w 3 3.Simplifythefollowing. a p )]TJ/F15 10.9091 Tf 8.485 0 Td [(49 b p )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 c p )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 )]TJ/F15 10.9091 Tf 8.484 0 Td [(16 d p 49 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 4.Findthecomplexsolutionsofthefollowingquadraticequations. a x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +13=0b3 x 2 +2 x +10=0 5.Foreachpolynomialgivenbelowndallofitszeros,completelyfactoritovertherealnumbers andcompletelyfactoritoverthecomplexnumbers. a x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +5 b x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 +9 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(18 c x 3 +6 x 2 +6 x +5 d3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 x 2 +43 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 e x 4 +9 x 2 +20 f4 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 +13 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x +3 g x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 +27 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +26Hint: x = i isoneofthezeros. h2 x 4 +5 x 3 +13 x 2 +7 x +5Hint: x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+2 i isazero. 6.Let z and w bearbitrarycomplexnumbers.Showthat z w = zw and z = z 7.Withthehelpofyourclassmates,buildapolynomial p withintegercoecientssuchthat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F53 10.9091 Tf 11.531 0 Td [(i isazeroof p p hasalocalmaximumatthepoint ; 0and p x as x !1

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218PolynomialFunctions 3.4.2Answers 1.a i 5 = i 4 i =1 i = i b i 304 = i 4 76 =1 76 =1 c i 3 =8 i 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 i d )]TJ/F53 10.9091 Tf 8.485 0 Td [(i 23 = )]TJ/F53 10.9091 Tf 8.485 0 Td [(i 23 = )]TJ/F53 10.9091 Tf 8.485 0 Td [(i 20 i 3 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F53 10.9091 Tf 8.485 0 Td [(i = i 2.a z + w =5+3 i b w )]TJ/F53 10.9091 Tf 10.909 0 Td [(z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 i c z w =10+5 i d z w = 2 5 + 11 5 i e w z = 2 25 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(11 25 i f w 3 =2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(11 i 3.a p )]TJ/F15 10.9091 Tf 8.485 0 Td [(49=7 i b p )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(16= i i =12 i 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 c p )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 )]TJ/F15 10.9091 Tf 8.484 0 Td [(16= p 144=12 d p 49 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(4=7 2 i =14 i 4.a x =2 3 i b x = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(1 3 p 29 3 i 5.a x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +5= x )]TJ/F15 10.9091 Tf 10.909 0 Td [(+2 i x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i Zeros: x =1 2 i b x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 +9 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(18= x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 +9 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 i x +3 i Zeros: x =2 ; 3 i c x 3 +6 x 2 +6 x +5= x +5 x 2 + x +1= x +5 x )]TJ/F55 10.9091 Tf 10.909 18.655 Td [( )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(1 2 + p 3 2 i !! x )]TJ/F55 10.9091 Tf 10.909 18.655 Td [( )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(1 2 )]TJ 12.104 16.405 Td [(p 3 2 i !! Zeros: x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 ;x = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(1 2 p 3 2 i d3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 x 2 +43 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(13= x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +13= x )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(+3 i x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 i Zeros: x = 1 3 ;x =2 3 i e x 4 +9 x 2 +20= )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 +4 )]TJ/F53 10.9091 Tf 11.818 -8.837 Td [(x 2 +5 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i x +2 i )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(i p 5 )]TJ/F53 10.9091 Tf 11.818 -8.837 Td [(x + i p 5 Zeros: x = 2 i; i p 5 f4 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 +13 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x +3= x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 x 2 +3= x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 x + p 3 i x )]TJ 10.909 9.024 Td [(p 3 i Zeros: x = 1 2 ;x = p 3 i g x 4 )]TJ/F15 10.9091 Tf 9.477 0 Td [(2 x 3 +27 x 2 )]TJ/F15 10.9091 Tf 9.477 0 Td [(2 x +26= x 2 )]TJ/F15 10.9091 Tf 9.478 0 Td [(2 x +26 x 2 +1= x )]TJ/F15 10.9091 Tf 9.478 0 Td [(+5 i x )]TJ/F15 10.9091 Tf 9.478 0 Td [( )]TJ/F15 10.9091 Tf 9.477 0 Td [(5 i x + i x )]TJ/F53 10.9091 Tf 9.477 0 Td [(i Zeros: x =1 5 i;x = i h2 x 4 +5 x 3 +13 x 2 +7 x +5= )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 +2 x +5 )]TJ/F15 10.9091 Tf 11.818 -8.836 Td [(2 x 2 + x +1 = 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+2 i x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 i x )]TJ/F55 10.9091 Tf 10.909 18.655 Td [( )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(1 4 + i p 7 4 !! x )]TJ/F55 10.9091 Tf 10.909 18.655 Td [( )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(1 4 )]TJ/F53 10.9091 Tf 10.909 0 Td [(i p 7 4 !! Zeros: x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 i; )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(1 4 i p 7 4

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Chapter4 RationalFunctions 4.1IntroductiontoRationalFunctions Ifweadd,subtractormultiplypolynomialfunctionsaccordingtothefunctionarithmeticrules denedinSection1.6,wewillproduceanotherpolynomialfunction.If,ontheotherhand,we dividetwopolynomialfunctions,theresultmaynotbeapolynomial.Inthischapterwestudy rationalfunctions -functionswhichareratiosofpolynomials. Definition 4.1 A rationalfunction isafunctionwhichistheratioofpolynomialfunctions. Saiddierently, r isarationalfunctionifitisoftheform r x = p x q x ; where p and q arepolynomialfunctions a a Accordingtothisdenition,allpolynomialfunctionsarealsorationalfunctions.Take q x =1. AswerecallfromSection1.5,wehavedomainissuesanytimethedenominatorofafractionis zero.Intheexamplebelow,wereviewthisconceptaswellassomeofthearithmeticofrational expressions. Example 4.1.1 Findthedomainofthefollowingrationalfunctions.Writethemintheform p x q x forpolynomialfunctions p and q andsimplify. 1. f x = 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x +1 2. g x =2 )]TJ/F15 10.9091 Tf 21.889 7.38 Td [(3 x +1 3. h x = 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F15 10.9091 Tf 12.105 7.381 Td [(3 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 4. r x = 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Solution. 1.Tondthedomainof f ,weproceedaswedidinSection1.5:wendthezerosofthe denominatorandexcludethemfromthedomain.Setting x +1=0resultsin x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Hence,

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220RationalFunctions ourdomainis ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 .Theexpression f x isalreadyintheformrequestedand whenwecheckforcommonfactorsamongthenumeratoranddenominatorwendnone,so wearedone. 2.Proceedingasbefore,wedeterminethedomainof g bysolving x +1=0.Asbefore,wend thedomainof g is ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 .Towrite g x intheformrequested,weneedtoget acommondenominator g x =2 )]TJ/F15 10.9091 Tf 21.888 7.38 Td [(3 x +1 = 2 1 )]TJ/F15 10.9091 Tf 21.888 7.38 Td [(3 x +1 = x +1 x +1 )]TJ/F15 10.9091 Tf 21.889 7.38 Td [(3 x +1 = x +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x +1 = 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x +1 Thisformulaisalsocompletelysimplied. 3.Thedenominatorsintheformulafor h x areboth x 2 )]TJ/F15 10.9091 Tf 11.468 0 Td [(1whosezerosare x = 1.Asa result,thedomainof h is ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 [ ; 1 .Wenowproceedtosimplify h x Sincewehavethesamedenominatorinbothterms,wesubtractthenumerators.Wethen factortheresultingnumeratoranddenominator,andcanceloutthecommonfactor. h x = 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x +2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x +1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x +1 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x +1 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 = 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x +1

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4.1IntroductiontoRationalFunctions221 4.Tondthedomainof r ,itmayhelptotemporarilyrewrite r x as r x = 2 x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Weneedtosetallofthedenominatorsequaltozerowhichmeansweneedtosolvenotonly x 2 )]TJ/F15 10.9091 Tf 10.958 0 Td [(1=0,butalso 3 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 =0.Wend x = 1fortheformerand x = 2 3 forthelatter.Our domainis ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 [ )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 2 3 [ )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(2 3 ; 1 [ ; 1 .Wesimplify r x byrewritingthedivisionas multiplicationbythereciprocalandthensimplifying r x = 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F53 10.9091 Tf 11.818 -8.837 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 AfewremarksaboutExample4.1.1areinorder.Notethattheexpressionsfor f x g x and h x workouttobethesame.However,onlytwoofthesefunctionsareactuallyequal.Recallthat functionsareultimatelysetsoforderedpairs, 1 andsofortwofunctionstobeequal,theyneed, amongotherthings,tohavethesamedomain.Since f x = g x and f and g havethesame domain,theyareequalfunctions.Eventhoughtheformula h x isthesameas f x ,thedomain of h isdierentthanthedomainof f ,andthustheyaredierentfunctions. Wenowturnourattentiontothegraphsofrationalfunctions.Considerthefunction f x = 2 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x +1 fromExample4.1.1.Usingagraphingcalculator,weobtain 1 YoushouldreviewSections1.2and1.4ifthisstatementcaughtyouoguard.

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222RationalFunctions Twobehaviorsofthegraphareworthyoffurtherdiscussion.First,notethatthegraphappears to`break'at x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Weknowfromourlastexamplethat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1isnotinthedomainof f whichmeans f )]TJ/F15 10.9091 Tf 8.484 0 Td [(1isundened.Whenwemakeatableofvaluestostudythebehaviorof f near x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1weseethatwecanget`near' x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1fromtwodirections.Wecanchoosevaluesalittle lessthan )]TJ/F15 10.9091 Tf 8.484 0 Td [(1,forexample x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 1, x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 : 01, x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 : 001,andsoon.Thesevaluesaresaidto `approach )]TJ/F15 10.9091 Tf 8.485 0 Td [(1fromthe left .'Similarly,thevalues x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 9, x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 99, x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 999,etc.,aresaid to`approach )]TJ/F15 10.9091 Tf 8.484 0 Td [(1fromthe right .'Ifwemaketwotables,wendthatthenumericalresultsconrm whatweseegraphically. x f x x;f x )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 1 32 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 : 1 ; 32 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 01 302 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 01 ; 302 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 001 3002 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 001 ; 3002 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 0001 30002 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 001 ; 30002 x f x x;f x )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(28 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 9 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(28 )]TJ/F15 10.9091 Tf 8.484 0 Td [(0 : 99 )]TJ/F15 10.9091 Tf 8.485 0 Td [(298 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 99 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(298 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 999 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2998 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 999 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2998 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 9999 )]TJ/F15 10.9091 Tf 8.484 0 Td [(29998 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 9999 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(29998 Asthe x valuesapproach )]TJ/F15 10.9091 Tf 8.485 0 Td [(1fromtheleft,thefunctionvaluesbecomelargerandlargerpositive numbers. 2 Weexpressthissymbolicallybystatingas x !)]TJ/F15 10.9091 Tf 24.54 0 Td [(1 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(, f x !1 .Similarly,using analogousnotation,weconcludefromthetablethatas x !)]TJ/F15 10.9091 Tf 23.189 0 Td [(1 + f x .Forthistypeof unboundedbehavior,wesaythegraphof y = f x hasa verticalasymptote of x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Roughly speaking,thismeansthatnear x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1,thegraphlooksverymuchliketheverticalline x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1. Anotherfeatureworthyofnoteaboutthegraphof y = f x isitseemsto`levelo'ontheleftand righthandsidesofthescreen.Thisisastatementabouttheendbehaviorofthefunction.Aswe discussedinSection3.1,theendbehaviorofafunctionisitsbehavioras x as x attainslarger 3 and largernegativevalueswithoutbound, x ,andas x becomeslargewithoutbound, x !1 Makingtablesofvalues,wend x f x x;f x )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 2 : 3333 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 ; 2 : 3333 )]TJ/F15 10.9091 Tf 8.485 0 Td [(100 2 : 0303 )]TJ/F15 10.9091 Tf 8.484 0 Td [(100 ; 2 : 0303 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1000 2 : 0030 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1000 ; 2 : 0030 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10000 2 : 0003 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10000 ; 2 : 0003 x f x x;f x 10 1 : 7273 ; 1 : 7273 100 1 : 9703 ; 1 : 9703 1000 1 : 9970 ; 1 : 9970 10000 1 : 9997 ; 1 : 9997 Fromthetables,weseeas x f x 2 + andas x !1 f x 2 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(.Herethe`+'means `fromabove'andthe` )]TJ/F15 10.9091 Tf 8.485 0 Td [('means`frombelow'.Inthiscase,wesaythegraphof y = f x hasa horizontalasymptote of y =2.Thismeansthattheendbehaviorof f resemblesthehorizontal line y =2,whichexplainsthe`levelingo'behaviorweseeinthecalculator'sgraph.Weformalize theconceptsofverticalandhorizontalasymptotesinthefollowingdenitions. Definition 4.2 Theline x = c iscalleda verticalasymptote ofthegraphofafunction y = f x ifas x c )]TJ/F15 10.9091 Tf 10.721 -3.959 Td [(oras x c + ,either f x !1 or f x 2 WewouldneedCalculustoconrmthisanalytically. 3 Here,theword`larger'meanslargerinabsolutevalue.

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4.1IntroductiontoRationalFunctions223 Definition 4.3 Theline y = c iscalleda horizontalasymptote ofthegraphofafunction y = f x ifas x oras x !1 ,either f x c )]TJ/F15 10.9091 Tf 10.721 -3.958 Td [(or f x c + InourdiscussionfollowingExample4.1.1,wedeterminedthat,despitethefactthattheformulafor h x reducedtothesameformulaas f x ,thefunctions f and h aredierent,since x =1isinthe domainof f ,but x =1isnotinthedomainof h .Ifwegraph h x = 2 x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 )]TJ/F34 7.9701 Tf 10.967 4.295 Td [(3 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 usingagraphing calculator,wearesurprisedtondthatthegraphlooksidenticaltothegraphof y = f x .There isaverticalasymptoteat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,butnear x =1,everythingseemne.Tablesofvaluesprovide numericalevidencewhichsupportsthegraphicalobservation. x h x x;h x 0 : 9 0 : 4210 : 9 ; 0 : 4210 0 : 99 0 : 4925 : 99 ; 0 : 4925 0 : 999 0 : 4992 : 999 ; 0 : 4992 0 : 9999 0 : 4999 : 9999 ; 0 : 4999 x h x x;h x 1 : 1 0 : 5714 : 1 ; 0 : 5714 1 : 01 0 : 5075 : 01 ; 0 : 5075 1 : 001 0 : 5007 : 001 ; 0 : 5007 1 : 0001 0 : 5001 : 0001 ; 0 : 5001 Weseethatas x 1 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(, h x 0 : 5 )]TJ/F15 10.9091 Tf 10.915 -3.959 Td [(andas x 1 + h x 0 : 5 + .Inotherwords,thepointson thegraphof y = h x areapproaching ; 0 : 5,butsince x =1isnotinthedomainof h ,itwould beinaccuratetollinapointat ; 0 : 5.Aswe'vedoneinpastsectionswhensomethinglikethis occurs, 4 weputanopencirclealsocalleda`hole'inthiscase 5 at ; 0 : 5.Belowisadetailed graphof y = h x ,withtheverticalandhorizontalasymptotesasdashedlines. x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(21234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 1 3 4 5 6 7 8 4 Forinstance,graphingpiecewisedenedfunctionsinSection1.7. 5 Staytuned.InCalculus,wewillseehowthese`holes'canbe`plugged'whenembarkingonamoreadvanced studyofcontintuity.

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224RationalFunctions Neither x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1nor x =1areinthedomainof h ,yetweseethebehaviorofthegraphof y = h x isdrasticallydierentnearthesepoints.Thereasonforthisliesinthesecondtolaststepwhenwe simpliedtheformulafor h x inExample4.1.1.Wehad h x = x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x +1 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 .Thereason x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1is notinthedomainof h isbecausethefactor x +1appearsinthedenominatorof h x ;similarly, x =1isnotinthedomainof h becauseofthefactor x )]TJ/F15 10.9091 Tf 9.837 0 Td [(1inthedenominatorof h x .Themajor dierencebetweenthesetwofactorsisthat x )]TJ/F15 10.9091 Tf 10.565 0 Td [(1cancelswithafactorinthenumeratorwhereas x +1doesnot.Looselyspeaking,thetroublecausedby x )]TJ/F15 10.9091 Tf 11.085 0 Td [(1inthedenominatoriscanceled awaywhilethefactor x +1remainstocausemischief.Thisiswhythegraphof y = h x has averticalasymptoteat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1butonlyaholeat x =1.Theseobservationsaregeneralizedand summarizedinthetheorembelow,whoseproofisfoundinCalculus. Theorem 4.1 LocationofVerticalAsymptotesandHoles: a Suppose r isarationalfunction whichcanbewrittenas r x = p x q x where p and q havenocommonzeros. b Let c bearealnumber whichisnotinthedomainof r If q c 6 =0,thenthegraphof y = r x hasaholeat c; p c q c If q c =0,thenthetheline x = c isaverticalasymptoteofthegraphof y = r x a Or,`Howtotellyourasymptotefromaholeinthegraph.' b Inotherwords, r x isinlowestterms. InEnglish,Theorem4.1saysif x = c isnotinthedomainof r but,whenwesimplify r x ,it nolongermakesthedenominator0,thenwehaveaholeat x = c .Otherwise,wehaveavertical asymptote. Example 4.1.2 Findtheverticalasymptotesof,and/orholesin,thegraphsofthefollowing rationalfunctions.Verifyyouranswersusingagraphingcalculator. 1. f x = 2 x x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2. g x = x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.91 0 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(9 3. h x = x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.91 0 Td [(6 x 2 +9 4. r x = x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +4 x +4 Solution. 1.TouseTheorem4.1,werstndalloftherealnumberswhicharen'tinthedomainof f .To doso,wesolve x 2 )]TJ/F15 10.9091 Tf 11.197 0 Td [(3=0andget x = p 3.Sincetheexpression f x isinlowestterms, thereisnocancellationpossible,andweconcludethatthelines x = )]TJ 8.485 9.024 Td [(p 3and x = p 3are verticalasymptotestothegraphof y = f x .Thecalculatorveriesthisclaim. 2.Solving x 2 )]TJ/F15 10.9091 Tf 11.083 0 Td [(9=0gives x = 3.Inlowestterms g x = x 2 )]TJ/F37 7.9701 Tf 6.587 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(9 = x )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 x +2 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 x +3 = x +2 x +3 .Since x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3continuestomaketroubleinthedenominator,weknowtheline x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3isavertical asymptoteofthegraphof y = g x .Since x =3nolongerproducesa0inthedenominator, wehaveaholeat x =3.Tondthe y -coordinateofthehole,wesubstitute x =3into x +2 x +3

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4.1IntroductiontoRationalFunctions225 andndtheholeisat )]TJ/F15 10.9091 Tf 5 -8.836 Td [(3 ; 5 6 .Whenwegraph y = g x usingacalculator,weclearlyseethe verticalasymptoteat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3,buteverythingseemscalmnear x =3. Thegraphof y = f x Thegraphof y = g x 3.Thedomainof h isallrealnumbers,since x 2 +9=0hasnorealsolutions.Accordingly,the graphof y = h x isdevoidofbothverticalasymptotesandholes. 4.Setting x 2 +4 x +4=0givesus x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2astheonlyrealnumberofconcern.Simplifying, wesee r x = x 2 )]TJ/F37 7.9701 Tf 6.587 0 Td [(x )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 x 2 +4 x +4 = x )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 x +2 x +2 2 = x )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 x +2 .Since x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2continuestoproducea0inthe denominatorofthereducedfunction,weknow x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2isaverticalasymptotetothegraph, whichthecalculatorconrms. Thegraphof y = h x Thegraphof y = r x Ournextexamplegivesusaphysicalinterpretationofaverticalasymptote.Thistypeofmodel arisesfromafamilyofequationscheerilynamed`doomsday'equations. 6 Theunfortunatename willmakesenseshortly. Example 4.1.3 Amathematicalmodelforthepopulation P ,inthousands,ofacertainspeciesof bacteria, t daysafteritisintroducedtoanenvironmentisgivenby P t = 100 )]TJ/F37 7.9701 Tf 6.587 0 Td [(t 2 ,0 t< 5. 1.Findandinterpret P 2.Whenwillthepopulationreach100 ; 000? 6 ThisisaclassofCalculusequationsinwhichapopulationgrowsveryrapidly.

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226RationalFunctions 3.Determinethebehaviorof P as t 5 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(.Interpretthisresultgraphicallyandwithinthe contextoftheproblem. Solution. 1.Substituting t =0gives P = 100 )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 2 =4,whichmeans4000bacteriaareinitiallyintroduced intotheenvironment. 2.Tondwhenthepopulationreaches100 ; 000,werstneedtorememberthat P t ismeasured in thousands .Inotherwords,100 ; 000bacteriacorrespondsto P t =100.Substituting for P t givestheequation 100 )]TJ/F37 7.9701 Tf 6.586 0 Td [(t 2 =100.Clearingdenominatorsanddividingby100gives )]TJ/F53 10.9091 Tf 11.478 0 Td [(t 2 =1,which,afterextractingsquareroots,produces t =4or t =6.Ofthesetwo solutions,only t =4ininourdomain,sothisisthesolutionwekeep.Hence,ittakes4days forthepopulationofbacteriatoreach100 ; 000. 3.Todeterminethebehaviorof P as t 5 )]TJ/F15 10.9091 Tf 7.084 -3.959 Td [(,wecanmakeatable t P t 4 : 9 10000 4 : 99 1000000 4 : 999 100000000 4 : 9999 10000000000 Inotherwords,as t 5 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(, P t !1 .Graphically,theline t =5isaverticalasymptoteof thegraphof y = P t .Physically,thismeansthepopulationofbacteriaisincreasingwithout boundaswenear5days,whichcannotphysicallyhappen.Forthisreason, t =5iscalled the`doomsday'forthispopulation.Thereisnowayanyenvironmentcansupportinnitely manybacteria,soshortlybefore t =5theenvironmentwouldcollapse. Nowthatwehavethoroughlyinvestigatedverticalasymptotes,wenowturnourattentionto horizontalasymptotes.Thenexttheoremtellsuswhentoexpecthorizontalasymptotes. Theorem 4.2 LocationofHorizontalAsymptotes: Suppose r isarationalfunctionand r x = p x q x ,where p and q arepolynomialfunctionswithleadingcoecients a and b ,respectively. Ifthedegreeof p x isthesameasthedegreeof q x ,then y = a b isthe a horizontalasymptote ofthegraphof y = r x Ifthedegreeof p x islessthanthedegreeof q x ,then y =0isthehorizontalasymptote ofthegraphof y = r x Ifthedegreeof p x isgreaterthanthedegreeof q x ,thenthegraphof y = r x hasno horizontalasymptotes. a Theuseofthedenitearticlewillbejustiedmomentarily.

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4.1IntroductiontoRationalFunctions227 LikeTheorem4.1,Theorem4.2isprovedusingCalculus.Nevertheless,wecanunderstandtheidea behinditusingourexample f x = 2 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x +1 .Ifweinterpret f x asadivisionproblem, x )]TJ/F15 10.9091 Tf 8.541 0 Td [(1 x +1, wendthequotientis2witharemainderof )]TJ/F15 10.9091 Tf 8.484 0 Td [(3.Usingwhatweknowaboutpolynomialdivision, specicallyTheorem3.4,weget2 x )]TJ/F15 10.9091 Tf 11.806 0 Td [(1=2 x +1 )]TJ/F15 10.9091 Tf 11.806 0 Td [(3.Dividingbothsidesby x +1gives 2 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x +1 =2 )]TJ/F34 7.9701 Tf 16.974 4.295 Td [(3 x +1 .Youmayrememberthisastheformulafor g x inExample4.1.1.As x becomes unboundedineitherdirection,thequantity 3 x +1 getscloserandclosertozerosothatthevaluesof f x becomecloserandcloserto2.Insymbols,as x f x 2,andwehavetheresult. 7 Noticethatthegraphgetsclosetothesame y valueas x or x !1 .Thismeansthat thegraphcanhaveonlyone horizontalasymptoteifitisgoingtohaveoneatall.Thuswewere justiedinusing`the'intheprevioustheorem.Bytheway,usinglongdivisiontodeterminethe asymptotewillserveuswellinthenextsectionsoyoumightwanttoreviewthattopic. Alternatively,wecanusewhatweknowaboutendbehaviorofpolynomialstohelpusunderstand thistheorem.FromTheorem3.2,weknowtheendbehaviorofapolynomialisdeterminedbyits leadingterm.Applyingthistothenumeratoranddenominatorof f x ,wegetthatas x f x = 2 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x +1 2 x x =2.ThislastapproachisusefulinCalculus,and,indeed,ismaderigorousthere. Keepthisinmindfortheremainderofthisparagraph.Applyingthisreasoningtothegeneral case,suppose r x = p x q x where a istheleadingcoecientof p x and b istheleadingcoecient of q x .As x r x ax n bx m ,where n and m arethedegreesof p x and q x ,respectively. Ifthedegreeof p x andthedegreeof q x arethesame,then n = m sothat r x a b ,which means y = a b isthehorizontalasymptoteinthiscase.Ifthedegreeof p x islessthanthedegree of q x ,then nm ,andhence n )]TJ/F53 10.9091 Tf 11.2 0 Td [(m isapositive numberand r x ax n )]TJ/F38 5.9776 Tf 5.756 0 Td [(m b ,whichbecomesunboundedas x .Aswesaidbefore,ifarational functionhasahorizontalasymptote,thenitwillhaveonlyone.Thisisnottrueforothertypes offunctionsweshallseeinlaterchapters. Example 4.1.4 Determinethehorizontalasymptotes,ifany,ofthegraphsofthefollowingfunctions.Verifyyouranswersusingagraphingcalculator. 1. f x = 5 x x 2 +1 2. g x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +1 3. h x = 6 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x +1 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 Solution. 1.Thenumeratorof f x is5 x ,whichisdegree1.Thedenominatorof f x is x 2 +1,which isdegree2.ApplyingTheorem4.2, y =0isthehorizontalasymptote.Sureenough,as x ,thegraphof y = f x getscloserandclosertothe x -axis. 2.Thenumeratorof g x x 2 )]TJ/F15 10.9091 Tf 9.924 0 Td [(4,isdegree2,butthedegreeofthedenominator, x +1,isdegree 1.ByTheorem4.2,thereisnohorizontalasymptote.Fromthegraph,weseethegraphof 7 Notethatas x f x 2 + ,whereasas x !1 f x 2 )]TJ/F63 8.9664 Tf 6.254 -3.809 Td [(.Wewrite f x 2ifweareunconcerned fromwhichdirectionthefunctionvalues f x approachthenumber2.

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228RationalFunctions y = g x doesn'tappeartolevelotoaconstantvalue,sothereisnohorizontalasymptote. 8 3.Thedegreesofthenumeratoranddenominatorof h x areboththree,soTheorem4.2tells us y = 6 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3isthehorizontalasymptote.Thecalculatorconrmsthis. Thegraphof y = f x Thegraphof y = g x Thegraphof y = h x Ourlastexampleofthesectiongivesusareal-worldapplicationofahorizontalasymptote.Though thepopulationbelowismoreaccuratelymodeledwiththefunctionsinChapter6,weapproximate it 9 usingarationalfunction. Example 4.1.5 Thenumberofstudents, N ,atlocalcollegewhohavehadtheu t monthsafter thesemesterbeginscanbemodeledbytheformula N t =500 )]TJ/F34 7.9701 Tf 14.809 4.295 Td [(450 1+3 t for t 0. 1.Findandinterpret N 2.Howlongwillittakeuntil300studentswillhavehadtheu? 3.Determinethebehaviorof N as t !1 .Interpretthisresultgraphicallyandwithinthe contextoftheproblem. Solution. 1. N =500 )]TJ/F34 7.9701 Tf 19.128 4.295 Td [(450 1+3 =50.Thismeansthatatthebeginningofthesemester,50students havehadtheu. 2.Weset N t =300toget500 )]TJ/F34 7.9701 Tf 14.111 4.295 Td [(450 1+3 t =300andsolve.Isolatingthefractiongives 450 1+3 t =200. Clearingdenominatorsgives450=200+3 t .Finally,weget t = 5 12 .Thismeansitwill take 5 12 months,orabout13days,for300studentstohavehadtheu. 3.Todeterminethebehaviorof N as t !1 ,wecanuseatable. 8 Thegraphdoes,however,seemtoresembleanon-constantlineas x .Wewilldiscussthisphenomenon inthenextsection. 9 Usingtechniquesyou'llseeinCalculus.

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4.1IntroductiontoRationalFunctions229 t N t 10 485 : 48 100 498 : 50 1000 499 : 85 10000 499 : 98 Thetablesuggeststhatas t !1 N t 500.Morespecically,500 )]TJ/F15 10.9091 Tf 7.084 -3.959 Td [(.Thismeansas timegoesby,onlyatotalof500studentswillhaveeverhadtheu.

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230RationalFunctions 4.1.1Exercises 1.Foreachrationalfunction f givenbelow: Findthedomainof f Identifyanyverticalasymptotesofthegraphof y = f x anddescribethebehaviorof thegraphnearthemusingpropernotation. Identifyanyholesinthegraph. Findthehorizontalasymptote,ifitexists,anddescribetheendbehaviorof f using propernotation. a f x = x 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 b f x = 3+7 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x c f x = x x 2 + x )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 d f x = x x 2 +1 e f x = x +7 x +3 2 f f x = x 3 +1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2.InExercise9inSection1.5,thepopulationofSasquatchinPortageCountywasmodeled bythefunction P t = 150 t t +15 ,where t =0representstheyear1803.Findthehorizontal asymptoteofthegraphof y = P t andexplainwhatitmeans. 3.InExercise8inSection3.1,wetafewpolynomialmodelstothefollowingelectriccircuit data.Thecircuitwasbuiltwithavariableresistor.Foreachofthefollowingresistancevalues measuredinkilo-ohms, k ,thecorrespondingpowertotheloadmeasuredinmilliwatts, mW isgiveninthetablebelow. 10 Resistance: k 1.012 2.199 3.275 4.676 6.805 9.975 Power: mW 1.063 1.496 1.610 1.613 1.505 1.314 Usingsomefundamentallawsofcircuitanalysismixedwithahealthydoseofalgebra,wecan derivetheactualformularelatingpowertoresistance.Forthiscircuit,itis P x = 25 x x +3 : 9 2 where x istheresistancevalue, x 0. aGraphthedataalongwiththefunction y = P x onyourcalculator. bApproximatethemaximumpowerthatcanbedeliveredtotheload.Whatisthecorrespondingresistancevalue? cFindandinterprettheendbehaviorof P x as x !1 4.Inhisnowfamous1919dissertationTheLearningCurveEquation ,LouisLeonThurstone presentsarationalfunctionwhichmodelsthenumberofwordsapersoncantypeinfour 10 TheauthorswishtothankDonAnthanandKenWhiteofLakelandCommunityCollegefordevisingthisproblem andgeneratingtheaccompanyingdataset.

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4.1IntroductiontoRationalFunctions231 minutesasafunctionofthenumberofpagesofpracticeonehascompleted.Thispaper, whichisnowinthepublicdomainandcanbefoundhere ,isfromabygoneerawhenstudents atbusinessschoolstooktypingclassesonmanualtypewriters.Usinghisoriginalnotation andoriginallanguage,wehave Y = L X + P X + P + R where L isthepredictedpracticelimitinterms ofspeedunits, X ispageswritten, Y iswritingspeedintermsofwordsinfourminutes, P is equivalentpreviouspracticeintermsofpagesand R istherateoflearning.InFigure5ofthe paper,hegraphsascatterplotandthecurve Y = 216 X +19 X +148 .Discussthisequationwithyour classmates.Howwouldyouupdatethenotation?Explainwhatthehorizontalasymptoteof thegraphmeans.Youshouldtakesometimetolookattheoriginalpaper.Skipoverthe computationsyoudon'tunderstandyetandtrytogetasenseofthetimeandplaceinwhich thestudywasconducted.

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232RationalFunctions 4.1.2Answers 1.a f x = x 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 Domain: ; 2 [ ; 1 Verticalasymptote: x =2 As x 2 )]TJ/F53 10.9091 Tf 7.085 -3.959 Td [(;f x As x 2 + ;f x !1 Noholesinthegraph Horizontalasymptote: y = 1 3 As x ;f x 1 3 )]TJ/F15 10.9091 Tf -108.398 -22.127 Td [(As x !1 ;f x 1 3 + b f x = 3+7 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x Domain: ; 5 2 [ 5 2 ; 1 Verticalasymptote: x = 5 2 As x 5 2 )]TJ/F53 10.9091 Tf 7.084 -6.276 Td [(;f x !1 As x 5 2 + ;f x Noholesinthegraph Horizontalasymptote: y = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(7 2 As x ;f x !)]TJ/F34 7.9701 Tf 23.62 4.295 Td [(7 2 + As x !1 ;f x !)]TJ/F34 7.9701 Tf 23.62 4.295 Td [(7 2 )]TJ0 g 0 G/F15 10.9091 Tf -127.186 -37.361 Td [(c f x = x x 2 + x )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 = x x +4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 Domain: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 3 [ ; 1 Verticalasymptotes: x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ;x =3 As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(4 )]TJ/F53 10.9091 Tf 7.085 -3.958 Td [(;f x As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(4 + ;f x !1 As x 3 )]TJ/F53 10.9091 Tf 7.085 -3.959 Td [(;f x As x 3 + ;f x !1 Noholesinthegraph Horizontalasymptote: y =0 As x ;f x 0 )]TJ/F15 10.9091 Tf -107.227 -17.508 Td [(As x !1 ;f x 0 + d f x = x x 2 +1 Domain: ; 1 Noverticalasymptotes Noholesinthegraph Horizontalasymptote: y =0 As x ;f x 0 )]TJ/F15 10.9091 Tf -107.227 -17.508 Td [(As x !1 ;f x 0 + e f x = x +7 x +3 2 Domain: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1 Verticalasymptote: x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(3 )]TJ/F53 10.9091 Tf 7.085 -3.959 Td [(;f x !1 As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(3 + ;f x !1 Noholesinthegraph Horizontalasymptote: y =0 As x ;f x 0 )]TJ0 0 1 rg 0 0 1 RG/F34 7.9701 Tf 7.085 0 Td [(11 As x !1 ;f x 0 + f f x = x 3 +1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Domain: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 [ ; 1 Verticalasymptote: x =1 As x 1 )]TJ/F53 10.9091 Tf 7.085 -3.959 Td [(;f x As x 1 + ;f x !1 Holeat )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 Nohorizontalasymptote As x ;f x As x !1 ;f x !1 2.Thehorizontalasymptoteofthegraphof P t = 150 t t +15 is y =150anditmeansthatthemodel predictsthepopulationofSasquatchinPortageCountywillneverexceed150. 11 Thisishardtoseeonthecalculator,buttrustme,thegraphisbelowthe x -axistotheleftof x = )]TJ/F63 8.9664 Tf 7.167 0 Td [(7.

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4.1IntroductiontoRationalFunctions233 3.a bThemaximumpowerisapproximately1 : 603 mW whichcorrespondsto3 : 9 k cAs x !1 ;P x 0 + whichmeansastheresistanceincreaseswithoutbound,the powerdiminishestozero.

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234RationalFunctions 4.2GraphsofRationalFunctions Inthissection,wetakeacloserlookatgraphingrationalfunctions.InSection4.1,welearnedthat thegraphsofrationalfunctionsmayincludeverticalasymptotes,holesinthegraph,andhorizontal asymptotes.Theorems4.1and4.2tellusexactlywhenandwherethesebehaviorswilloccur,and ifwecombinetheseresultswithwhatwealreadyknowaboutgraphingfunctions,wewillquickly beabletogeneratereasonablegraphsofrationalfunctions. OneofthestandardtoolswewilluseisthesigndiagramwhichwasrstintroducedinSection2.4, andthenrevisitedinSection3.1.Inthosesections,weoperatedunderthebeliefthatafunction couldn'tchangeitssignwithoutitsgraphcrossingthroughthe x -axis.Themajortheoremwe usedtojustifythisbeliefwastheIntermediateValueTheorem,Theorem3.1.Itturnsoutthe IntermediateValueTheoremappliestoall continuous functions, 1 notjustpolynomials.Although rationalfunctionsarecontinuousontheirdomains, 2 Theorem4.1tellsusverticalasymptotesand holesoccuratthevaluesexcludedfromtheirdomains.Inotherwords,rationalfunctionsaren't continuousattheseexcludedvalueswhichleavesopenthepossibilitythatthefunctioncouldchange sign without crossingthroughthe x -axis.Considerthegraphof y = h x fromExample4.1.1, recordedbelowforconvenience.Wehaveaddedits x -interceptat )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 ; 0 forthediscussionthat follows.Supposewewishtoconstructasigndiagramfor h x .Recallthattheintervalswhere h x > 0,or+,correspondtothe x -valueswherethegraphof y = h x is above the x -axis;the intervalsonwhich h x < 0,or )]TJ/F15 10.9091 Tf 8.485 0 Td [(correspondtowherethegraphis below the x -axis. x y 1 2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(21234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 1 3 4 5 6 7 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 2 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + + Asweexaminethegraphof y = h x ,readingfromlefttoright,wenotethatfrom ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, 1 Recallthat,forourpurposes,thismeansthegraphsaredevoidofanybreaks,jumpsorholes 2 AnotherresultfromCalculus.

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4.2GraphsofRationalFunctions235 thegraphisabovethe x -axis,so h x is+there.At x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1,wehaveaverticalasymptote,at whichpointthegraph`jumps'acrossthe x -axis.Ontheinterval )]TJ/F54 10.9091 Tf 5.001 -8.836 Td [()]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 1 2 ,thegraphisbelowthe x -axis,so h x is )]TJ/F15 10.9091 Tf 8.485 0 Td [(there.Thegraphcrossesthroughthe x -axisat )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 2 ; 0 andremainsabovethe x -axisuntil x =1,wherewehavea`hole'inthegraph.Since h isundened,thereisnosign here.Sowehave h x as+ontheinterval )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 2 ; 1 .Continuing,weseethaton ; 1 ,thegraph of y = h x isabovethe x -axis,andsowemark+there.Toconstructasigndiagramfromthis information,wenotonlyneedtodenotethezeroof h ,butalsotheplacesnotinthedomainof h .Asisourcustom,wewrite`0'above 1 2 onthesigndiagramtoremindusthatitisazeroof h Weneedadierentnotationfor )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and1,andwehavechosentouse` '-anonstandardsymbol calledtheinterrobang .Weusethissymboltoconveyasenseofsurprise,caution,andwonderment -anappropriateattitudetotakewhenapproachingthesepoints.Themoralofthestoryisthat whenconstructingsigndiagramsforrationalfunctions,weincludethezerosaswellasthevalues excludedfromthedomain. StepsforConstructingaSignDiagramforaRationalFunction Suppose r isarationalfunction. 1.Placeanyvaluesexcludedfromthedomainof r onthenumberlinewithan` 'abovethem. 2.Findthezerosof r andplacethemonthenumberlinewiththenumber0abovethem. 3.Chooseatestvalueineachoftheintervalsdeterminedinsteps1and2. 4.Determinethesignof r x foreachtestvalueinstep3,andwritethatsignabovethe correspondinginterval. Wenowpresentourprocedureforgraphingrationalfunctionsandapplyittoafewexhaustive examples.Pleasenotethatwedecreasetheamountofdetailgivenintheexplanationsaswemove throughtheexamples.Thereadershouldbeabletollinanydetailsinthosestepswhichwehave abbreviated. StepsforGraphingRationalFunctions Suppose r isarationalfunction. 1.Findthedomainof r 2.Reduce r x tolowestterms,ifapplicable. 3.Findthe x -and y -interceptsofthegraphof y = r x ,iftheyexist. 4.Determinethelocationofanyverticalasymptotesorholesinthegraph,iftheyexist. Analyzethebehaviorof r oneithersideoftheverticalasymptotes,ifapplicable. 5.Analyzetheendbehaviorof r .Uselongdivision,asneeded. 6.Useasigndiagramandplotadditionalpoints,asneeded,tosketchthegraphof y = r x

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236RationalFunctions Example 4.2.1 Sketchadetailedgraphof f x = 3 x x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 Solution. Wefollowthesixstepprocedureoutlinedabove. 1.Asusual,wesetthedenominatorequaltozerotoget x 2 )]TJ/F15 10.9091 Tf 11.21 0 Td [(4=0.Wend x = 2,soour domainis ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 2 [ ; 1 2.Toreduce f x tolowestterms,wefactorthenumeratoranddenominatorwhichyields f x = 3 x x )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x +2 .Therearenocommonfactorswhichmeans f x isalreadyinlowest terms. 3.Tondthe x -interceptsofthegraphof y = f x ,weset y = f x =0.Solving 3 x x )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x +2 =0 resultsin x =0.Since x =0isinourdomain, ; 0isthe x -intercept.Tondthe y -intercept, weset x =0andnd y = f =0,sothat ; 0isour y -interceptaswell. 3 4.Thetwonumbersexcludedfromthedomainof f are x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x =2.Since f x didn't reduceatall,bothofthesevaluesof x stillcausetroubleinthedenominator,andso,by Theorem4.1, x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x =2areverticalasymptotesofthegraph.Wecanactuallygo astepfartheratthispointanddetermineexactlyhowthegraphapproachestheasymptote neareachofthesevalues.Thoughnotabsolutelynecessary, 4 itisgoodpracticeforthose headingotoCalculus.Forthediscussionthatfollows,itisbesttousethefactoredformof f x = 3 x x )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x +2 Thebehaviorof y = f x as x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(2 : Suppose x !)]TJ/F15 10.9091 Tf 22.425 0 Td [(2 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(.Ifweweretobuildatableof values,we'duse x -valuesalittlelessthan )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,say )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 : 1, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 : 01and )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 : 001.Whilethere isnoharminactuallybuildingatablelikewedidinSection4.1,wewanttodevelopa `numbersense'here.Let'sthinkabouteachfactorintheformulaof f x asweimagine substitutinganumberlike x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 : 000001into f x .Thequantity3 x wouldbevery closeto )]TJ/F15 10.9091 Tf 8.485 0 Td [(6,thequantity x )]TJ/F15 10.9091 Tf 10.047 0 Td [(2wouldbeverycloseto )]TJ/F15 10.9091 Tf 8.485 0 Td [(4,andthefactor x +2would beverycloseto0.Morespecically, x +2wouldbealittlelessthan0,inthiscase, )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 000001 : Wewillcallsuchanumbera`verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [(',`verysmall'meaningcloseto zeroinabsolutevalue.So,mentally,as x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(2 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(,weestimate f x = 3 x x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [( = 3 2verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [( Now,thecloser x getsto )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,thesmaller x +2willbecome,andsoeventhoughwe aremultiplyingour`verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [('by2,thedenominatorwillcontinuetogetsmaller andsmaller,andremainnegative.Theresultisafractionwhosenumeratorispositive, butwhosedenominatorisverysmallandnegative.Mentally, f x 3 2verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [( 3 verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [( verybig )]TJ/F15 10.9091 Tf 8.485 0 Td [( 3 Aswementionedatleastonceearlier,sincefunctionscanhaveatmostone y -intercept,oncewend ; 0ison thegraph,weknowitisthe y -intercept. 4 Thesigndiagraminstep6willalsodeterminethebehaviorneartheverticalasymptotes.

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4.2GraphsofRationalFunctions237 Theterm`verybig )]TJ/F15 10.9091 Tf 8.485 0 Td [('meansanumberwithalargeabsolutevaluewhichisnegative. 5 Whatallofthismeansisthatas x !)]TJ/F15 10.9091 Tf 23.173 0 Td [(2 )]TJ/F15 10.9091 Tf 7.085 -3.958 Td [(, f x .Nowsupposewewantedto determinethebehaviorof f x as x !)]TJ/F15 10.9091 Tf 23.74 0 Td [(2 + .Ifweimaginesubstitutingsomethinga littlelargerthan )]TJ/F15 10.9091 Tf 8.485 0 Td [(2infor x ,say )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 999999,wementallyestimate f x )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4verysmall+ = 3 2verysmall+ 3 verysmall+ verybig+ Weconcludethatas x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(2 + f x !1 Thebehaviorof y = f x as x 2 : Consider x 2 )]TJ/F15 10.9091 Tf 7.084 -3.959 Td [(.Weimaginesubstituting x =1 : 999999.Approximating f x aswedidabove,weget f x 6 verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [( = 3 2verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [( 3 verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [( verybig )]TJ/F15 10.9091 Tf 8.485 0 Td [( Weconcludethatas x 2 )]TJ/F15 10.9091 Tf 7.085 -3.958 Td [(, f x .Similarly,as x 2 + ,weimaginesubstituting x =2 : 000001,weget f x 3 verysmall+ verybig+.Soas x 2 + ;f x !1 Graphically,wehavethatnear x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x =2thegraphof y = f x lookslike 6 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(113 5.Next,wedeterminetheendbehaviorofthegraphof y = f x .Sincethedegreeofthe numeratoris1,andthedegreeofthedenominatoris2,Theorem4.2tellsusthat y =0 isthehorizontalasymptote.Aswiththeverticalasymptotes,wecangleanmoredetailed informationusing`numbersense'.Forthediscussionbelow,weusetheformula f x = 3 x x 2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 Thebehaviorof y = f x as x : Ifweweretomakeatableofvaluestodiscuss thebehaviorof f as x ,wewouldsubstitutevery`large'negativenumbersinfor x ,say,forexample, x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1billion.Thenumerator3 x wouldthenbe )]TJ/F15 10.9091 Tf 8.485 0 Td [(3billion,whereas 5 Theactualretailvalueof f )]TJ/F63 8.9664 Tf 7.167 0 Td [(2 : 000001isapproximately )]TJ/F63 8.9664 Tf 7.168 0 Td [(1 ; 500 ; 000. 6 Wehavedeliberatelyleftothelabelsonthe y -axisbecauseweknowonlythebehaviornear x = 2,notthe actualfunctionvalues.

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238RationalFunctions thedenominator x 2 )]TJ/F15 10.9091 Tf 11.283 0 Td [(4wouldbe )]TJ/F15 10.9091 Tf 8.485 0 Td [(1billion 2 )]TJ/F15 10.9091 Tf 11.283 0 Td [(4,whichisprettymuchthesameas 1billion 2 .Hence, f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1billion )]TJ/F15 10.9091 Tf 8.485 0 Td [(3billion 1billion 2 )]TJ/F15 10.9091 Tf 33.317 7.38 Td [(3 billion verysmall )]TJ/F15 10.9091 Tf 8.484 0 Td [( Noticethatifwesubstitutedin x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1trillion,essentiallythesamekindofcancellation wouldhappen,andwewouldbeleftwithaneven`smaller'negativenumber.Thisnot onlyconrmsthefactthatas x f x 0,ittellsusthat f x 0 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(.Inother words,thegraphof y = f x isalittlebit below the x -axisaswemovetothefarleft. Thebehaviorof y = f x as x !1 : Ontheipside,wecanimaginesubstitutingvery largepositivenumbersinfor x andlookingatthebehaviorof f x .Forexample,let x =1billion.Proceedingasbefore,weget f billion 3billion 1billion 2 3 billion verysmall+ Thelargerthenumberweputin,thesmallerthepositivenumberwewouldgetout.In otherwords,as x !1 f x 0 + ,sothegraphof y = f x isalittlebit above the x -axisaswelooktowardthefarright. Graphically,wehave 7 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 6.Lastly,weconstructasigndiagramfor f x .The x -valuesexcludedfromthedomainof f are x = 2,andtheonlyzeroof f is x =0.Displayingtheseappropriatelyonthenumber linegivesusfourtestintervals,andwechoosethetestvalues 8 we x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3, x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, x =1, and x =3.Wend f )]TJ/F15 10.9091 Tf 8.485 0 Td [(3is )]TJ/F15 10.9091 Tf 8.485 0 Td [(, f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1is+, f is )]TJ/F15 10.9091 Tf 8.485 0 Td [(,and f is+.Combiningthis withourpreviouswork,wegetthegraphof y = f x below. 7 Aswiththeverticalasymptotesinthepreviousstep,weknowonlythebehaviorofthegraphas x .For thatreason,weprovideno x -axislabels. 8 Inthisparticularcase,wecaneschewtestvalues,sinceouranalysisofthebehaviorof f nearthevertical asymptotesandourendbehavioranalysishavegivenusthesignsoneachofthetestintervals.Ingeneral,however, thiswon'talwaysbethecase,sofordemonstrationpurposes,wecontinuewithourusualconstruction.

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4.2GraphsofRationalFunctions239 )]TJ/F15 10.9091 Tf 8.485 0 Td [(202 )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 1 + 3 x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11345 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 Acoupleofnotesareinorder.First,thegraphof y = f x certainlyseemstopossesssymmetry withrespecttotheorigin.Infact,wecancheck f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(f x toseethat f isanoddfunction. Insometextbooks,checkingforsymmetryispartofthestandardprocedureforgraphingrational functions;butsinceithappenscomparativelyrarely 9 we'lljustpointitoutwhenweseeit.Also notethatwhile y =0isthehorizontalasymptote,thegraphof f neverthelesscrossesthe x -axis at ; 0.Themyththatgraphsofrationalfunctionscan'tcrosstheirhorizontalasymptotesis completelyfalse,asweshallseeagaininournextexample. Example 4.2.2 Sketchadetailedgraphof g x = 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 Solution. 1.Setting x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6=0gives x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x =3.Ourdomainis ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3 [ ; 1 2.Factoring g x gives g x = x )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 x +1 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 x +2 .Thereisnocancellation,so g x isinlowestterms. 3.Tondthe x -interceptweset y = g x =0.Usingthefactoredformof g x above,wend thezerostobethesolutionsof x )]TJ/F15 10.9091 Tf 11.288 0 Td [(5 x +1=0.Weobtain x = 5 2 and x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Since bothofthesenumbersareinthedomainof g ,wehavetwo x -intercepts, )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(5 2 ; 0 and )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0. Tondthe y -intercept,weset x =0andnd y = g = 5 6 ,soour y -interceptis )]TJ/F15 10.9091 Tf 5 -8.836 Td [(0 ; 5 6 4.Since g x wasgiventousinlowestterms,wehave,onceagainbyTheorem4.1vertical asymptotes x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2and x =3.Keepinginmind g x = x )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 x +1 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 x +2 ,weproceedtoour analysisneareachofthesevalues. Thebehaviorof y = g x as x !)]TJ/F15 10.9091 Tf 22.754 0 Td [(2 : As x !)]TJ/F15 10.9091 Tf 22.754 0 Td [(2 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(,weimaginesubstitutinganumber alittlebitlessthan )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Wehave g x )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5verysmall )]TJ/F15 10.9091 Tf 8.484 0 Td [( 9 verysmall+ verybig+ 9 AndJedoesn'tthinkmuchofittobeginwith...

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240RationalFunctions soas x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(2 )]TJ/F15 10.9091 Tf 7.084 -3.959 Td [(, g x !1 .Ontheipside,as x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(2 + ,weget g x 9 verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [( verybig )]TJ/F15 10.9091 Tf 8.485 0 Td [( so g x Thebehaviorof y = g x as x 3 : As x 3 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(,weimagineplugginginanumberjust shyof3.Wehave g x verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [( 4 verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [( verybig )]TJ/F15 10.9091 Tf 8.485 0 Td [( Hence,as x 3 )]TJ/F15 10.9091 Tf 7.085 -3.958 Td [(, g x .As x 3 + ,weget g x 4 verysmall+ verybig+ so g x !1 Graphically,wehaveagain,withoutlabelsonthe y -axis x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1124 5.Sincethedegreesofthenumeratoranddenominatorof g x arethesame,weknowfrom Theorem4.2thatwecanndthehorizontalasymptoteofthegraphof g bytakingthe ratiooftheleadingtermscoecients, y = 2 1 =2.However,ifwetakethetimetodoa moredetailedanalysis,wewillbeabletorevealsome`hidden'behaviorwhichwouldbelost otherwise. 10 AsinthediscussionfollowingTheorem4.2,weusetheresultofthelongdivision )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 torewrite g x = 2 x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 x 2 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 as g x =2 )]TJ/F37 7.9701 Tf 19.067 4.295 Td [(x )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 x 2 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 : Wefocusour attentionontheterm x )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 x 2 )]TJ/F37 7.9701 Tf 6.587 0 Td [(x )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 10 Thatis,ifyouuseacalculatortograph.Onceagain,Calculusistheultimategraphingpowertool.

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4.2GraphsofRationalFunctions241 Thebehaviorof y = g x as x : Ifimaginesubstituting x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1billioninto x )]TJ/F34 7.9701 Tf 6.587 0 Td [(7 x 2 )]TJ/F37 7.9701 Tf 6.587 0 Td [(x )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 ,weestimate x )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 x 2 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1billion 1billion 2 verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [(. 11 Hence, g x =2 )]TJ/F53 10.9091 Tf 24.255 7.38 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [(=2+verysmall+ Inotherwords,as x ,thegraphof y = g x isalittlebit above theline y =2. Thebehaviorof y = g x as x !1 Toconsider x )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 x 2 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 as x !1 ,weimagine substituting x =1billionand,goingthroughtheusualmentalroutine,nd x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 verysmall+ Hence, g x 2 )]TJ/F15 10.9091 Tf 14.689 0 Td [(verysmall+,inotherwords,thegraphof y = g x isjust below theline y =2as x !1 On y = g x ,wehaveagain,withoutlabelsonthe x -axis x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 6.Finallyweconstructoursigndiagram.Weplacean` 'above x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x =3,anda`0' above x = 5 2 and x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Choosingtestvaluesinthetestintervalsgivesus f x is+on theintervals ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2, )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 5 2 ,and ; 1 ,and )]TJ/F15 10.9091 Tf 8.485 0 Td [(ontheintervals )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(5 2 ; 3 Aswepiecetogetheralloftheinformation,wenotethatthegraphmustcrossthehorizontal asymptoteatsomepointafter x =3inorderforittoapproach y =2fromunderneath.This isthesubtletythatwewouldhavemissedhadweskippedthelongdivisionandsubsequent endbehavioranalysis.Wecan,infact,ndexactlywhenthegraphcrosses y =2.Asaresult ofthelongdivision,wehave g x =2 )]TJ/F37 7.9701 Tf 20.111 4.295 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(7 x 2 )]TJ/F37 7.9701 Tf 6.587 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 .For g x =2,wewouldneed x )]TJ/F34 7.9701 Tf 6.587 0 Td [(7 x 2 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 =0. Thisgives x )]TJ/F15 10.9091 Tf 10.343 0 Td [(7=0,or x =7.Notethat x )]TJ/F15 10.9091 Tf 10.343 0 Td [(7istheremainderwhen2 x 2 )]TJ/F15 10.9091 Tf 10.343 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.344 0 Td [(5isdivided by x 2 )]TJ/F53 10.9091 Tf 10.13 0 Td [(x )]TJ/F15 10.9091 Tf 10.13 0 Td [(6,andsoitmakessensethatfor g x toequalthequotient2,theremainderfrom thedivisionmustbe0.Sureenough,wend g =2.Moreover,itstandstoreasonthat g mustattainarelativeminimumatsomepointpast x =7.Calculusveriesthatat x =13, wehavesuchaminimumatexactly ; 1 : 96.Thereaderischallengedtondcalculator windowswhichshowthegraphcrossingitshorizontalasymptoteononewindow,andthe relativeminimumintheother. 11 Inthedenominator,wewouldhavebillion 2 )]TJ/F63 8.9664 Tf 9.439 0 Td [(1billion )]TJ/F63 8.9664 Tf 9.439 0 Td [(6.It'seasytoseewhythe6isinsignicant,butto ignorethe1billionseemscriminal.However,comparedtobillion 2 ,it'sontheinsignicantside;it's10 18 versus 10 9 .Weareonceagainusingthefactthatforpolynomials,endbehaviorisdeterminedbytheleadingterm,soin thedenominator,the x 2 termwinsoutoverthe x term.

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242RationalFunctions )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 5 2 3 + )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( + x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112456789 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 3 4 5 6 7 8 Ournextexamplegivesusnotonlyaholeinthegraph,butalsosomeslightlydierentendbehavior. Example 4.2.3 Sketchadetailedgraphof h x = 2 x 3 +5 x 2 +4 x +1 x 2 +3 x +2 Solution. 1.Fordomain,youknowthedrill.Solving x 2 +3 x +2=0gives x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Our answeris ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 1 2.Toreduce h x ,weneedtofactorthenumeratoranddenominator.Tofactorthenumerator, weusethetechniques 12 setforthinSection3.3andweget h x = 2 x 3 +5 x 2 +4 x +1 x 2 +3 x +2 = x +1 x +1 2 x +2 x +1 = x +1 x +1 1 2 x +2 x +1 = x +1 x +1 x +2 Wewillusethisreducedformulafor h x aslongaswe'renotsubstituting x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1.Tomake thisexclusionspecic,wewrite h x = x +1 x +1 x +2 x 6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1. 3.Tondthe x -intercepts,asusual,weset h x =0andsolve.Solving x +1 x +1 x +2 =0yields x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 and x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Thelatterisn'tinthedomainof h ,soweexcludeit.Ouronly x interceptis )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 ; 0 .Tondthe y -intercept,weset x =0.Since0 6 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1,wecanusethe reducedformulafor h x andweget h = 1 2 fora y -interceptof )]TJ/F15 10.9091 Tf 5 -8.836 Td [(0 ; 1 2 4.FromTheorem4.1,weknowthatsince x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2stillposesathreatinthedenominatorof thereducedfunction,wehaveaverticalasymptotethere.Asfor x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,wenotethefactor x +1wascanceledfromthedenominatorwhenwereduced h x ,andsoitnolongercauses troublethere.Thismeanswegetaholewhen x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Tondthe y -coordinateofthehole, wesubstitute x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1into x +1 x +1 x +2 ,perTheorem4.1andget0.Hence,wehaveaholeon 12 Betyouneverthoughtyou'dneversee that stuagainbeforetheFinalExam!

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4.2GraphsofRationalFunctions243 the x -axisat )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0.Itshouldmakeyouuncomfortableplugging x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1intothereduced formulafor h x ,especiallysincewe'vemadesuchabigdealconcerningthestipulationabout notletting x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1forthatformula.WhatwearereallydoingistakingaCalculusshort-cut tothemoredetailedkindofanalysisnear x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1whichwewillshowbelow.Speakingof which,forthediscussionthatfollows,wewillusetheformula h x = x +1 x +1 x +2 x 6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1. Thebehaviorof y = h x as x !)]TJ/F15 10.9091 Tf 22.704 0 Td [(2 : As x !)]TJ/F15 10.9091 Tf 22.703 0 Td [(2 )]TJ/F15 10.9091 Tf 7.084 -3.959 Td [(,weimaginesubstitutinganumber alittlebitlessthan )]TJ/F15 10.9091 Tf 8.484 0 Td [(2.Wehave h x )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 verysmall )]TJ/F34 7.9701 Tf 6.586 0 Td [( 3 verysmall )]TJ/F34 7.9701 Tf 6.587 0 Td [( verybig )]TJ/F15 10.9091 Tf 8.485 0 Td [( andsoas x !)]TJ/F15 10.9091 Tf 23.027 0 Td [(2 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(, h x .Ontheothersideof )]TJ/F15 10.9091 Tf 8.484 0 Td [(2,as x !)]TJ/F15 10.9091 Tf 23.026 0 Td [(2 + ,wendthat h x 3 verysmall+ verybig+,so h x !1 Thebehaviorof y = h x as x !)]TJ/F15 10.9091 Tf 22.975 0 Td [(1 As x !)]TJ/F15 10.9091 Tf 22.976 0 Td [(1 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(,weimagineplugginginanumber abitlessthan x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Wehave h x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1verysmall )]TJ/F34 7.9701 Tf 6.587 0 Td [( 1 =verysmall+Hence,as x !)]TJ/F15 10.9091 Tf 23.752 0 Td [(1 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(, h x 0 + .Thismeans,as x !)]TJ/F15 10.9091 Tf 23.752 0 Td [(1 )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(,thegraphisabitabovethepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0.As x !)]TJ/F15 10.9091 Tf 22.806 0 Td [(1 + ,weget h x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1verysmall+ 1 =verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [(.Thisgivesus thatas x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(1 + h x 0 )]TJ/F15 10.9091 Tf 7.084 -3.959 Td [(,sothegraphisalittlebitlowerthan )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0here. Graphically,wehave x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 5.Forendbehavior,wenotethatthedegreeofthenumeratorof h x ,2 x 3 +5 x 2 +4 x +1is3, andthedegreeofthedenominator, x 2 +3 x +2,is2.Theorem4.2isofnohelphere,sincethe degreeofthenumeratorisgreaterthanthedegreeofthedenominator.Thatwon'tstopus, however,inouranalysis.Sinceforendbehaviorweareconsideringvaluesof x as x wearefarenoughawayfrom x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1tousethereducedformula, h x = x +1 x +1 x +2 x 6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1. Toperformlongdivision,wemultiplyoutthenumeratorandget h x = 2 x 2 +3 x +1 x +2 x 6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, and,asaresult,werewrite h x =2 x )]TJ/F15 10.9091 Tf 11.215 0 Td [(1+ 3 x +2 x 6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Asinthepreviousexample,we focusourattentiononthetermgeneratedfromtheremainder, 3 x +2 Thebehaviorof y = h x as x : Substituting x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1billioninto 3 x +2 ,wegetthe estimate 3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1billion verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [(.Hence, h x =2 x )]TJ/F15 10.9091 Tf 8.523 0 Td [(1+ 3 x +2 2 x )]TJ/F15 10.9091 Tf 8.524 0 Td [(1+verysmall )]TJ/F15 10.9091 Tf 8.485 0 Td [(. Thismeansthegraphof y = h x isalittlebit below theline y =2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1as x .

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244RationalFunctions Thebehaviorof y = h x as x !1 : If x !1 ,then 3 x +2 verysmall+.Thismeans h x 2 x )]TJ/F15 10.9091 Tf 11.03 0 Td [(1+verysmall+,orthatthegraphof y = h x isalittlebit above the line y =2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1as x !1 Thisisendbehaviorunlikeanywe'veeverseen.Insteadofapproachingahorizontalline, thegraphisapproachingaslantedline.Forthisreason, y =2 x )]TJ/F15 10.9091 Tf 12.117 0 Td [(1iscalleda slant asymptote 13 ofthegraphof y = h x .Aslantasymptotewillalwaysarisewhenthedegree ofthenumeratorisexactlyonemorethanthedegreeofthedenominator,andthere'snoway todetermineexactlywhatitiswithoutgoingthroughthelongdivision.Graphicallywehave x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 6.Tomakeoursigndiagram,weplacean` 'above x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1anda`0'above x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 Onourfourtestintervals,wend h x is+on )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 ; 1 and h x is )]TJ/F15 10.9091 Tf 8.485 0 Td [(on ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 .Puttingallofourworktogetheryieldsthegraphbelow. )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [( + )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(14 )]TJ/F35 5.9776 Tf 5.756 0 Td [(13 )]TJ/F35 5.9776 Tf 5.756 0 Td [(12 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.756 0 Td [(10 )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 8 9 13 Alsocalledan`oblique'asymptoteinsometexts.

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4.2GraphsofRationalFunctions245 Wecouldaskwhetherthegraphof y = h x crossesitsslantasymptote.Fromtheformula h x =2 x )]TJ/F15 10.9091 Tf 10.654 0 Td [(1+ 3 x +2 x 6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,weseethatif h x =2 x )]TJ/F15 10.9091 Tf 10.653 0 Td [(1,wewouldhave 3 x +2 =0.Sincethiswill neverhappen,weconcludethegraphnevercrossesitsslantasymptote. 14 Weendthissectionwithanexamplethatshowsit'snotallpathologicalweirdnesswhenitcomes torationalfunctionsandtechnologystillhasaroletoplayinstudyingtheirgraphsatthislevel. Example 4.2.4 Sketchthegraphof r x = x 4 +1 x 2 +1 Solution. 1.Thedenominator x 2 +1isneverzerosothedomainis ; 1 2.Withnorealzerosinthedenominator, x 2 +1isanirreduciblequadratic.Ouronlyhopeof reducing r x isif x 2 +1isafactorof x 4 +1.Performinglongdivisiongivesus x 4 +1 x 2 +1 = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ 2 x 2 +1 Theremainderisnotzeroso r x isalreadyreduced. 3.Tondthe x -intercept,we'dset r x =0.Sincetherearenorealsolutionsto x 4 +1 x 2 +1 =0,we haveno x -intercepts.Since r =1,soweget ; 1forthe y -intercept. 4.Thisstepdoesn'tapplyto r ,sinceitsdomainisallrealnumbers. 5.Forendbehavior,onceagain,sincethedegreeofthenumeratorisgreaterthanthatofthe denominator,Theorem4.2doesn'tapply.Weknowfromourattempttoreduce r x thatwe canrewrite r x = x 2 )]TJ/F15 10.9091 Tf 11.122 0 Td [(1+ 2 x 2 +1 ,andsowefocusourattentiononthetermcorresponding totheremainder, 2 x 2 +1 Itshouldbeclearthatas x 2 x 2 +1 verysmall+,which means r x x 2 )]TJ/F15 10.9091 Tf 10.854 0 Td [(1+verysmall+.Sothegraph y = r x isalittlebit above thegraph oftheparabola y = x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(1as x .Graphically, 1 2 3 4 5 x y 14 Butrestassured,somegraphsdo!

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246RationalFunctions 6.Thereisn'tmuchworktodoforasigndiagramfor r x ,sinceitsdomainisallrealnumbers andithasnozeros.Oursoletestintervalis ; 1 ,andsinceweknow r =1,we conclude r x is+forallrealnumbers.Atthispoint,wedon'thavemuchtogoonfor agraph.Belowisacomparisonofwhatwehavedeterminedanalyticallyversuswhatthe calculatorshowsus.Wehavenowaytodetecttherelativeextremaanalytically 15 apartfrom bruteforceplottingofpoints,whichisdonemoreecientlybythecalculator. x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 1 2 3 4 5 6 15 WithoutappealingtoCalculus,ofcourse.

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4.2GraphsofRationalFunctions247 4.2.1Exercises 1.Findtheslantasymptoteofthegraphoftherationalfunction. a f x = x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x +1 x 2 +1 b f x = 2 x 2 +5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 3 x +2 c f x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 + x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 2 +3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 d f x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 3 +4 x x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 2.Usethesix-stepproceduretographeachrationalfunctiongiven.Besuretodrawany asymptotesasdashedlines. a f x = 4 x +2 b f x = 5 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x c f x = 1 x 2 d f x = 1 x 2 + x )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 e f x = 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x +3 f f x = x x 2 + x )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 g f x = )]TJ/F53 10.9091 Tf 8.484 0 Td [(x 3 +4 x x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 h f x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 x 3 + x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 16 3.Example4.2.4showedusthatthesix-stepprocedurecannottelluseverythingofimportance aboutthegraphofarationalfunction.WithoutCalculus,weneedtouseourgraphing calculatorstorevealthehiddenmysteriesofrationalfunctionbehavior.Workingwithyour classmates,useagraphingcalculatortoexaminethegraphsofthefollowingrationalfunctions. Compareandcontrasttheirfeatures.Whichfeaturescanthesix-stepprocessrevealandwhich featurescannotbedetectedbyit? a f x = 1 x 2 +1 b f x = x x 2 +1 c f x = x 2 x 2 +1 d f x = x 3 x 2 +1 4.Graphthefollowingrationalfunctionsbyapplyingtransformationstothegraphof y = 1 x a f x = 1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g x =1 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(3 x c h x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1 x Hint:Divide d j x = 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 Hint:Longdivision Discusswithyourclassmateshowyouwouldgraph f x = ax + b cx + d .Whatrestrictionsmust beplacedon a;b;c and d sothatthegraphisindeedatransformationof y = 1 x ? 16 Onceyou'vedonethesix-stepprocedure,useyourcalculatortographthisfunctionontheviewingwindow [0 ; 12] [0 ; 0 : 25].Whatdoyousee?

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248RationalFunctions 5.InExample3.1.1inSection3.1weshowedthat p x = 4 x + x 3 x isnotapolynomialeventhough itsformulareducedto4+ x 2 for x 6 =0.However,itisarationalfunctionsimilartothose studiedinthesection.Withthehelpofyourclassmates,graph p x 6.Let g x = x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x 3 +24 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 x +135 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x 2 +15 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(7 : Withthehelpofyourclassmates,ndthe x -and y -interceptsofthegraphof g .Findtheintervalsonwhichthefunctionisincreasing,the intervalsonwhichitisdecreasingandthelocalextrema.Findalloftheasymptotesofthe graphof g andanyholesinthegraph,iftheyexist.Besuretoshowallofyourworkincluding anypolynomialorsyntheticdivision.Sketchthegraphof g ,usingmorethanonepictureif necessarytoshowalloftheimportantfeaturesofthegraph.

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4.2GraphsofRationalFunctions249 4.2.2Answers 1.a y = x b y = 2 3 x + 11 9 c y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(18 d y = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2.a f x = 4 x +2 Domain: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 No x -intercepts y -intercept: ; 0 Verticalasymptote: x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(2 )]TJ/F53 10.9091 Tf 7.085 -3.959 Td [(;f x As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(2 + ;f x !1 Horizontalasymptote: y =0 As x ;f x 0 )]TJ/F15 10.9091 Tf -110.257 -17.508 Td [(As x !1 ;f x 0 + x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 b f x = 5 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x Domain: ; 3 [ ; 1 x -intercept: ; 0 y -intercept: ; 0 Verticalasymptote: x =3 As x 3 )]TJ/F53 10.9091 Tf 7.085 -3.959 Td [(;f x !1 As x 3 + ;f x Horizontalasymptote: y = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(5 2 As x ;f x !)]TJ/F34 7.9701 Tf 23.62 4.296 Td [(5 2 + As x !1 ;f x !)]TJ/F34 7.9701 Tf 23.62 4.295 Td [(5 2 )]TJ0 g 0 G0 g 0 GET1 0 0 1 336.618 307.489 cmq1 0 0 1 0 0 cm1 0 0 1 0 120.05 cmq 1 0 0 1 0.24907 0.24907 cm1 0 0 1 0 -120.05 cm0 0 0 rg 0 0 0 RG0.49814 w[ ]0 d1 J1 j10 M0 65.4686 m0.99632 65.62335 l1.99265 65.78252 l2.98897 65.94653 l3.98529 66.11559 l4.98161 66.2898 l5.97794 66.46933 l6.97426 66.65479 l7.96967 66.84587 l8.96599 67.0435 l9.96231 67.24765 l10.95863 67.45866 l11.95496 67.67711 l12.95128 67.903 l13.9476 68.13712 l14.94392 68.37973 l15.94025 68.63132 l16.93657 68.89249 l17.93289 69.16353 l18.92921 69.44537 l19.92554 69.73846 l20.92186 70.04341 l21.91727 70.36082 l22.25003 70.46921 22.5822 70.57957 22.91359 70.69206 c23.2465 70.80518 23.57867 70.92009 23.90991 71.0376 c24.24283 71.15572 24.575 71.27597 24.90623 71.39865 c25.2393 71.52208 25.57146 71.64764 25.90256 71.77596 c26.23563 71.90501 26.56778 72.03651 26.89888 72.1709 c27.2321 72.3062 27.56425 72.44392 27.8952 72.58469 c28.22858 72.72652 28.56088 72.87094 28.89168 73.01869 c29.2252 73.16753 29.55736 73.31923 29.888 73.47429 c30.22168 73.63072 30.55383 73.79019 30.88432 73.9533 c31.21815 74.11809 31.55031 74.28592 31.88065 74.45753 c32.21463 74.63115 32.54694 74.80809 32.87697 74.989 c33.21126 75.17233 33.54341 75.35901 33.87329 75.5501 c34.20773 75.74376 34.54004 75.94124 34.86961 76.14311 c35.20436 76.34819 35.53667 76.5572 35.86594 76.7711 c36.20068 76.98848 36.53284 77.20998 36.86134 77.43648 c37.1967 77.6677 37.52916 77.90332 37.85767 78.14412 c38.19347 78.39023 38.52594 78.64107 38.85399 78.89737 c39.19025 79.16005 39.52272 79.42744 39.85031 79.70078 c40.18703 79.9817 40.51949 80.26765 40.84663 80.55968 c41.18396 80.86082 41.51642 81.16728 41.84296 81.47998 c42.1809 81.80363 42.51335 82.1326 42.83928 82.4684 c43.17783 82.81714 43.51044 83.1715 43.8356 83.53284 c44.17491 83.90984 44.50752 84.29262 44.83192 84.68239 c45.17198 85.09116 45.50476 85.50586 45.82825 85.92787 c46.16922 86.37251 46.50198 86.8234 46.82457 87.28159 c47.16646 87.76727 47.49937 88.25905 47.82089 88.75844 c48.16385 89.29095 48.49677 89.8297 48.81721 90.3759 c49.16122 90.96223 49.4943 91.55495 49.81354 92.1551 c50.15846 92.80316 50.49123 93.45775 50.80894 94.11963 c51.1554 94.84126 51.48862 95.56927 51.80527 96.30458 c52.15309 97.11209 52.48631 97.92569 52.80159 98.74643 c53.15077 99.6558 53.4843 100.57126 53.79791 101.49355 c54.14877 102.52528 54.4823 103.5628 54.79424 104.60701 c55.14676 105.78728 55.48059 106.973 55.79056 108.16513 c56.14491 109.52812 56.47905 110.89642 56.78688 112.27081 c57.14336 113.86229 57.47552 115.45923 57.7832 117.06088 cS55.75183 114.40768 m56.80878 114.9231 57.56137 115.90607 57.7832 117.06088 c57.56137 115.90607 57.89618 114.7142 58.68695 113.84387 cS3.18071 64.44437 m2.34076 65.26733 1.16235 65.64679 0 65.4686 c1.16235 65.64679 2.17297 66.36185 2.7278 67.39865 cS83.68591 1.42303 m83.99739 2.6144 84.32956 3.80028 84.68193 4.98024 c84.99402 6.02505 85.32785 7.06319 85.67886 8.09569 c85.99232 9.01767 86.3257 9.93282 86.67488 10.84204 c86.99 11.66263 87.32324 12.47592 87.6709 13.28313 c87.98755 14.0183 88.32062 14.746 88.66692 15.46747 c88.98509 16.13026 89.31831 16.78577 89.6637 17.43488 c89.98277 18.03474 90.31586 18.6273 90.65971 19.21349 c90.98001 19.75954 91.31293 20.29813 91.65573 20.83034 c91.97755 21.33003 92.31062 21.82211 92.65266 22.3081 c92.9751 22.76613 93.30786 23.21687 93.64868 23.66136 c93.97217 24.0832 94.3048 24.49777 94.6447 24.90639 c94.96895 25.29616 95.30157 25.67879 95.64072 26.05563 c95.96603 26.41714 96.2988 26.77164 96.6375 27.12053 c96.96327 27.45618 97.29588 27.78514 97.63367 28.1088 c97.96005 28.4215 98.29251 28.7278 98.62968 29.0288 c98.95699 29.32097 99.2896 29.60707 99.62646 29.88815 c99.9539 30.16133 100.28638 30.42873 100.62248 30.69125 c100.95053 30.9474 101.28285 31.19824 101.6185 31.44435 c101.94685 31.68515 102.27917 31.92062 102.61452 32.15184 c102.94363 32.37865 103.27625 32.60045 103.61145 32.81813 c103.94057 33.03188 104.27287 33.24089 104.60747 33.44597 c104.93689 33.64784 105.26904 33.84517 105.60349 34.03883 c105.93352 34.22992 106.26582 34.41675 106.60027 34.60008 c106.9303 34.78099 107.26245 34.95778 107.59628 35.1314 c107.92647 35.30301 108.25862 35.47084 108.5923 35.63548 c108.92264 35.79843 109.25479 35.9579 109.58832 36.11433 c109.91911 36.26955 110.25157 36.42125 110.58525 36.57024 c110.9159 36.71783 111.24805 36.86226 111.58127 37.00394 c111.91206 37.1447 112.24422 37.28244 112.57729 37.41757 c112.90854 37.55196 113.24084 37.68361 113.57422 37.81282 c113.9053 37.94113 114.23732 38.0667 114.57024 38.19012 c114.90134 38.3128 115.23349 38.43306 115.56625 38.55103 c115.8975 38.66853 116.2295 38.7836 116.56227 38.89656 c116.89383 39.0092 117.22614 39.11942 117.55905 39.2278 c117.89044 39.33574 118.22246 39.44154 118.55507 39.54552 c119.55109 39.85031 l120.54802 40.14355 l121.54404 40.42525 l122.54005 40.69629 l123.53607 40.9573 l124.53285 41.20905 l125.52902 41.45166 l126.52504 41.68562 l127.52182 41.91182 l128.51784 42.13011 l129.51385 42.34111 l130.50987 42.54512 l131.5068 42.7429 l132.50282 42.93414 l133.49884 43.11945 l134.49562 43.29913 l135.49164 43.47319 l136.48766 43.64224 l137.48367 43.8061 l138.4806 43.96542 l139.47662 44.12018 lS136.29591 45.14441 m137.13586 44.32144 138.31427 43.94199 139.47662 44.12018 c138.31427 43.94199 137.30365 43.22693 136.74883 42.19012 cS85.88771 3.93663 m84.79909 3.49197 83.98335 2.56073 83.68591 1.42303 c83.98335 2.56073 83.7278 3.77206 82.99611 4.69264 cS39.85046 81.19533 m39.4541 81.19533 39.07404 81.03784 38.7938 80.7576 c38.51355 80.47736 38.35606 80.09729 38.35606 79.70093 c38.35606 79.30457 38.51355 78.9245 38.7938 78.64426 c39.07404 78.36401 39.4541 78.20653 39.85046 78.20653 c40.24683 78.20653 40.62689 78.36401 40.90714 78.64426 c41.18738 78.9245 41.34486 79.30457 41.34486 79.70093 c41.34486 80.09729 41.18738 80.47736 40.90714 80.7576 c40.62689 81.03784 40.24683 81.19533 39.85046 81.19533 chf0.49814 w69.73831 0 m69.73831 1.99205 lS69.73831 5.97794 m69.73831 9.96208 lS69.73831 13.94792 m69.73831 17.93124 lS69.73831 21.91786 m69.73831 25.9015 lS69.73831 29.88785 m69.73831 33.8719 lS69.73831 37.85783 m69.73831 41.8418 lS69.73831 45.82777 m69.73831 49.81343 lS69.73831 53.79776 m69.73831 57.78397 lS69.73831 61.76775 m69.73831 65.75395 lS69.73831 69.73769 m69.73831 73.72362 lS69.73831 77.70767 m69.73831 81.69374 lS69.73831 85.67766 m69.73831 89.66377 lS69.73831 93.6476 m69.73831 97.63339 lS69.73831 101.61758 m69.73831 105.60356 lS69.73831 109.58757 m69.73831 113.5735 lS69.73831 117.55934 m69.73831 119.55139 lS0.49814 w0 54.79439 m2.05165 54.79439 lS6.15276 54.79439 m10.25606 54.79439 lS14.35716 54.79439 m18.45996 54.79439 lS22.56152 54.79439 m26.66469 54.79439 lS30.76805 54.79439 m34.86852 54.79439 lS38.97241 54.79439 m43.0732 54.79439 lS47.1768 54.79439 m51.278 54.79439 lS55.38118 54.79439 m59.48228 54.79439 lS63.58556 54.79439 m67.68721 54.79439 lS71.78993 54.79439 m75.89127 54.79439 lS79.99431 54.79439 m84.0977 54.79439 lS88.19868 54.79439 m92.3018 54.79439 lS96.40308 54.79439 m100.50629 54.79439 lS104.60747 54.79439 m108.71072 54.79439 lS112.81183 54.79439 m116.91513 54.79439 lS121.01622 54.79439 m125.1195 54.79439 lS129.22061 54.79439 m133.32391 54.79439 lS137.42497 54.79439 m139.47662 54.79439 lS0 79.70093 m139.47662 79.70093 lS134.49532 82.19159 m135.67125 80.62367 137.51674 79.70093 139.47662 79.70093 c137.51674 79.70093 135.67125 78.77818 134.49532 77.21027 cS0.49814 w39.85046 0 m39.85046 119.55139 lS0.49814 w37.3598 114.57008 m38.92772 115.74602 39.85046 117.5915 39.85046 119.55139 c39.85046 117.5915 40.77321 115.74602 42.34113 114.57008 cS0.49814 w9.96262 77.7084 m9.96262 81.69345 lS19.92523 77.7084 m19.92523 81.69345 lS29.88785 77.7084 m29.88785 81.69345 lS39.85046 77.7084 m39.85046 81.69345 lS49.81308 77.7084 m49.81308 81.69345 lS59.7757 77.7084 m59.7757 81.69345 lS69.73831 77.7084 m69.73831 81.69345 lS79.70093 77.7084 m79.70093 81.69345 lS89.66354 77.7084 m89.66354 81.69345 lS99.62616 77.7084 m99.62616 81.69345 lS109.58878 77.7084 m109.58878 81.69345 lS119.55139 77.7084 m119.55139 81.69345 lS129.514 77.7084 m129.514 81.69345 lS0.49814 w41.84299 9.96262 m37.85794 9.96262 lS41.84299 19.92523 m37.85794 19.92523 lS41.84299 29.88785 m37.85794 29.88785 lS41.84299 39.85046 m37.85794 39.85046 lS41.84299 49.81308 m37.85794 49.81308 lS41.84299 59.7757 m37.85794 59.7757 lS41.84299 69.73831 m37.85794 69.73831 lS41.84299 79.70093 m37.85794 79.70093 lS41.84299 89.66354 m37.85794 89.66354 lS41.84299 99.62616 m37.85794 99.62616 lS41.84299 109.58878 m37.85794 109.58878 lS1 0 0 1 0 120.05 cmQ1 0 0 1 0 -120.05 cmQ1 0 0 1 -336.618 -307.489 cmBT/F37 7.9701 Tf 474.711 381.493 Td [(x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123456789 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 c f x = 1 x 2 Domain: ; 0 [ ; 1 No x -intercepts No y -intercepts Verticalasymptote: x =0 As x 0 )]TJ/F53 10.9091 Tf 7.085 -3.958 Td [(;f x !1 As x 0 + ;f x !1 Horizontalasymptote: y =0 As x ;f x 0 + As x !1 ;f x 0 + x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 1 2 3 4 5

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250RationalFunctions d f x = 1 x 2 + x )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 Domain: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 3 [ ; 1 No x -intercepts y -intercept: ; )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(1 12 Verticalasymptotes: x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4and x =3 As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(4 )]TJ/F53 10.9091 Tf 7.085 -3.959 Td [(;f x !1 As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(4 + ;f x As x 3 )]TJ/F53 10.9091 Tf 7.085 -3.959 Td [(;f x As x 3 + ;f x !1 Horizontalasymptote: y =0 As x ;f x 0 + As x !1 ;f x 0 + x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 e f x = 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x +3 Domain: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1 2 [ 1 2 ; 1 No x -intercepts y -intercept: ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x +3 ;x 6 = 1 2 Holeinthegraphat 1 2 ; )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(2 7 Verticalasymptote: x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(3 )]TJ/F53 10.9091 Tf 7.085 -3.959 Td [(;f x !1 As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(3 + ;f x Horizontalasymptote: y =0 As x ;f x 0 + As x !1 ;f x 0 + x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 f f x = x x 2 + x )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 Domain: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 3 [ ; 1 x -intercept: ; 0 y -intercept: ; 0 Verticalasymptotes: x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4and x =3 As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(4 )]TJ/F53 10.9091 Tf 7.085 -3.958 Td [(;f x As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(4 + ;f x !1 As x 3 )]TJ/F53 10.9091 Tf 7.085 -3.959 Td [(;f x As x 3 + ;f x !1 Horizontalasymptote: y =0 As x ;f x 0 )]TJ/F15 10.9091 Tf -110.257 -17.508 Td [(As x !1 ;f x 0 + x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1

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4.2GraphsofRationalFunctions251 g f x = )]TJ/F53 10.9091 Tf 8.484 0 Td [(x 3 +4 x x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 Domain: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 3 [ ; 1 x -intercepts: )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0 ; ; 0 ; ; 0 y -intercept: ; 0 Verticalasymptotes: x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ;x =3 As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(3 )]TJ/F53 10.9091 Tf 7.085 -3.958 Td [(;f x !1 As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(3 + ;f x As x 3 )]TJ/F53 10.9091 Tf 7.085 -3.959 Td [(;f x !1 As x 3 + ;f x Slantasymptote: y = )]TJ/F53 10.9091 Tf 8.484 0 Td [(x As x ;f x !1 As x !1 ;f x x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123456 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 h f x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 x 3 + x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x Domain: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0 [ ; 1 [ ; 1 f x = x )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 x x +2 ;x 6 =1 No x -intercepts No y -intercepts Verticalasymptotes: x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x =0 As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(2 )]TJ/F53 10.9091 Tf 7.085 -3.958 Td [(;f x As x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(2 + ;f x !1 As x 0 )]TJ/F53 10.9091 Tf 7.085 -3.959 Td [(;f x !1 As x 0 + ;f x Holeinthegraphat ; 0 Horizontalasymptote: y =0 As x ;f x 0 )]TJ/F15 10.9091 Tf -110.257 -17.508 Td [(As x !1 ;f x 0 + x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 4.a f x = 1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 Shiftthegraphof y = 1 x totheright2units. x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3

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252RationalFunctions b g x =1 )]TJ/F15 10.9091 Tf 12.495 7.38 Td [(3 x Verticallystretchthegraphof y = 1 x byafactorof3. Reectthegraphof y = 3 x aboutthe x -axis. Shiftthegraphof y = )]TJ/F15 10.9091 Tf 10.071 7.38 Td [(3 x up1unit. x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123456 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 c h x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x +1 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+ 1 x Shiftthegraphof y = 1 x down2units. x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 d j x = 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 =3 )]TJ/F15 10.9091 Tf 21.889 7.381 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 Shiftthegraphof y = 1 x totheright2units. Reectthegraphof y = 1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 aboutthe x -axis. Shiftthegraphof y = )]TJ/F15 10.9091 Tf 19.465 7.38 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 up3units. x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 1 2 3 4 5 6 7

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4.3RationalInequalitiesandApplications253 4.3RationalInequalitiesandApplications Inthissection,weusesigndiagramstosolverationalinequalitiesincludingsomethatarisefrom real-worldapplications.Ourrstexampleshowcasesthecriticaldierenceinprocedurebetween solvingarationalequationandarationalinequality. Example 4.3.1 1.Solve x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 1 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1. 2.Solve x 3 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 x +1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 1 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1. 3.Useyourcalculatortographicallycheckyouranswersto1and2. Solution. 1.Tosolvetheequation,wecleardenominators x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 1 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1= 1 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +2= x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 x +2expand 2 x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x =0 x x +1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1=0factor x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 ; 0 ; 1 Sincewecleareddenominators,weneedtocheckforextraneoussolutions.Sureenough,we seethat x =1doesnotsatisfytheoriginalequationandmustbediscarded.Oursolutions are x = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 and x =0. 2.Tosolvetheinequality,itmaybetemptingtobeginaswedidwiththeequation )]TJ/F15 10.9091 Tf 12.402 0 Td [(namely bymultiplyingbothsidesbythequantity x )]TJ/F15 10.9091 Tf 11.558 0 Td [(1.Theproblemisthat,dependingon x x )]TJ/F15 10.9091 Tf 11.525 0 Td [(1maybepositivewhichdoesn'taecttheinequalityor x )]TJ/F15 10.9091 Tf 11.525 0 Td [(1couldbenegative whichwouldreversetheinequality.Insteadofworkingbycases,wecollectalloftheterms ononesideoftheinequalitywith0ontheotherandmakeasigndiagramusingthetechnique givenonpage235inSection4.2.

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254RationalFunctions x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 1 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 x +1 0 2 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 0getacommondenominator 2 x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 0expand Viewingthelefthandsideasarationalfunction r x wemakeasigndiagram.Theonly valueexcludedfromthedomainof r is x =1whichisthesolutionto2 x )]TJ/F15 10.9091 Tf 10.956 0 Td [(2=0.Thezeros of r arethesolutionsto2 x 3 )]TJ/F53 10.9091 Tf 10.737 0 Td [(x 2 )]TJ/F53 10.9091 Tf 10.737 0 Td [(x =0,whichwehavealreadyfoundtobe x =0, x = )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 2 and x =1,thelatterwasdiscountedasazerobecauseitisnotinthedomain.Choosingtest valuesineachtestinterval,weconstructthesigndiagrambelow. )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 01 + 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + + Weareinterestedinwhere r x 0.Wend r x > 0,or+,ontheintervals )]TJ/F54 10.9091 Tf 5 -8.837 Td [( ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 ; 1and ; 1 .Weaddtotheseintervalsthezerosof r )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 and0,togetournalsolution: )]TJ/F54 10.9091 Tf 5 -8.836 Td [( ; )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 [ [0 ; 1 [ ; 1 3.Geometrically,ifweset f x = x 3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x +1 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 and g x = 1 2 x )]TJ/F15 10.9091 Tf 10.964 0 Td [(1,thesolutionsto f x = g x are the x -coordinatesofthepointswherethegraphsof y = f x and y = g x intersect.The solutionto f x g x representsnotonlywherethegraphsmeet,buttheintervalsover whichthegraphof y = f x isabove > thegraphof g x .Weobtainthegraphsbelow. The`Intersect'commandconrmsthatthegraphscrosswhen x = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 and x =0.Itisclear fromthecalculatorthatthegraphof y = f x isabovethegraphof y = g x on )]TJ/F54 10.9091 Tf 5 -8.836 Td [( ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 aswellason ; 1 .Accordingtothecalculator,oursolutionisthen )]TJ/F54 10.9091 Tf 5 -8.836 Td [( ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 [ [0 ; 1

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4.3RationalInequalitiesandApplications255 which almost matchestheanswerwefoundanalytically.Wehavetorememberthat f is notdenedat x =1,and,eventhoughitisn'tshownonthecalculator,thereisahole 1 in thegraphof y = f x when x =1whichiswhy x =1needstobeexcludedfromournal answer. Ournextexampledealswiththe averagecost functionofPortaBoyGamesystemsfromExample 2.1.5inSection2.1. Example 4.3.2 Givenacostfunction C x ,whichreturnsthetotalcostofproducing x products, the averagecost function, AC x = C x x ,computesthecostperitem.Recallthatthecost C ,in dollars,toproduce x PortaBoygamesystemsforalocalretaileris C x =80 x +150, x 0. 1.Findanexpressionfortheaveragecostfunction AC x .Determineanappropriateapplied domainfor AC 2.Findandinterpret AC 3.Solve AC x < 100andinterpret. 4.Determinethebehaviorof AC x as x !1 andinterpret. Solution. 1.From AC x = C x x ,weobtain AC x = 80 x +150 x .Thedomainof C is x 0,butsince x =0 causesproblemsfor AC x ,wegetourdomaintobe x> 0,or ; 1 2.Wend AC = 80+150 10 =95,sotheaveragecosttoproduce10gamesystemsis$95 persystem. 3.Solving AC x < 100meanswesolve 80 x +150 x < 100.Weproceedasinthepreviousexample. 80 x +150 x < 100 80 x +150 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(100 < 0 80 x +150 )]TJ/F15 10.9091 Tf 10.91 0 Td [(100 x x < 0commondenominator 150 )]TJ/F15 10.9091 Tf 10.909 0 Td [(20 x x < 0 Ifwetakethelefthandsidetobearationalfunction r x ,weneedtokeepinmindthethe applieddomainoftheproblemis x> 0.Thismeansweconsideronlythepositivehalfofthe numberlineforoursigndiagram.On ; 1 r isdenedeverywheresoweneedonlylook forzerosof r .Setting r x =0gives150 )]TJ/F15 10.9091 Tf 10.928 0 Td [(20 x =0,sothat x = 15 2 =7 : 5.Thetestintervals onourdomainare ; 7 : 5and : 5 ; 1 .Wend r x < 0on : 5 ; 1 1 Thereisnoasymptoteat x =1sincethegraphiswellbehavednear x =1.AccordingtoTheorem4.1,there mustbeaholethere.

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256RationalFunctions 0 7 : 5 + 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( Inthecontextoftheproblem, x representsthenumberofPortaBoygamessystemsproduced and AC x istheaveragecosttoproduceeachsystem.Solving AC x < 100meansweare tryingtondhowmanysystemsweneedtoproducesothattheaveragecostislessthan$100 persystem.Oursolution, : 5 ; 1 tellsusthatweneedtoproducemorethan7 : 5systemsto achievethis.Sinceitdoesn'tmakesensetoproducehalfasystem,ournalansweris[8 ; 1 4.WecanapplyTheorem4.2to AC x andwend y =80isahorizontalasymptotetothe graphof y = AC x .Tomorepreciselydeterminethebehaviorof AC x as x !1 ,we rstuselongdivision 2 andrewrite AC x =80+ 150 x .As x !1 150 x 0 + ,whichmeans AC x 80+verysmall+.Thustheaveragecostpersystemisgettingcloserto$80 persystem.Ifweset AC x =80,weget 150 x =0,whichisimpossible,soweconclude that AC x > 80forall x> 0.Thismeanstheaveragecostpersystemisalwaysgreater than$80persystem,buttheaveragecostisapproachingthisamountasmoreandmore systemsareproduced.LookingbackatExample2.1.5,werealize$80isthevariablecost persystem )]TJ/F15 10.9091 Tf 12.473 0 Td [(thecostpersystemaboveandbeyondthexedinitialcostof$150.Another waytointerpretouransweristhat`innitely'manysystemswouldneedtobeproducedto eectivelycounterbalancethexedcost. Ournextexampleisanotherclassic`boxwithnotop'problem. Example 4.3.3 Aboxwithasquarebaseandnotopistobeconstructedsothatithasavolume of1000cubiccentimeters.Let x denotethewidthofthebox,incentimeters.Refertothegure below. width, x height depth 1.Expresstheheight h incentimetersasafunctionofthewidth x andstatetheapplieddomain. 2.Solve h x x andinterpret. 3.Findandinterpretthebehaviorof h x as x 0 + andas x !1 4.Expressthesurfacearea S oftheboxasafunctionof x andstatetheapplieddomain. 5.Useacalculatortoapproximatetotwodecimalplacesthedimensionsoftheboxwhich minimizethesurfacearea. 2 Inthiscase,longdivisionamountstoterm-by-termdivision.

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4.3RationalInequalitiesandApplications257 Solution. 1.Wearetoldthevolumeoftheboxis1000cubiccentimetersandthat x representsthewidth, incentimeters.Fromgeometry,weknowVolume=width height depth.Sincethebase oftheboxistobeasquare,thewidthandthedepthareboth x centimeters.Using h forthe height,wehave1000= x 2 h ,sothat h = 1000 x 2 .Usingfunctionnotation, 3 h x = 1000 x 2 Asfor theapplieddomain,inorderfortheretobeaboxatall, x> 0,andsinceeverysuchchoice of x willreturnapositivenumberfortheheight h wehavenootherrestrictionsandconclude ourdomainis ; 1 2.Tosolve h x x ,weproceedasbeforeandcollectallnonzerotermsononesideofthe inequalityanduseasigndiagram. h x x 1000 x 2 x 1000 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 0 1000 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 x 2 0commondenominator Weconsiderthelefthandsideoftheinequalityasourrationalfunction r x .Wesee r is undenedat x =0,but,asinthepreviousexample,theapplieddomainoftheproblemis x> 0,soweareconsideringonlythebehaviorof r on ; 1 .Thesolezeroof r comeswhen 1000 )]TJ/F53 10.9091 Tf 10.057 0 Td [(x 3 =0,whichis x =10.Choosingtestvaluesintheintervals ; 10and ; 1 gives thefollowingdiagram. 0 + 10 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( Wesee r x > 0on ; 10,andsince r x =0at x =10,oursolutionis ; 10].Inthecontext oftheproblem, h representstheheightoftheboxwhile x representsthewidthanddepth ofthebox.Solving h x x istantamounttondingthevaluesof x whichresultinabox wheretheheightisatleastasbigasthewidthand,inthiscase,depth.Ouranswertells usthewidthoftheboxcanbeatmost10centimetersforthistohappen. 3.As x 0 + h x = 1000 x 2 !1 .Thismeansthesmallerthewidth x and,inthiscase,depth, thelargertheheight h hastobeinordertomaintainavolumeof1000cubiccentimeters.As x !1 ,wend h x 0 + ,whichmeanstomaintainavolumeof1000cubiccentimeters, thewidthanddepthmustgetbiggerthesmallertheheightbecomes. 3 Thatis, h x means` h of x ',not` h times x 'here.

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258RationalFunctions 4.Sincetheboxhasnotop,thesurfaceareacanbefoundbyaddingtheareaofeachofthe sidestotheareaofthebase.Thebaseisasquareofdimensions x by x ,andeachsidehas dimensions x by h .Wegetthesurfacearea, S = x 2 +4 xh .Toget S asafunctionof x ,we substitute h = 1000 x 2 toobtain S = x 2 +4 x )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1000 x 2 .Hence,asafunctionof x S x = x 2 + 4000 x Thedomainof S isthesameas h ,namely ; 1 ,forthesamereasonsasabove. 5.Arstattemptatthegraphof y = S x onthecalculatormayleadtofrustration.Chances aregoodthattherstwindowchosentoviewthegraphwillsuggest y = S x hasthe x -axis asahorizontalasymptote.Fromtheformula S x = x 2 + 4000 x ,however,weget S x x 2 as x !1 ,so S x !1 .Readjustingthewindow,wend S doespossessarelativeminimum at x 12 : 60.Asfaraswecantell, 4 thisistheonlyrelativeextremum,andsoitisthe absoluteminimumaswell.Thismeansthewidthanddepthoftheboxshouldeachmeasure approximately12 : 60centimeters.Todeterminetheheight,wend h : 60 6 : 30,sothe heightoftheboxshouldbeapproximately6 : 30centimeters. Inmanyinstancesinthesciences,rationalfunctionsareencounteredasaresultoffundamental naturallawswhicharetypicallyaresultofassumingcertainbasicrelationshipsbetweenvariables. Thesebasicrelationshipsaresummarizedinthedenitionbelow. Definition 4.4 Suppose x y ,and z arevariablequantities.Wesay y variesdirectlywith oris directlyproportionalto x ifthereisaconstant k such that y = kx y variesinverselywith oris inverselyproportionalto x ifthereisaconstant k such that y = k x z variesjointlywith oris jointlyproportionalto x and y ifthereisaconstant k suchthat z = kxy Theconstant k intheabovedenitionsiscalledthe constantofproportionality Example 4.3.4 TranslatethefollowingintomathematicalequationsusingDenition4.4. 4 withoutCalculus,thatis...

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4.3RationalInequalitiesandApplications259 1.Hooke'sLaw :Theforce F exertedonaspringisdirectlyproportionaltheextension x ofthe spring. 2.Boyle'sLaw :Ataconstanttemperature,thepressure P ofanidealgasisinverselyproportionaltoitsvolume V 3.Thevolume V ofarightcircularconevariesjointlywiththeheight h oftheconeandthe squareoftheradius r ofthebase. 4.Ohm'sLaw :Thecurrent I throughaconductorbetweentwopointsisdirectlyproportionalto thevoltage V betweenthetwopointsandinverselyproportionaltotheresistance R between thetwopoints. 5.Newton'sLawofUniversalGravitation :Supposetwoobjects,oneofmass m andoneofmass M ,arepositionedsothatthedistancebetweentheircentersofmassis r .Thegravitational force F exertedonthetwoobjectsvariesdirectlywiththeproductofthetwomassesand inverselywiththesquareofthedistancebetweentheircentersofmass. Solution. 1.Applyingthedenitionofdirectvariation,weget F = kx forsomeconstant k 2.Since P and V areinverselyproportional,wewrite P = k V 3.Thereisabitofambiguityhere.It'sclearthevolumeandheightoftheconeisrepresentedby thequantities V and h ,respectively,butdoes r representtheradiusofthebaseorthesquare oftheradiusofthebase?Itistheformer.Usually,ifanalgebraicoperationisspecied likesquaring,itismeanttobeexpressedintheformula.WeapplyDenition4.4toget V = khr 2 4.Eventhoughtheproblemdoesn'tusethephrase`variesjointly',thefactthatthecurrent I isgivenasrelatingtotwodierentquantitiesimpliesthis.Since I variesdirectlywith V but inverselywith R ,wewrite I = kV R 5.Wewritetheproductofthemasses mM andthesquareofthedistanceas r 2 .Wehave F variesdirectlywith mM andinverselywith r 2 ,sothat F = kmM r 2 Inmanyoftheformulasinthepreviousexample,morethantwovaryingquantitiesarerelated.In practice,however,usuallyallbuttwoquantitiesareheldconstantinanexperimentandthedata collectedisusedtorelatejusttwoofthevariables.Comparingjusttwovaryingquantitiesallows ustoviewtherelationshipbetweenthemasfunctional,asthenextexampleillustrates. Example 4.3.5 Accordingtothiswebsite theactualdatarelatingthevolume V ofagasandits pressure P usedbyBoyleandhisassistantin1662toverifythegaslawthatbearshisnameis givenbelow.

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260RationalFunctions V 48 46 44 42 40 38 36 34 32 30 28 26 24 P 29 : 13 30 : 56 31 : 94 33 : 5 35 : 31 37 39 : 31 41 : 63 44 : 19 47 : 06 50 : 31 54 : 31 58 : 81 V 23 22 21 20 19 18 17 16 15 14 13 12 P 61 : 31 64 : 06 67 : 06 70 : 69 74 : 13 77 : 88 82 : 75 87 : 88 93 : 06 100 : 44 107 : 81 117 : 56 1.Useyourcalculatortogenerateascatterdiagramforthesedatausing V astheindependent variableand P asthedependentvariable.Doesitappearfromthegraphthat P isinversely proportionalto V ?Explain. 2.Assuming P and V dovaryinversely,usethedatatoapproximatetheconstantofproportionality. 3.Useyourcalculatortodeterminea`PowerRegression'tothisdata 5 anduseitverifyyour resultsin1and2. Solution. 1.If P reallydoesvaryinverselywith V ,then P = k V forsomeconstant k .Fromthedataplot, thepointsdoseemtolikealongacurvelike y = k x 2.Todeterminetheconstantofproportionality,wenotethatfrom P = k V ,weget k = PV Multiplyingeachofthevolumenumberstimeseachofthepressurenumbers, 6 weproducea numberwhichisalwaysapproximately1400.Wesuspectthat P = 1400 V .Graphing y = 1400 x alongwiththedatagivesusgoodreasontobelieveourhypothesesthat P and V are,infact, inverselyrelated. ThegraphofthedataThedatawith y = 1400 x 3.Afterperforminga`PowerRegression',thecalculatortsthedatatothecurve y = ax b where a 1400and b )]TJ/F15 10.9091 Tf 20.828 0 Td [(1withacorrelationcoecientwhichisdarnednearperfect 7 .Inother words, y =1400 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 or y = 1400 x ,asweguessed. 5 Wewilltalkmoreaboutthisinthecomingchapters. 6 Youcanusetellthecalculatortodothisalgebraonthelistsandsaveyourselfsometime. 7 WewillrevisitthisexampleoncewehavedevelopedlogarithmsinChapter6toseehowwecanactually`linearize' thisdataanddoalinearregressiontoobtainthesameresult.

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4.3RationalInequalitiesandApplications261

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262RationalFunctions 4.3.1Exercises 1.Solveeachrationalequation.Besuretocheckforextraneoussolutions. a x 5 x +4 =3 b 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 +1 =1 c 1 x +3 + 1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 d 2 x +17 x +1 = x +5 e x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 x 3 + x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x =1 f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 3 +4 x x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(9 =4 x 2.Solveeachrationalinequality.Expressyouranswerusingintervalnotation. a 1 x +2 0 b x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x +2 0 c x x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 < 0 d x 2 +5 x +6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 > 0 e 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 +1 1 f 2 x +17 x +1 >x +5 g )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 3 +4 x x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 4 x h 1 x 2 +1 < 0 i x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 + x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(15 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 2 x j 5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 2 +9 x +10 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 3.AnotherClassicProblem:Acanismadeintheshapeofarightcircularcylinderandisto holdonepint.Fordrygoods,onepintisequalto33 : 6cubicinches. 8 aFindanexpressionforthevolume V ofthecanbasedontheheight h andthebase radius r bFindanexpressionforthesurfacearea S ofthecanbasedontheheight h andthebase radius r .Hint:Thetopandbottomofthecanarecirclesofradius r andthesideof thecanisreallyjustarectanglethathasbeenbentintoacylinder. cUsingthefactthat V =33 : 6,write S asafunctionof r andstateitsapplieddomain. dUseyourgraphingcalculatortondthedimensionsofthecanwhichhasminimalsurface area. 4.InExercise9inSection1.5,thepopulationofSasquatchinPortageCountywasmodeledby thefunction P t = 150 t t +15 ,where t =0representstheyear1803.Whenweretherefewerthan 100Sasquatchinthecounty? 5.Thecost C indollarstoremove p %oftheinvasivespeciesofIppizutishfromSasquatch Pondisgivenby C p = 1770 p 100 )]TJ/F37 7.9701 Tf 6.587 0 Td [(p where0 p< 100. 8 Accordingtowww.dictionary.com ,therearedierentvaluesgivenforthisconversion.Wewillstickwith33 : 6in 3 forthisproblem.

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4.3RationalInequalitiesandApplications263 aFindandinterpret C and C bWhatdoestheverticalasymptoteat x =100meanwithinthecontextoftheproblem? cIfyouhave $ 40000tospendforIppizutiremoval,whatpercentageoftheshcanyou remove? 6.Translatethefollowingintomathematicalequations. aAtaconstantpressure,thetemperature T ofanidealgasisdirectlyproportionaltoits volume V .ThisisCharles'sLaw bThefrequencyofawave f isinverselyproportionaltothewavelengthofthewave cThedensity d ofamaterialisdirectlyproportionaltothemassoftheobject m and inverselyproportionaltoitsvolume V dThesquareoftheorbitalperiodofaplanet P isdirectlyproportionaltothecubeofthe semi-majoraxisofitsorbit a .ThisisKepler'sThirdLawofPlanetaryMotion eThedragofanobjecttravelingthroughauid D variesjointlywiththedensityofthe uid andthesquareofthevelocityoftheobject fSupposetwoelectricpointcharges,onewithcharge q andonewithcharge Q ,arepositioned r unitsapart.Theelectrostaticforce F exertedonthechargesvariesdirectlywith theproductofthetwochargesandinverselywiththesquareofthedistancebetween thecharges.ThisisCoulomb'sLaw 7.Accordingtothiswebpage ,thefrequency f ofavibratingstringisgivenby f = 1 2 L r T where T isthetension, isthelinearmass 9 ofthestringand L isthelengthofthevibrating partofthestring.Expressthisrelationshipusingthelanguageofvariation. 8.AccordingtotheCentersforDiseaseControlandPreventionwww.cdc.gov ,aperson'sBody MassIndex B isdirectlyproportionaltohisweight W inpoundsandinverselyproportional tothesquareofhisheight h ininches. aExpressthisrelationshipasamathematicalequation. bIfapersonwhowas5feet,10inchestallweighed235poundshadaBodyMassIndex of33.7,whatisthevalueoftheconstantofproportionality? cRewritethemathematicalequationfoundinpart8atoincludethevalueoftheconstant foundinpart8bandthenndyourBodyMassIndex. 9.Weknowthatthecircumferenceofacirclevariesdirectlywithitsradiuswith2 asthe constantofproportionality.Thatis,weknow C =2 r: Withthehelpofyourclassmates, compilealistofotherbasicgeometricrelationshipswhichcanbeseenasvariations. 9 Alsoknownasthelineardensity.Itissimplyameasureofmassperunitlength.

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264RationalFunctions 4.3.2Answers 1.a x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(6 7 b x =1 ;x =2 c x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 d x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 ;x =2 eNosolution f x =0 ;x = p 2 2.a )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 b )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3] c )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 [ ; 1 d ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ ; 1 e ; 1] [ [2 ; 1 f ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 2 g ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 [ )]TJ 8.485 9.024 Td [(p 8 ; 0 [ p 8 ; 3 hNosolution i[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 0 [ ; 4 [ [5 ; 1 j )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 [ ; 1 3.a V = r 2 h b S =2 r 2 +2 rh c S r =2 r 2 + 67 : 2 r ; Domain r> 0 d r 1 : 749in.and h 3 : 498in. 4. P =100sobefore1903therewerefewerthan100SasquatchinPortageCounty. 5.a C =590meansitcosts $ 590toremove25%oftheshandand C =33630 meansitwouldcost $ 33630toremove95%oftheshfromthepond. bTheverticalasymptoteat x =100meansthataswetrytoremove100%ofthesh fromthepond,thecostincreaseswithoutbound.Thusitisimpossibletoremove100% ofthesh. cFor $ 40000youcouldremoveabout95.76%ofthesh. 6.a T = kV b 10 f = k c d = km V d P 2 = ka 3 e 11 D = k 2 f 12 F = kqQ r 2 7.Rewriting f = 1 2 L r T as f = 1 2 p T L p weseethatthefrequency f variesdirectlywiththe squarerootofthetensionandvariesinverselywiththelengthandthesquarerootofthe linearmass. 8.a B = kW h 2 b k =702 : 68TheCDCuses703. c B = 703 W h 2 10 Thecharacter isthelowercaseGreekletter`lambda.' 11 Note:Thecharacters and arethelowercaseGreekletters`rho'and`nu,'respectively. 12 NotethesimilaritytothisformulaandNewton'sLawofUniversalGravitationasdiscussedinExample5.

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Chapter5 FurtherTopicsinFunctions 5.1FunctionComposition Beforeweembarkuponanyfurtheradventureswithfunctions,weneedtotakesometimetogather ourthoughtsandgainsomeperspective.Chapter1rstintroducedustofunctionsinSection1.4. Atthattime,functionswerespecickindsofrelations-setsofpointsintheplanewhichpassedthe VerticalLineTest,Theorem1.1.InSection1.5,wedevelopedtheideathatfunctionsareprocesses -ruleswhichmatchinputstooutputs-andthisgaverisetotheconceptsofdomainandrange. WespokeabouthowfunctionscouldbecombinedinSection1.6usingthefourbasicarithmetic operations,tookamoredetailedlookattheirgraphsinSection1.7andstudiedhowtheirgraphs behavedundercertainclassesoftransformationsinSection1.8.InChapter2,wetookacloser lookatthreefamiliesoffunctions:linearfunctionsSection2.1,absolutevaluefunctions 1 Section 2.2,andquadraticfunctionsSection2.3.Linearandquadraticfunctionswerespecialcasesof polynomialfunctions,whichwestudiedingeneralityinChapter3.Chapter3culminatedwith theRealFactorizationTheorem,Theorem3.16,whichsaysthatallpolynomialfunctionswithreal coecientscanbethoughtofasproductsoflinearandquadraticfunctions.Ournextstepwasto enlargeoureld 2 ofstudytorationalfunctionsinChapter4.Beingquotientsofpolynomials,we canultimatelyviewthisfamilyoffunctionsasbeingbuiltupoflinearandquadraticfunctionsas well.Soinsomesense,Chapters2,3,and4canbethoughtofasanexhaustivestudyoflinearand quadratic 3 functionsandtheirarithmeticcombinationsasdescribedinSection1.6.Wenowwish tostudyotheralgebraicfunctions,suchas f x = p x and g x = x 2 = 3 ,andthepurposeofthe rsttwosectionsofthischapteristoseehowthesekindsoffunctionsarisefrompolynomialand rationalfunctions.Tothatend,werststudyanewwaytocombinefunctionsasdenedbelow. Definition 5.1 Suppose f and g aretwofunctions.The composite of g with f ,denoted g f isdenedbytheformula g f x = g f x ,provided x isanelementofthedomainof f and f x isanelementofthedomainof g 1 Thesewereintroduced,asyoumayrecall,aspiecewise-denedlinearfunctions. 2 Thisisareallybadmathpun. 3 Ifwebroadenourconceptoffunctionstoallowforcomplexvaluedcoecients,theComplexFactorization Theorem,Theorem3.14,tellsuseveryfunctionwehavestudiedthusfarisacombinationoflinearfunctions.

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266FurtherTopicsinFunctions Thequantity g f isalsoread` g composedwith f 'or,moresimply` g of f .'Atitsmostbasiclevel, Denition5.1tellsustoobtaintheformulafor g f x ,wereplaceeveryoccurrenceof x inthe formulafor g x withtheformulawehavefor f x .Ifwetakeastepbackandlookatthisfroma procedural,`inputsandoutputs'perspective,Dention5.1tellsustheoutputfrom g f isfound bytakingtheoutputfrom f f x ,andthenmakingthattheinputto g .Theresult, g f x ,is theoutputfrom g f .Fromthisperspective,wesee g f asatwostepprocesstakinganinput x andrstapplyingtheprocedure f thenapplyingtheprocedure g .Abstractly,wehave f g g f x f x g f x Intheexpression g f x ,thefunction f isoftencalledthe`inside'functionwhile g isoftencalled the`outside'function.Therearetwowaystogoaboutevaluatingcompositefunctions-`inside out'and`outsidein'-dependingonwhichfunctionwereplacewithitsformularst.Bothways aredemonstratedinthefollowingexample. Example 5.1.1 Let f x = x 2 )]TJ/F15 10.9091 Tf 10.622 0 Td [(4 x g x =2 )]TJ 10.622 8.569 Td [(p x +3,and h x = 2 x x +1 .Findandsimplifythe indicatedcompositefunctions.Statethedomainofeach. 1. g f x 2. f g x 3. g h x 4. h g x 5. h h x 6. h g f x 7. h g f x Solution. 1.Bydenition, g f x = g f x .Wenowillustratethetwowaystoevaluatethis. insideout :Weinserttheexpression f x into g rsttoget g f x = g f x = g )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x =2 )]TJ/F55 10.9091 Tf 10.909 9.913 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3=2 )]TJ/F55 10.9091 Tf 10.909 10.822 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 Hence, g f x =2 )]TJ 10.909 9.198 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3.

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5.1FunctionComposition267 outsidein :Weusetheformulafor g rsttoget g f x = g f x =2 )]TJ/F55 10.9091 Tf 10.909 9.86 Td [(p f x +3=2 )]TJ/F55 10.9091 Tf 10.909 9.913 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3=2 )]TJ/F55 10.9091 Tf 10.909 10.822 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 Wegetthesameanswerasbefore, g f x =2 )]TJ 10.909 9.198 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3. Tondthedomainof g f ,weneedtondtheelementsinthedomainof f whoseoutputs f x areinthedomainof g .WeaccomplishthisbyfollowingtherulesetforthinSection 1.5,thatis,wendthedomain before wesimplify.Tothatend,weexamine g f x = 2 )]TJ/F55 10.9091 Tf 10.931 9.38 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3.Tokeepthesquareroothappy,wesolvetheinequality x 2 )]TJ/F15 10.9091 Tf 10.931 0 Td [(4 x +3 0 bycreatingasigndiagram.Ifwelet r x = x 2 )]TJ/F15 10.9091 Tf 11.074 0 Td [(4 x +3,wendthezerosof r tobe x =1 and x =3.Weobtain 13 + 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + Oursolutionto x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 0,andhencethedomainof g f ,is ; 1] [ [3 ; 1 2.Tond f g x ,wend f g x insideout :Weinserttheexpression g x into f rsttoget f g x = f g x = f )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 )]TJ 10.909 8.569 Td [(p x +3 = )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 )]TJ 10.909 8.57 Td [(p x +3 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(4 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 )]TJ 10.909 8.57 Td [(p x +3 =4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 p x +3+ )]TJ/F54 10.9091 Tf 5 -0.267 Td [(p x +3 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8+4 p x +3 =4+ x +3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 = x )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 outsidein :Weusetheformulafor f x rsttoget f g x = f g x = g x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 g x = )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 )]TJ 10.909 8.57 Td [(p x +3 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 )]TJ 10.909 8.57 Td [(p x +3 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1samealgebraasbefore Thusweget f g x = x )]TJ/F15 10.9091 Tf 11.352 0 Td [(1.Tondthedomainof f g ,welooktothestepbefore wedidanysimplicationandnd f g x = )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 )]TJ 10.909 8.57 Td [(p x +3 2 )]TJ/F15 10.9091 Tf 10.868 0 Td [(4 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 )]TJ 10.909 8.57 Td [(p x +3 .Tokeepthe squareroothappy,weset x +3 0andndourdomaintobe[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1 3.Tond g h x ,wecompute g h x .

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268FurtherTopicsinFunctions insideout :Weinserttheexpression h x into g rsttoget g h x = g h x = g 2 x x +1 =2 )]TJ/F55 10.9091 Tf 10.909 19.145 Td [(s 2 x x +1 +3 =2 )]TJ/F55 10.9091 Tf 10.909 16.731 Td [(r 2 x x +1 + 3 x +1 x +1 getcommondenominators =2 )]TJ/F55 10.9091 Tf 10.909 16.155 Td [(r 5 x +3 x +1 outsidein :Weusetheformulafor g x rsttoget g h x = g h x =2 )]TJ/F55 10.9091 Tf 10.909 9.328 Td [(p h x +3 =2 )]TJ/F55 10.9091 Tf 10.909 19.145 Td [(s 2 x x +1 +3 =2 )]TJ/F55 10.9091 Tf 10.909 16.155 Td [(r 5 x +3 x +1 getcommondenominatorsasbefore Hence, g h x =2 )]TJ/F55 10.9091 Tf 10.963 12.293 Td [(q 5 x +3 x +1 .Tondthedomain,welooktothestepbeforewebeganto simplify: g h x =2 )]TJ/F55 10.9091 Tf 10.91 15.872 Td [(r 2 x x +1 +3.Toavoiddivisionbyzero,weneed x 6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Tokeep theradicalhappy,weneedtosolve 2 x x +1 +3 0.Gettingcommondenominatorsasbefore, thisreducesto 5 x +3 x +1 0.Dening r x = 5 x +3 x +1 ,wehavethat r isundenedat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and r x =0at x = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(3 5 .Weget )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + Ourdomainis ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(3 5 ; 1 4.Wend h g x bynding h g x insideout :Weinserttheexpression g x into h rsttoget h g x = h g x

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5.1FunctionComposition269 = h )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 )]TJ 10.909 8.57 Td [(p x +3 = 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 )]TJ 10.909 8.57 Td [(p x +3 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 )]TJ 10.909 8.57 Td [(p x +3 +1 = 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 p x +3 3 )]TJ 10.909 8.569 Td [(p x +3 outsidein :Weusetheformulafor h x rsttoget h g x = h g x = 2 g x g x +1 = 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 )]TJ 10.909 8.569 Td [(p x +3 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 )]TJ 10.909 8.57 Td [(p x +3 +1 = 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 p x +3 3 )]TJ 10.909 8.57 Td [(p x +3 Hence, h g x = 4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 p x +3 3 )]TJ 6.586 6.183 Td [(p x +3 .Tondthedomainof h g ,welooktothestepbeforeany simplication: h g x = 2 2 )]TJ 6.586 6.183 Td [(p x +3 2 )]TJ 6.587 6.183 Td [(p x +3 +1 .Tokeepthesquareroothappy,werequire x +3 0 or x )]TJ/F15 10.9091 Tf 20 0 Td [(3.Settingthedenominatorequaltozerogives )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 )]TJ 10.909 8.57 Td [(p x +3 +1=0or p x +3=3. Squaringbothsidesgivesus x +3=9,or x =6.Since x =6checksintheoriginalequation, )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 )]TJ 10.909 8.57 Td [(p x +3 +1=0,weknow x =6istheonlyzeroofthedenominator.Hence,thedomain of h g is[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 6 [ ; 1 5.Tond h h x ,wesubstitutethefunction h intoitself, h h x insideout :Weinserttheexpression h x into h toget h h x = h h x = h 2 x x +1 = 2 2 x x +1 2 x x +1 +1 = 4 x x +1 2 x x +1 +1 x +1 x +1

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270FurtherTopicsinFunctions = 4 x x +1 x +1 2 x x +1 x +1+1 x +1 = 4 x : 1 x +1 x +1 2 x : 1 x +1 x +1+ x +1 = 4 x 3 x +1 outsidein :Thisapproachyields h h x = h h x = 2 h x h x +1 = 2 2 x x +1 2 x x +1 +1 = 4 x 3 x +1 samealgebraasbefore Tondthedomainof h h ,weanalyze h h x = 2 2 x x +1 2 x x +1 +1 .Tokeepthedenominator x +1happy,weneed x 6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Settingthedenominator 2 x x +1 +1=0gives x = )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 3 .Our domainis ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 3 [ )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 3 ; 1 6.Theexpression h g f x indicatesthatwerstndthecomposite, g f andcompose thefunction h withtheresult.Weknowfromnumber1that g f x =2 )]TJ 10.873 9.198 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3. Wenowproceedasusual. insideout :Weinserttheexpression g f x into h rsttoget h g f x = h g f x = h 2 )]TJ 10.909 9.198 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 = 2 2 )]TJ 10.909 9.198 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 2 )]TJ 10.909 9.198 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 +1

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5.1FunctionComposition271 = 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 3 )]TJ 10.909 9.199 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 outsidein :Weusetheformulafor h x rsttoget h g f x = h g f x = 2 g f x g f x +1 = 2 2 )]TJ 10.909 9.198 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 2 )]TJ 10.909 9.198 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 +1 = 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 3 )]TJ 10.909 9.198 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 Soweget h g f x = 4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 p x 2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 x +3 3 )]TJ 6.586 6.675 Td [(p x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 x +3 .Tondthedomain,welookatthestepbefore webegantosimplify, h g f x = 2 2 )]TJ 6.587 6.676 Td [(p x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 x +3 2 )]TJ 6.587 6.676 Td [(p x 2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 x +3 +1 .Forthesquareroot,weneed x 2 )]TJ/F15 10.9091 Tf 11.541 0 Td [(4 x +3 0,whichwedeterminedinnumber1tobe ; 1] [ [3 ; 1 .Next,weset thedenominatortozeroandsolve: 2 )]TJ 10.909 9.198 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 +1=0.Weget p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3=3, and,aftersquaringbothsides,wehave x 2 )]TJ/F15 10.9091 Tf 11.216 0 Td [(4 x +3=9.Tosolve x 2 )]TJ/F15 10.9091 Tf 11.216 0 Td [(4 x )]TJ/F15 10.9091 Tf 11.216 0 Td [(6=0,weuse thequadraticformulaandget x =2 p 10.Thereaderisencouragedtocheckthatboth ofthesenumberssatisfytheoriginalequation, 2 )]TJ 10.909 9.199 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 +1=0.Hencewemust excludethesenumbersfromthedomainof h g f .Ournaldomainfor h f g is ; 2 )]TJ 10.909 9.024 Td [(p 10 [ )]TJ 10.909 9.024 Td [(p 10 ; 1] [ 3 ; 2+ p 10 [ )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2+ p 10 ; 1 7.Theexpression h g f x indicatesthatwerstndthecomposite h g andthencompose thatwith f .Fromnumber4,wegave h g x = 4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 p x +3 3 )]TJ 6.586 6.183 Td [(p x +3 .Wenowproceedasbefore. insideout :Weinserttheexpression f x into h g rsttoget h g f x = h g f x = h g )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x = 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 3 )]TJ/F55 10.9091 Tf 10.909 9.38 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 = 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 3 )]TJ 10.909 9.198 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 outsidein :Weusetheformulafor h g x rsttoget h g f x = h g f x

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272FurtherTopicsinFunctions = 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 p f x +3 3 )]TJ/F55 10.9091 Tf 10.909 9.327 Td [(p f x +3 = 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 3 )]TJ/F55 10.9091 Tf 10.909 9.38 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 = 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 3 )]TJ 10.909 9.198 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +3 Wenotethattheformulafor h g f x beforesimplicationisidenticaltothatof h g f x beforewesimpliedit.Hence,thetwofunctionshavethesamedomain, h f g is ; 2 )]TJ 10.909 9.024 Td [(p 10 [ )]TJ 10.909 9.024 Td [(p 10 ; 1] [ 3 ; 2+ p 10 [ )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2+ p 10 ; 1 ItshouldbeclearfromExample5.1.1that,ingeneral,whenyoucomposetwofunctions,suchas f and g above,theordermatters. 4 Wefoundthatthefunctions f g and g f weredierentas were g h and h g .Thinkingoffunctionsasprocesses,thisisn'tallthatsurprising.Ifwethinkof oneprocessasputtingonoursocks,andtheotherasputtingonourshoes,theorderinwhichwe dothesetwotasksdoesmatter. 5 Alsonotetheimportanceofndingthedomainofthecomposite function before simplifying.Forinstance,thedomainof f g ismuchdierentthanitssimplied formulawouldindicate.Composingafunctionwithitself,asinthecaseof h h ,mayseemodd. Lookingatthisfromaproceduralperspective,however,thismerelyindicatesperformingatask h andthendoingitagain-likesettingthewashingmachinetodoa`doublerinse'.Composinga functionwithitselfiscalled`iterating'thefunction,andwecouldeasilyspendanentirecourseon justthat.ThelasttwoproblemsinExample5.1.1servetodemonstratethe associative property offunctions.Thatis,whencomposingthreeormorefunctions,aslongaswekeeptheorderthe same,itdoesn'tmatterwhichtwofunctionswecomposerst.Thispropertyaswellasanother importantpropertyarelistedinthetheorembelow. Theorem 5.1 PropertiesofFunctionComposition: Suppose f g ,and h arefunctions. h g f = h g f ,providedthecompositefunctionsaredened. If I isdenedas I x = x forallrealnumbers x ,then I f = f I = f ByrepeatedapplicationsofDenition5.1,wend h g f x = h g f x = h g f x Similarly, h g f x = h g f x = h g f x .Thisestablishesthattheformulasforthe twofunctionsarethesame.Weleaveittothereadertothinkaboutwhythedomainsofthese twofunctionsareidentical,too.Thesetwofactsestablishtheequality h g f = h g f Aconsequenceoftheassociativityoffunctioncompositionisthatthereisnoneedforparentheses 4 Thisshowsusfunctioncompositionisn't commutative .Anexampleofanoperationweperformontwofunctions whichiscommutativeisfunctionaddition,whichwedenedinSection1.6.Inotherwords,thefunctions f + g and g + f arealwaysequal.Whichoftheremainingoperationsonfunctionswehavediscussedarecommutative? 5 AmoremathematicalexampleinwhichtheorderoftwoprocessesmatterscanbefoundinSection1.8.Infact, allofthetransformationsinthatsectioncanbeviewedintermsofcomposingfunctionswithlinearfunctions.

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5.1FunctionComposition273 whenwewrite h g f .ThesecondpropertycanalsobeveriedusingDenition5.1.Recallthat thefunction I x = x iscalledthe identityfunction andwasintroducedinExercise12inSection 2.1.Ifwecomposethefunction I withafunction f ,thenwehave I f x = I f x = f x andasimilarcomputationshows f I x = f x .Thisestablishesthatwehaveanidentity forfunctioncompositionmuchinthesamewaytherealnumber1isanidentityforrealnumber multiplication.Thatis,justasforanyrealnumber x ,1 x = x 1= x ,wehaveforanyfunction f I f = f I = f .Weshallseetheconceptofanidentitytakeongreatsignicanceinthenext section.Outinthewild,functioncompositionisoftenusedtorelatetwoquantitieswhichmaynot bedirectlyrelated,buthaveavariableincommon,asillustratedinournextexample. Example 5.1.2 Thesurfacearea S ofasphereisafunctionofitsradius r andisgivenbythe formula S r =4 r 2 .Supposethesphereisbeinginatedsothattheradiusofthesphereis increasingaccordingtotheformula r t =3 t 2 ,where t ismeasuredinseconds, t 0,and r is measuredininches.Findandinterpret S r t Solution. Ifwelookatthefunctions S r and r t individually,weseetheformergivesthe surfaceareaofasphereofagivenradiuswhilethelattergivestheradiusatagiventime.So, givenaspecictime, t ,wecouldndtheradiusatthattime, r t andfeedthatinto S r tond thesurfaceareaatthattime.Fromthisweseethatthesurfacearea S isultimatelyafunctionof time t andwend S r t = S r t =4 r t 2 =4 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(3 t 2 2 =36 t 4 .Thisformulaallowsusto computethesurfaceareadirectlygiventhetimewithoutgoingthroughthe`middleman' r AusefulskillinCalculusistobeabletotakeacomplicatedfunctionandbreakitdownintoa compositionofeasierfunctionswhichourlastexampleillustrates. Example 5.1.3 Writeeachofthefollowingfunctionsasacompositionoftwoormorenon-identity functions.Checkyouranswerbyperformingthefunctioncomposition. 1. F x = j 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 j 2. G x = 2 x 2 +1 3. H x = p x +1 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Solution. Therearemanyapproachestothiskindofproblem,andweshowcaseadierent methodologyineachofthesolutionsbelow. 1.Ourgoalistoexpressthefunction F as F = g f forfunctions g and f .FromDenition 5.1,weknow F x = g f x ,andwecanthinkof f x asbeingthe`inside'functionand g asbeingthe`outside'function.Lookingat F x = j 3 x )]TJ/F15 10.9091 Tf 11.279 0 Td [(1 j froman`insideversusoutside' perspective,wecanthinkof3 x )]TJ/F15 10.9091 Tf 11.671 0 Td [(1beinginsidetheabsolutevaluesymbols.Takingthis cue,wedene f x =3 x )]TJ/F15 10.9091 Tf 11.419 0 Td [(1.Atthispoint,wehave F x = j f x j .Whatistheoutside function?Thefunctionwhichtakestheabsolutevalueofitsinput, g x = j x j .Sureenough, g f x = g f x = j f x j = j 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 j = F x ,sowearedone.

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274FurtherTopicsinFunctions 2.Weattackdeconstructing G fromanoperationalapproach.Givenaninput x ,therststep istosquare x ,thenadd1,thendividetheresultinto2.Wewillassigneachofthesestepsa functionsoastowrite G asacompositeofthreefunctions: f g and h .Ourrstfunction, f ,isthefunctionthatsquaresitsinput, f x = x 2 .Thenextfunctionisthefunctionthat adds1toitsinput, g x = x +1.Ourlastfunctiontakesitsinputanddividesitinto2, h x = 2 x .Theclaimisthat G = h g f .Wend h g f x = h g f x = h g )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 = h )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 +1 = 2 x 2 +1 = G x 3.Ifwelook H x = p x +1 p x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 withaneyetowardsbuildingacomplicatedfunctionfromsimpler functions,weseetheexpression p x isasimplepieceofthelargerfunction.Ifwedene f x = p x ,wehave H x = f x +1 f x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 .Ifwewanttodecompose H = g f ,thenwecanglean theformulafrom g x bylookingatwhatisbeingdoneto f x .Wend g x = x +1 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 .We check g f x = g f x = f x +1 f x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 = p x +1 p x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = H x ,asrequired.

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5.1FunctionComposition275 5.1.1Exercises 1.Let f x =3 x )]TJ/F15 10.9091 Tf 11.344 0 Td [(6 ;g x = j x j ;h x = p x and k x = 1 x .Findandsimplifytheindicated compositefunctions.Statethedomainofeach. a f g x b g f x c f h x d h f x e g h x f h g x g f k x h k f x i h k x j k h x k f g h x l h g k x m k h f x n h k g f x 2.Let f x =2 x +1and g x = x 2 )]TJ/F53 10.9091 Tf 10.851 0 Td [(x )]TJ/F15 10.9091 Tf 10.852 0 Td [(6and h x = x +6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 .Findandsimplifytheindicated compositefunctions.Findthedomainofeach. a g f x b h f x c h g x d h h x 3.Let f x = p 9 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 and g x = x 2 )]TJ/F15 10.9091 Tf 10.469 0 Td [(9.Findandsimplifytheindicatedcompositefunctions. Statethedomainofeach. a f f x b g g x c g f x d f g x 4.Let f bethefunctiondenedby f = f )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 4 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 ; 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; ; 1 ; ; 3 ; ; 4 ; ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 g and let g bethefunctiondened g = f )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; ; 0 ; ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; ; 1 ; ; 2 g Findtheeachofthefollowingvaluesifitexists. a f g b f g )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 c f f d f g )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 e g f f g f )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 g g g )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 h g f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 i g f g j f f f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 k f f f f f l n times z }| { g g g 5.Let g x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x;h x = x +2 ;j x =3 x and k x = x )]TJ/F15 10.9091 Tf 9.779 0 Td [(4.Inwhatordermustthesefunctions becomposedwith f x = p x tocreate F x =3 p )]TJ/F53 10.9091 Tf 8.484 0 Td [(x +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4?

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276FurtherTopicsinFunctions 6.Whatlinearfunctionscouldbeusedtotransform f x = x 3 into F x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 x )]TJ/F15 10.9091 Tf 11.164 0 Td [(7 3 +1? Whatistheproperorderofcomposition? 7.Writethefollowingasacompositionoftwoormorenon-identityfunctions. a h x = p 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 b r x = 2 5 x +1 c F x = )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 3 d R x = 2 x 3 +1 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 8.Writethefunction F x = r x 3 +6 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 asacompositionofthreeormorenon-identityfunctions. 9.Thevolume V ofacubeisafunctionofitssidelength x .Let'sassumethat x = t +1is alsoafunctionoftime t ,where x ismeasuredininchesand t ismeasuredinminutes.Find aformulafor V asafunctionof t 10.Supposealocalvendorcharges$2perhotdogandthatthenumberofhotdogssoldperhour x isgivenby x t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 t 2 +20 t +92,where t isthenumberofhourssince10AM,0 t 4. aFindanexpressionfortherevenueperhour R asafunctionof x bFindandsimplify R x t .Whatdoesthisrepresent? cWhatistherevenueperhouratnoon? 11.Discusswithyourclassmateshow`real-world'processessuchasllingoutfederalincometax formsorcomputingyournalcoursegradecouldbeviewedasauseoffunctioncomposition. Findaprocessforwhichcompositionwithitselfiterationmakessense.

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5.1FunctionComposition277 5.1.2Answers 1.a f g x =3 j x j)]TJ/F15 10.9091 Tf 16.364 0 Td [(6 Domain: ; 1 b g f x = j 3 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(6 j Domain: ; 1 c f h x =3 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 Domain:[0 ; 1 d h f x = p 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 Domain:[2 ; 1 e g h x = p x Domain:[0 ; 1 f h g x = p j x j Domain: ; 1 g f k x = 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 Domain: ; 0 [ ; 1 h k f x = 1 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 Domain: ; 2 [ ; 1 i h k x = r 1 x Domain: ; 1 j k h x = 1 p x Domain: ; 1 k f g h x =3 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 Domain:[0 ; 1 l h g k x = s 1 x Domain: ; 0 [ ; 1 m k h f x = 1 p 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 Domain: ; 1 n h k g f x = r 1 j 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 j Domain: ; 2 [ ; 1 2.a g f x =4 x 2 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 Domain: ; 1 b h f x = 2 x +7 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 Domain: )]TJ/F54 10.9091 Tf 5 -8.836 Td [( ; 5 2 [ )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(5 2 ; 1 c h g x = x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 Domain: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 4 [ ; 1 d h h x = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(7 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(30 5 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(42 Domain: ; 6 [ )]TJ/F15 10.9091 Tf 5 -8.836 Td [(6 ; 42 5 [ )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(42 5 ; 1 3.a f f x = j x j Domain:[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 3] b g g x = x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(18 x 2 +72 Domain: ; 1 c g f x = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 2 Domain:[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 3] d f g x = p )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 4 +18 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 Domain:[ )]TJ 8.485 9.024 Td [(p 12 ; )]TJ 8.485 9.024 Td [(p 6] [ [ p 6 ; p 12] 6 4.a f g = f g = f =4 b f g )]TJ/F15 10.9091 Tf 8.484 0 Td [(1= f )]TJ/F15 10.9091 Tf 8.485 0 Td [(4whichisundened c f f = f f = f =3 d f g )]TJ/F15 10.9091 Tf 8.485 0 Td [(3= f g )]TJ/F15 10.9091 Tf 8.485 0 Td [(3= f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2=2 e g f = g f = g )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 f g f )]TJ/F15 10.9091 Tf 8.485 0 Td [(3= g whichisundened g g g )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= g g )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= g =0 h g f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= g f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= g =1 i g f g = g f = g = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 j f f f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= f f = f =3 k f f f f f = f f f f = f f f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= f f = f =3 l n times z }| { g g g =0 6 Thequantity )]TJ/F64 8.9664 Tf 7.167 0 Td [(x 4 +18 x 2 )]TJ/F63 8.9664 Tf 9.137 0 Td [(72isa`quadraticindisguise'whichfactorsnicely.SeeExample3.3.4isSection3.3.

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278FurtherTopicsinFunctions 5. F x =3 p )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4= k j f h g x 6.Onepossiblesolutionis F x = )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 2 x )]TJ/F15 10.9091 Tf 10.249 0 Td [(7 3 +1= k j f h g x where g x =2 x;h x = x )]TJ/F15 10.9091 Tf 11.958 0 Td [(7 ;j x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 x and k x = x +1.Youcouldalsohave F x = H f G x where G x =2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7and H x = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 x +1. 7.a h x = g f x where f x =2 x )]TJ/F15 10.9091 Tf 11.373 0 Td [(1 and g x = p x b r x = g f x where f x =5 x +1 and g x = 2 x c F x = g f x where f x = x 2 )]TJ/F15 10.9091 Tf 11.271 0 Td [(1 and g x = x 3 d R x = g f x where f x = x 3 and g x = 2 x +1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 8. F x = r x 3 +6 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 = h g f x where f x = x 3 ;g x = x +6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 and h x = p x 9. V x = x 3 so V x t = t +1 3 10.a R x =2 x b R x t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 t 2 +40 t +184,0 t 4.Thisgivestherevenueperhourasafunction oftime. cNooncorrespondsto t =2,so R x =232.Thehourlyrevenueatnoonis$232 perhour.

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5.2InverseFunctions279 5.2InverseFunctions ThinkingofafunctionasaprocesslikewedidinSection1.5,inthissectionweseekanother functionwhichmightreversethatprocess.Asinreallife,wewillndthatsomeprocesseslike puttingonsocksandshoesarereversiblewhilesomelikecookingasteakarenot.Westartby discussingaverybasicfunctionwhichisreversible, f x =3 x +4.Thinkingof f asaprocess,we startwithaninput x andapplytwosteps,aswesawinSection1.5 1.multiplyby3 2.add4 Toreversethisprocess,weseekafunction g whichwillundoeachofthesestepsandtaketheoutput from f ,3 x +4,andreturntheinput x .Ifwethinkofthereal-worldreversibletwo-stepprocessof rstputtingonsocksthenputtingonshoes,toreversetheprocess,wersttakeotheshoes,and thenwetakeothesocks.Inmuchthesameway,thefunction g shouldundothesecondstepof f rst.Thatis,thefunction g should 1. subtract 4 2. divide by3 Followingthisprocedure,weget g x = x )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 3 .Let'schecktoseeifthefunction g doesthejob. If x =5,then f =3+4=15+4=19.Takingtheoutput19from f ,wesubstituteit into g toget g = 19 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 3 = 15 3 =5,whichisouroriginalinputto f .Tocheckthat g does thejobforall x inthedomainof f ,wetakethegenericoutputfrom f f x =3 x +4,and substitutethatinto g .Thatis, g f x = g x +4= x +4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 3 = 3 x 3 = x ,whichisouroriginal inputto f .Ifwecarefullyexaminethearithmeticaswesimplify g f x ,weactuallysee g rst `undoing'theadditionof4,andthen`undoing'themultiplicationby3.Notonlydoes g undo f ,but f alsoundoes g .Thatis,ifwetaketheoutputfrom g g x = x )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 3 ,andputthatinto f ,weget f g x = f )]TJ/F37 7.9701 Tf 6.196 -4.541 Td [(x )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 3 =3 )]TJ/F37 7.9701 Tf 6.195 -4.541 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 3 +5= x )]TJ/F15 10.9091 Tf 11.487 0 Td [(5+5= x .Usingthelanguageoffunction compositiondevelopedinSection5.1,thestatements g f x = x and f g x = x canbewritten as g f x = x and f g x = x ,respectively.Abstractly,wecanvisualizetherelationship between f and g inthediagrambelow. f g x = g f x y = f x

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280FurtherTopicsinFunctions Themainideatogetfromthediagramisthat g takestheoutputsfrom f andreturnsthemto theirrespectiveinputs,andconversely, f takesoutputsfrom g andreturnsthemtotheirrespective inputs.Wenowhaveenoughbackgroundtostatethecentraldenitionofthesection. Definition 5.2 Suppose f and g aretwofunctionssuchthat 1. g f x = x forall x inthedomainof f and 2. f g x = x forall x inthedomainof g Then f and g aresaidtobe inverses ofeachother.Thefunctions f and g aresaidtobe invertible Ourrstresultofthesectionformalizestheconceptsthatinversefunctionsexchangeinputsand outputsandisaconsequenceofDenition5.2andtheFundamentalGraphingPrincipleforFunctions. Theorem 5.2 PropertiesofInverseFunctions: Suppose f and g areinversefunctions. Therange a of f isthedomainof g andthedomainof f istherangeof g f a = b ifandonlyif g b = a a;b isonthegraphof f ifandonlyif b;a isonthegraphof g a Recallthisisthesetofalloutputsofafunction. ThethirdpropertyinTheorem5.2tellsusthatthegraphsofinversefunctionsarereectionsabout theline y = x .Foraproofofthis,wereferthereadertoExample1.1.6inSection1.1.Aplotof theinversefunctions f x =3 x +4and g x = x )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 3 conrmsthistobethecase. x y y = f x y = g x y = x )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 1 2 Ifweabstractonestepfurther,wecanexpressthesentimentinDenition5.2bysayingthat f and g areinversesifandonlyif g f = I 1 and f g = I 2 where I 1 istheidentityfunctionrestricted 1 tothedomainof f and I 2 istheidentityfunctionrestrictedtothedomainof g .Inotherwords, I 1 x = x forall x inthedomainof f and I 2 x = x forall x inthedomainof g .Usingthis descriptionofinversesalongwiththepropertiesoffunctioncompositionlistedinTheorem5.1, 1 Theidentityfunction I ,whichwasintroducedinSection2.1andmentionedinTheorem5.1,hasadomainofall realnumbers.Ingeneral,thedomainsof f and g arenotallrealnumbers,whichnecessitatestherestrictionslisted here.

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5.2InverseFunctions281 wecanshowthatfunctioninversesareunique. 2 Suppose g and h arebothinversesofafunction f .ByTheorem5.2,thedomainof g isequaltothedomainof h ,sincebotharetherangeof f Thismeanstheidentityfunction I 2 appliesbothtothedomainof h andthedomainof g .Thus h = h I 2 = h f g = h f g = I 1 g = g ,asrequired. 3 Wesummarizethediscussionofthe lasttwoparagraphsinthefollowingtheorem. 4 Theorem 5.3 UniquenessofInverseFunctionsandTheirGraphs: Suppose f isan invertiblefunction. Thereisexactlyoneinversefunctionfor f ,denoted f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 read f -inverse Thegraphof y = f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x isthereectionofthegraphof y = f x acrosstheline y = x Thenotation f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 isanunfortunatechoicesinceyou'vebeenprogrammedsinceElementaryAlgebra tothinkofthisas 1 f .Thisismostdenitely not thecasesince,forinstance, f x =3 x +4hasas itsinverse f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = x )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 3 ,whichiscertainlydierentthan 1 f x = 1 3 x +4 .Whydoesthisconfusing notationpersist?AswementionedinSection5.1,theidentityfunction I istofunctioncomposition whattherealnumber1istorealnumbermultiplication.Thechoiceofnotation f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 alludestothe propertythat f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 f = I 1 and f f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 = I 2 ,inmuchthesamewayas3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 3=1and3 3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 =1. Let'sturnourattentiontothefunction f x = x 2 .Is f invertible?Alikelycandidatefortheinverse isthefunction g x = p x .Checkingthecompositionyields g f x = g f x = p x 2 = j x j ,which isnotequalto x forall x inthedomain ; 1 .Forexample,when x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2, f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 2 =4, but g = p 4=2,whichmeans g failedtoreturntheinput )]TJ/F15 10.9091 Tf 8.485 0 Td [(2fromitsoutput4.What g did, however,ismatchtheoutput4toa dierent input,namely2,whichsatises f =4.Thisissue ispresentedschematicallyinthepicturebelow. f g x = )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 x =2 4 Weseefromthediagramthatsinceboth f )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and f are4,itisimpossibletoconstructa function whichtakes4backto both x =2and x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Bydenition,afunctionmatches 2 Inotherwords,invertiblefunctionshaveexactlyoneinverse. 3 Itisanexcellentexercisetoexplaineachstepinthisstringofequalities. 4 Intheinterestsoffulldisclosure,theauthorswouldliketoadmitthatmuchofthediscussionintheprevious paragraphscouldhaveeasilybeenavoidedhadweappealedtothedescriptionofafunctionasasetoforderedpairs. Wemakenoapologyforourdiscussionfromafunctioncompositionstandpoint,however,sinceitexposesthereader tomoreabstractwaysofthinkingoffunctionsandinverses.WewillrevisitthisconceptagaininChapter8.

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282FurtherTopicsinFunctions arealnumberwithexactlyoneotherrealnumber.Fromagraphicalstandpoint,weknowthat if y = f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x exists,itsgraphcanbeobtainedbyreecting y = x 2 abouttheline y = x ,in accordancewithTheorem5.3.Doingsoproduces )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 ; 4 ; 4 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 1 2 3 4 5 6 7 y = f x = x 2 reectacross y = x )462()222()223()222()222()222()223()222()222()222()222()223()222()222()462(! switch x and y coordinates ; )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 ; 2 x y 1234567 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 y = f )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x ? Weseethattheline x =4intersectsthegraphofthesupposedinversetwice-meaningthegraph failstheVerticalLineTest,Theorem1.1,andassuch,doesnotrepresent y asafunctionof x .The verticalline x =4onthegraphontherightcorrespondstothe horizontalline y =4onthegraph of y = f x .Thefactthatthehorizontalline y =4intersectsthegraphof f twicemeanstwo dierent inputs,namely x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2and x =2,arematchedwiththe same output,4,whichisthe causeofallofthetrouble.Ingeneral,forafunctiontohaveaninverse, dierent inputsmustgo to dierent outputs,orelsewewillrunintothesameproblemwedidwith f x = x 2 .Wegive thispropertyaname. Definition 5.3 Afunction f issaidtobe one-to-one if f matchesdierentinputstodierent outputs.Equivalently, f isone-to-oneifandonlyifwhenever f c = f d ,then c = d Graphically,wedetectone-to-onefunctionsusingthetestbelow. Theorem 5.4 TheHorizontalLineTest: Afunction f isone-to-oneifandonlyifno horizontallineintersectsthegraphof f morethanonce. Wesaythatthegraphofafunction passes theHorizontalLineTestifnohorizontallineintersects thegraphmorethanonce;otherwise,wesaythegraphofthefunction fails theHorizontalLine Test.Wehavearguedthatif f isinvertible,then f mustbeone-to-one,otherwisethegraphgiven byreectingthegraphof y = f x abouttheline y = x willfailtheVerticalLineTest.Itturns outthatbeingone-to-oneisalsoenoughtoguaranteeinvertibility.Toseethis,wethinkof f as thesetoforderedpairswhichconstituteitsgraph.Ifswitchingthe x -and y -coordinatesofthe pointsresultsinafunction,then f isinvertibleandwehavefound f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 .Thisispreciselywhatthe HorizontalLineTestdoesforus:itcheckstoseewhetherornotasetofpointsdescribes x asa functionof y .Wesummarizetheseresultsbelow.

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5.2InverseFunctions283 Theorem 5.5 EquivalentConditionsforInvertibility: Suppose f isafunction.Thefollowingstatementsareequivalent. f isinvertible. f isone-to-one. Thegraphof f passestheHorizontalLineTest. Weputthisresulttoworkinthenextexample. Example 5.2.1 Determineifthefollowingfunctionsareone-to-oneintwoways:aanalytically usingDenition5.3andbgraphicallyusingtheHorizontalLineTest. 1. f x = 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 5 2. g x = 2 x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3. h x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +4 4. F = f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; ; 2 ; ; 1 g Solution. 1.aTodetermineif f isone-to-oneanalytically,weassume f c = f d andattemptto deducethat c = d f c = f d 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 c 5 = 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 d 5 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 c =1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 d )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 c = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 d c = d X Hence, f isone-to-one. bTocheckif f isone-to-onegraphically,welooktoseeifthegraphof y = f x passesthe HorizontalLineTest.Wehavethat f isanon-constantlinearfunction,whichmeansits graphisanon-horizontalline.Thusthegraphof f passestheHorizontalLineTestas seenbelow. 2.aWebeginwiththeassumptionthat g c = g d andtrytoshow c = d g c = g d 2 c 1 )]TJ/F53 10.9091 Tf 10.91 0 Td [(c = 2 d 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(d 2 c )]TJ/F53 10.9091 Tf 10.91 0 Td [(d =2 d )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 c )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 cd =2 d )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 dc 2 c =2 d c = d X Wehaveshownthat g isone-to-one.

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284FurtherTopicsinFunctions bWecangraph g usingthesixstepprocedureoutlinedinSection4.2.Wegetthesole interceptat ; 0,averticalasymptote x =1andahorizontalasymptotewhichthe graphnevercrosses y = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2.Weseefromthatthegraphof g passestheHorizontal LineTest. x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(112 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 1 2 y = f x x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 1 2 3 4 y = g x 3.aWebeginwith h c = h d .Asweworkourwaythroughtheproblem,weencountera nonlinearequation.Wemovethenon-zerotermstotheleft,leavea0ontherightand factoraccordingly. h c = h d c 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 c +4= d 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 d +4 c 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 c = d 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 d c 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(d 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 c +2 d =0 c + d c )]TJ/F53 10.9091 Tf 10.909 0 Td [(d )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 c )]TJ/F53 10.9091 Tf 10.909 0 Td [(d =0 c )]TJ/F53 10.9091 Tf 10.909 0 Td [(d c + d )]TJ/F15 10.9091 Tf 10.909 0 Td [(2=0factorbygrouping c )]TJ/F53 10.9091 Tf 10.909 0 Td [(d =0or c + d )]TJ/F15 10.9091 Tf 10.909 0 Td [(2=0 c = d or c =2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(d Weget c = d asonepossibility,butwealsogetthepossibilitythat c =2 )]TJ/F53 10.9091 Tf 11.469 0 Td [(d .This suggeststhat f maynotbeone-to-one.Taking d =0,weget c =0or c =2.With f =4and f =4,wehaveproducedtwodierentinputswiththesameoutput meaning f isnotone-to-one. bWenotethat h isaquadraticfunctionandwegraph y = h x usingthetechniques presentedinSection2.3.Thevertexis ; 3andtheparabolaopensupwards.Wesee immediatelyfromthegraphthat h isnotone-to-one,sincethereareseveralhorizontal lineswhichcrossthegraphmorethanonce. 4.aThefunction F isgiventousasasetoforderedpairs.Thecondition F c = F d meanstheoutputsfromthefunctionthe y -coordinatesoftheorderedpairsarethe same.Weseethatthepoints )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1and ; 1arebothelementsof F with F )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=1 and F =1.Since )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 6 =2,wehaveestablishedthat F is not one-to-one. bGraphically,weseethehorizontalline y =1crossesthegraphmorethanonce.Hence, thegraphof F failstheHorizontalLineTest.

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5.2InverseFunctions285 x y 12 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 1 2 3 4 5 6 y = h x x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 1 2 y = F x Wehaveshownthatthefunctions f and g inExample5.2.1areone-to-one.Thismeanstheyare invertible,soitisnaturaltowonderwhat f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x and g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x wouldbe.For f x = 1 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 x 5 ,wecan thinkourwaythroughtheinversesincethereisonlyoneoccurrenceof x .Wecantrackstep-by-step whatisdoneto x andreversethosestepsaswedidatthebeginningofthechapter.Thefunction g x = 2 x 1 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x isabittrickiersince x occursintwoplaces.Whenoneevaluates g x foraspecic valueof x ,whichisrst,the2 x orthe1 )]TJ/F53 10.9091 Tf 11.208 0 Td [(x ?Wecanimaginefunctionsmorecomplicatedthan thesesoweneedtodevelopageneralmethodologytoattackthisproblem.Theorem5.2tellsus equation y = f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x isequivalentto f y = x andthisisthebasisofouralgorithm. StepsforndingtheInverseofaOne-to-oneFunction 1.Write y = f x 2.Interchange x and y 3.Solve x = f y for y toobtain y = f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x Notethatwecouldhavesimplywritten`Solve x = f y for y 'andbedonewithit.Theactof interchangingthe x and y istheretoremindusthatwearendingtheinversefunctionbyswitching theinputsandoutputs. Example 5.2.2 Findtheinverseofthefollowingone-to-onefunctions.Checkyouranswersanalyticallyusingfunctioncompositionandgraphically. 1. f x = 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 5 2. g x = 2 x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x Solution. 1.Aswementionedearlier,itispossibletothinkourwaythroughtheinverseof f byrecording thestepsweapplyto x andtheorderinwhichweapplythemandthenreversingthosesteps inthereverseorder.Weencouragethereadertodothis.We,ontheotherhand,willpractice thealgorithm.Wewrite y = f x andproceedtoswitch x and y

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286FurtherTopicsinFunctions y = f x y = 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 5 x = 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 y 5 switch x and y 5 x =1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 y 5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 y 5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 = y y = )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(5 2 x + 1 2 Wehave f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(5 2 x + 1 2 .Tocheckthisansweranalytically,werstcheckthat )]TJ/F53 10.9091 Tf 5 -8.837 Td [(f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 f x = x forall x inthedomainof f ,whichisallrealnumbers. )]TJ/F53 10.9091 Tf 5 -8.837 Td [(f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 f x = f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 f x = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(5 2 f x + 1 2 = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(5 2 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 5 + 1 2 = )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(1 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x + 1 2 = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 + x + 1 2 = x X Wenowcheckthat )]TJ/F53 10.9091 Tf 5 -8.837 Td [(f f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = x forall x intherangeof f whichisalsoallrealnumbers. Recallthatthedomainof f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 istherangeof f )]TJ/F53 10.9091 Tf 5 -8.837 Td [(f f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = f f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x 5 = 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(5 2 x + 1 2 5 = 1+5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 5 = 5 x 5 = x X Tocheckouranswergraphically,wegraph y = f x and y = f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x onthesamesetofaxes. 5 Theyappeartobereectionsacrosstheline y = x 5 Notethatifyouperformyourcheckonacalculatorformoresophisticatedfunctions,you'llneedtotakeadvantage ofthe`ZoomSquare'featuretogetthecorrectgeometricperspective.

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5.2InverseFunctions287 x y y = f x y = f )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 x y = x )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 1 2 2.Tond g )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x ,westartwith y = g x .Wenotethatthedomainof g is ; 1 [ ; 1 y = g x y = 2 x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x x = 2 y 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y switch x and y x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y =2 y x )]TJ/F53 10.9091 Tf 10.909 0 Td [(xy =2 y x = xy +2 y x = y x +2factor y = x x +2 Weobtain g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = x x +2 .Tocheckthisanalytically,werstcheck )]TJ/F53 10.9091 Tf 5 -8.836 Td [(g )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 g x = x forall x inthedomainof g ,thatis,forall x 6 =1. )]TJ/F53 10.9091 Tf 5 -8.836 Td [(g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 g x = g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 g x = g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 2 x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x = 2 x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x +2 = 2 x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x +2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x cleardenominators

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288FurtherTopicsinFunctions = 2 x 2 x +2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x = 2 x 2 x +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x = 2 x 2 = x X Next,wecheck g )]TJ/F53 10.9091 Tf 5 -8.836 Td [(g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = x forall x intherangeof g .Fromthegraphof g inExample 5.2.1,wehavethattherangeof g is ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 .Thismatchesthedomainweget fromtheformula g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = x x +2 ,asitshould. )]TJ/F53 10.9091 Tf 5 -8.837 Td [(g g )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = g )]TJ/F53 10.9091 Tf 5 -8.837 Td [(g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = g x x +2 = 2 x x +2 1 )]TJ/F55 10.9091 Tf 10.909 15.382 Td [( x x +2 = 2 x x +2 1 )]TJ/F55 10.9091 Tf 10.909 15.382 Td [( x x +2 x +2 x +2 cleardenominators = 2 x x +2 )]TJ/F53 10.9091 Tf 10.91 0 Td [(x = 2 x 2 = x X Graphing y = g x and y = g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x onthesamesetofaxesisbusy,butwecanseethesymmetricrelationshipifwethickenthecurvefor y = g )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x .Notethattheverticalasymptote x =1ofthegraphof g correspondstothehorizontalasymptote y =1ofthegraphof g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 asitshouldsince x and y areswitched.Similarly,thehorizontalasymptote y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2ofthe graphof g correspondstotheverticalasymptote x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2ofthegraphof g )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 .

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5.2InverseFunctions289 x y y = x )]TJ/F35 5.9776 Tf 5.757 0 Td [(6 )]TJ/F35 5.9776 Tf 5.757 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123456 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 1 2 3 4 5 6 y = g x and y = g )]TJ/F89 6.9738 Tf 7.046 0 Td [(1 x Wenowreturnto f x = x 2 .Weknowthat f isnotone-to-one,andthus,isnotinvertible. However,ifwerestrictthedomainof f ,wecanproduceanewfunction g whichisone-to-one.If wedene g x = x 2 x 0,thenwehave x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 1 2 3 4 5 6 7 y = f x = x 2 restrictdomainto x 0 )462()222()223()222()222()222()223()222()222()222()222()223()222()222()462(! x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 1 2 3 4 5 6 7 y = g x = x 2 x 0 Thegraphof g passestheHorizontalLineTest.Tondaninverseof g ,weproceedasusual y = g x y = x 2 ;x 0 x = y 2 ;y 0switch x and y y = p x y = p x since y 0

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290FurtherTopicsinFunctions Weget g )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = p x .Atrstitlookslikewe'llrunintothesametroubleasbefore,butwhen wecheckthecomposition,thedomainrestrictionon g savestheday.Weget )]TJ/F53 10.9091 Tf 5 -8.836 Td [(g )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 g x = g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 g x = g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 = p x 2 = j x j = x ,since x 0.Checking )]TJ/F53 10.9091 Tf 5 -8.837 Td [(g g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = g )]TJ/F53 10.9091 Tf 5 -8.837 Td [(g )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = g p x = p x 2 = x .Graphing 6 g and g )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 onthesamesetofaxesshowsthattheyarereections abouttheline y = x y = x y = g x y = g )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 x x y 12345678 1 2 3 4 5 6 7 8 Ournextexamplecontinuesthethemeofdomainrestriction. Example 5.2.3 Graphthefollowingfunctionstoshowtheyareone-to-oneandndtheirinverses. Checkyouranswersanalyticallyusingfunctioncompositionandgraphically. 1. j x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +4, x 1.2. k x = p x +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Solution. 1.Thefunction j isarestrictionofthefunction h fromExample5.2.1.Sincethedomainof j isrestrictedto x 1,weareselectingonlythe`lefthalf'oftheparabola.Weseethatthe graphof j passestheHorizontalLineTestandthus j isinvertible. x y 12 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 y = j x 6 Wegraphed y = p x inSection1.8.

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5.2InverseFunctions291 Wenowuseouralgorithmtond j )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x y = j x y = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +4 ;x 1 x = y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 y +4 ;y 1switch x and y 0= y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 y +4 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x y = 2 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 quadraticformula, c =4 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x y = 2 p 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 2 y = 2 p 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 y = 2 2 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 y = 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 y =1 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 y =1 )]TJ 10.909 8.569 Td [(p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3since y 1. Wehave j )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x =1 )]TJ 11.292 8.569 Td [(p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3.Whenwesimplify )]TJ/F53 10.9091 Tf 5 -8.836 Td [(j )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 j x ,weneedtorememberthat thedomainof j is x 1. )]TJ/F53 10.9091 Tf 5 -8.837 Td [(j )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 j x = j )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 j x = j )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +4 ;x 1 =1 )]TJ/F55 10.9091 Tf 10.909 9.38 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 =1 )]TJ 10.909 9.199 Td [(p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 =1 )]TJ/F55 10.9091 Tf 10.909 9.381 Td [(p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 =1 )-222(j x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 j =1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1since x 1 = x X Checking j j )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ,weget )]TJ/F53 10.9091 Tf 5 -8.836 Td [(j j )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = j )]TJ/F53 10.9091 Tf 5 -8.836 Td [(j )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = j )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ 10.909 8.57 Td [(p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 = )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ 10.909 8.569 Td [(p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ 10.909 8.569 Td [(p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 +4 =1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3+ )]TJ/F54 10.9091 Tf 5 -0.266 Td [(p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2+2 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3+4 =3+ x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 = x X WecanusewhatweknowfromSection1.8tograph y = j )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x onthesameaxesas y = j x toget

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292FurtherTopicsinFunctions y = j x y = j )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 x y = x x y 123456 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 2.Wegraph y = k x = p x +2 )]TJ/F15 10.9091 Tf 9.588 0 Td [(1usingwhatwelearnedinSection1.8andsee k isone-to-one. x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 y = k x Wenowtrytond k )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 y = k x y = p x +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x = p y +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1switch x and y x +1= p y +2 x +1 2 = )]TJ/F54 10.9091 Tf 5 -0.872 Td [(p y +2 2 x 2 +2 x +1= y +2 y = x 2 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Wehave k )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = x 2 +2 x )]TJ/F15 10.9091 Tf 9.878 0 Td [(1.Basedonourexperience,weknowsomethingisn'tquiteright. Wedetermined k )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 isaquadraticfunction,andwehaveseenseveraltimesinthissection thatthesearenotone-to-oneunlesstheirdomainsaresuitablyrestricted.Theorem5.2tells usthatthedomainof k )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 istherangeof k .Fromthegraphof k ,weseethattherange is[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ,whichmeanswerestrictthedomainof k )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 to x )]TJ/F15 10.9091 Tf 20.669 0 Td [(1.Wenowcheckthatthis worksinourcompositions.

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5.2InverseFunctions293 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(k )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 k x = k )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 k x = k )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 )]TJ/F54 10.9091 Tf 5 -0.267 Td [(p x +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;x )]TJ/F15 10.9091 Tf 20 0 Td [(2 = )]TJ/F54 10.9091 Tf 5 -0.266 Td [(p x +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 +2 )]TJ/F54 10.9091 Tf 5 -0.266 Td [(p x +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = )]TJ/F54 10.9091 Tf 5 -0.267 Td [(p x +2 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 p x +2+1+2 p x +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = x +2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 = x X and )]TJ/F53 10.9091 Tf 5 -8.836 Td [(k k )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = k )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x )]TJ/F15 10.9091 Tf 20 0 Td [(1 = p x 2 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 = p x 2 +2 x +1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = p x +1 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = j x +1 j)]TJ/F15 10.9091 Tf 16.363 0 Td [(1 = x +1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1since x )]TJ/F15 10.9091 Tf 20 0 Td [(1 = x X Graphically,everythingchecksoutaswell,providedthatwerememberthedomainrestriction on k )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 meanswetaketherighthalfoftheparabola. y = k x y = k )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 x x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 Ourlastexampleofthesectiongivesanapplicationofinversefunctions. Example 5.2.4 RecallfromSection2.1thattheprice-demandequationforthePortaBoygame systemis p x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x +250for0 x 166,where x representsthenumberofsystemssold weeklyand p isthepricepersystemindollars. 1.Explainwhy p isone-to-oneandndaformulafor p )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x .Statetherestricteddomain. 2.Findandinterpret p )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 3.RecallfromSection2.3thattheweeklyprot P ,indollars,asaresultofselling x systemsis givenby P x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x 2 +170 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(150.Findandinterpret )]TJ/F53 10.9091 Tf 5 -8.836 Td [(P p )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x .

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294FurtherTopicsinFunctions 4.Useyouranswertopart3todeterminethepriceperPortaBoywhichwouldyieldthemaximumprot.ComparewithExample2.3.3. Solution. 1.Weleavetothereadertoshowthegraphof p x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 x +250,0 x 166,isaline segmentfrom ; 250to ; 1,andassuchpassestheHorizontalLineTest.Hence, p is one-to-one.Wendtheexpressionfor p )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x asusualandget p )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = 500 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x 3 .Thedomain of p )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 shouldmatchtherangeof p ,whichis[1 ; 250],andassuch,werestrictthedomainof p )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 to1 x 250. 2.Wend p )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = 500 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 3 =20.Sincethefunction p tookasinputstheweeklysalesand furnishedthepricepersystemastheoutput, p )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 takesthepricepersystemandreturnsthe weeklysalesasitsoutput.Hence, p )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 =20means20systemswillbesoldinaweekif thepriceissetat$220persystem. 3.Wecompute )]TJ/F53 10.9091 Tf 5 -8.837 Td [(P p )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = P )]TJ/F53 10.9091 Tf 5 -8.837 Td [(p )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = P )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(500 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(500 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x 3 2 +170 )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(500 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 x 3 )]TJ/F15 10.9091 Tf 9.795 0 Td [(150. AfteraheftyamountofElementaryAlgebra, 7 weobtain )]TJ/F53 10.9091 Tf 5 -8.837 Td [(P p )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(2 3 x 2 +220 x )]TJ/F34 7.9701 Tf 10.641 4.295 Td [(40450 3 Tounderstandwhatthismeans,recallthattheoriginalprotfunction P gaveustheweekly protasafunctionoftheweeklysales.Thefunction p )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 givesustheweeklysalesasa functionoftheprice.Hence, P p )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 takesasitsinputaprice.Thefunction p )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 returnsthe weeklysales,whichinturnisfedinto P toreturntheweeklyprot.Hence, )]TJ/F53 10.9091 Tf 5 -8.837 Td [(P p )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x givesustheweeklyprotindollarsasafunctionofthepricepersystem, x ,usingtheweekly sales p )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x asthe`middleman'. 4.WeknowfromSection2.3thatthegraphof y = )]TJ/F53 10.9091 Tf 5 -8.837 Td [(P p )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x isaparabolaopeningdownwards.Themaximumprotisrealizedatthevertex.Sinceweareconcernedonlywiththe pricepersystem,weneedonlyndthe x -coordinateofthevertex.Identifying a = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(2 3 and b =220,weget,bytheVertexFormula,Equation2.4, x = )]TJ/F37 7.9701 Tf 12.235 4.295 Td [(b 2 a =165.Hence,weeklyprot ismaximizedifwesetthepriceat$165persystem.Comparingthiswithouranswerfrom Example2.3.3,thereisaslightdiscrepancytothetuneof$0 : 50.Weleaveittothereaderto balancethebooksappropriately. 7 Itisgoodreviewtoactuallydothis!

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5.2InverseFunctions295 5.2.1Exercises 1.Showthatthefollowingfunctionsareone-to-oneandndtheinverse.Checkyouranswers algebraicallyandgraphically.Verifytherangeof f isthedomainof f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 andvice-versa. a f x =6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b f x =1 )]TJ/F34 7.9701 Tf 12.104 4.296 Td [(4+3 x 5 c f x = )]TJ 8.485 8.57 Td [(p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(5+2 d f x = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 x x 5 e f x =3 x +4 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 ;x )]TJ/F15 10.9091 Tf 20 0 Td [(4 f f x =4 x 2 +4 x +1, x< )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 g f x = 3 4 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x h f x = x 1 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 x i f x = 4 x +2 3 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 j f x = )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 x +3 2.ShowthattheFahrenheittoCelsiusconversionfunctionfoundinExercise3inSection2.1is invertibleandthatitsinverseistheCelsiustoFahrenheitconversionfunctionfoundinthat sameexercise. 3.Analyticallyshowthatthefunction f x = x 3 +3 x +1isone-to-one.Sincendingaformula foritsinverseisbeyondthescopeofthistextbook,useTheorem5.2tohelpyoucompute f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ;f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; and f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3. 4.Withthehelpofyourclassmates,ndaformulafortheinverseofthefollowing. a f x = ax + b;a 6 =0 b f x = a p x )]TJ/F53 10.9091 Tf 10.91 0 Td [(h + k;a 6 =0 ;x h c f x = ax + b cx + d ;a 6 =0 ;b 6 =0 ;c 6 =0 ;d 6 =0 d f x = ax 2 + bx + c where a 6 =0 ;x )]TJ/F37 7.9701 Tf 23.75 4.295 Td [(b 2 a 5.Let f x = 2 x x 2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 .UsingthetechniquesinSection4.2,graph y = f x .Verify f isone-to-one ontheinterval )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1.UsetheprocedureoutlinedonPage5.2tondthepossibilitiesfor f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x anduseyourgraphingcalculatortohelpchoosethecorrectformulafor f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x .Note thatsince f =0,itshouldbethecasethat f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 =0.Whatgoeswrongwhenyou attempttosubstitute x =0into f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x ?Discusswithyourclassmateshowthisproblem arosealgebraically,andpossibleremedies. 6.Suppose f isaninvertiblefunction.Provethatifgraphsof y = f x and y = f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x intersect atall,theydosoontheline y = x 7.Withthehelpofyourclassmates,explainwhyafunctionwhichiseitherstrictlyincreasing orstrictlydecreasingonitsentiredomainwouldhavetobeone-to-one,henceinvertible. 8.Let f and g beinvertiblefunctions.Withthehelpofyourclassmatesshowthat f g is one-to-one,henceinvertible,andthat f g )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = g )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x 9.Whatgraphicalfeaturemustafunction f possessforittobeitsowninverse?

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296FurtherTopicsinFunctions 5.2.2Answers 1.a f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = x +2 6 b f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(5 3 x + 1 3 c f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 +5 ;x 2 d f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x =5+ p x +25 e f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = q x +5 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 f f x = )]TJ/F40 7.9701 Tf 9.68 10.993 Td [(p x +1 2 x> 1 g f x = 4 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 x h f x = x 3 x +1 i f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = 6 x +2 3 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 j f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x +3 3.Giventhat f =1,wehave f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 =0.Similarly f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 =1and f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3= )]TJ/F15 10.9091 Tf 8.485 0 Td [(1

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5.3OtherAlgebraicFunctions297 5.3OtherAlgebraicFunctions Thissectionservesasawatershedforfunctionswhicharecombinationsofpolynomial,andmore generally,rationalfunctions,withtheoperationsofradicals.ItisbusinessofCalculustodiscuss thesefunctionsinallthedetailtheydemandsoouraiminthissectionistohelpshoreupthe requisiteskillsneededsothatthereadercananswerCalculus'scallwhenthetimecomes.We brieyrecallthedenitionandsomeofthebasicpropertiesofradicalsfromIntermediateAlgebra. 1 Definition 5.4 Let x bearealnumberand n anaturalnumber. a If n isodd,the principal n th root of x ,denoted n p x istheuniquerealnumbersatisfying n p x n = x .If n iseven, n p x is denedsimilarly b provided x 0and n p x 0.The index isthenumber n andthe radicand is thenumber x .For n =2,wewrite p x insteadof 2 p x a Recallthismeans n =1 ; 2 ; 3 ;::: b Recallboth x = )]TJ/F63 8.9664 Tf 7.168 0 Td [(2and x =2satisfy x 4 =16,but 4 p 16=2,not )]TJ/F63 8.9664 Tf 7.167 0 Td [(2. Itisworthremarkingthat,inlightofSection5.2,wecoulddene f x = n p x functionallyasthe inverseof g x = x n withthestipulationthatwhen n iseven,thedomainof g isrestrictedto[0 ; 1 Fromwhatweknowabout g x = x n fromSection3.1alongwithTheorem5.3,wecanproduce thegraphsof f x = n p x byreectingthegraphsof g x = x n acrosstheline y = x .Belowarethe graphsof y = p x y = 4 p x and y = 6 p x .Thepoint ; 0isindicatedasareference.Theaxesare hiddensowecanseetheverticalsteepeningnear x =0andthehorizontalatteningas x !1 y = p x y = 4 p x y = 6 p x Theodd-indexedradicalfunctionsalsofollowapredictabletrend-steepeningnear x =0and atteningas x .Intheexercises,you'llhaveachancetographsomebasicradicalfunctions usingthetechniquespresentedinSection1.8. y = 3 p x y = 5 p x y = 7 p x Wehaveusedallofthefollowingpropertiesatsomepointinthetextbookforthecase n =2the squareroot,butwelistthemhereingeneralityforcompleteness. 1 AlthoughwediscussedimaginarynumbersinSection3.4,werestrictourattentiontorealnumbersinthissection. Seetheepilogueonpage216formoredetails.

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298FurtherTopicsinFunctions Theorem 5.6 PropertiesofRadicals: Let x and y berealnumbersand m and n benatural numbers.If n p x n p y arerealnumbers,then ProductRule: n p xy = n p x n p y PowersofRadicals: n p x m = n p x m QuotientRule: n r x y = n p x n p y ,provided y 6 =0. If n isodd, n p x n = x ;if n iseven, n p x n = j x j TheproofofTheorem5.6isbasedonthedenitionoftheprincipalrootsandpropertiesofexponents.Toestablishtheproductrule,considerthefollowing.If n isodd,thenbydenition n p xy istheuniquerealnumbersuchthat n p xy n = xy .Giventhat )]TJ/F38 5.9776 Tf 8.03 -5.168 Td [(n p x n p y n = n p x n )]TJ/F38 5.9776 Tf 8.03 -6.44 Td [(n p y n = xy itmustbethecasethat n p xy = n p x n p y .If n iseven,then n p xy istheuniquenon-negativereal numbersuchthat n p xy n = xy .Alsonotethatsince n iseven, n p x and n p y arealsonon-negative andhencesois n p x n p y .Proceedingasabove,wendthat n p xy = n p x n p y .Thequotientruleis provedsimilarlyandisleftasanexercise.Thepowerruleresultsfromrepeatedapplicationofthe productrule,solongas n p x isarealnumbertostartwith. 2 Thelastpropertyisanapplicationof thepowerrulewhen n isodd,andtheoccurrenceoftheabsolutevaluewhen n isevenisdueto therequirementthat n p x 0inDenition5.4.Forinstance, 4 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 4 = 4 p 16=2= j)]TJ/F15 10.9091 Tf 15.704 0 Td [(2 j ,not )]TJ/F15 10.9091 Tf 8.485 0 Td [(2. It'sthislastpropertywhichmakescompositionsofrootsandpowersdelicate.Thisisespecially truewhenweuseexponentialnotationforradicals.Recallthefollowingdenition. Definition 5.5 Let x bearealnumber, m aninteger a and n anaturalnumber. x 1 n = n p x andisdenedwhenever n p x isdened. x m n = n p x m = n p x m ,whenever n p x m isdened. a Recallthismeans m =0 ; 1 ; 2 ;::: TherationalexponentsdenedinDenition5.5behaveverysimilarlytotheusualintegerexponents fromElementaryAlgebrawithonecriticalexception.Considertheexpression )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 = 3 3 = 2 .Applying theusuallawsofexponents,we'dbetemptedtosimplifythisas )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 = 3 3 = 2 = x 2 3 3 2 = x 1 = x However,ifwesubstitute x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1andapplyDenition5.5,wend )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 = 3 = )]TJ/F35 5.9776 Tf 8.03 -4.313 Td [(3 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 =1 sothat )]TJ/F15 10.9091 Tf 5 -8.836 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 = 3 3 = 2 =1 3 = 2 = )]TJ/F54 10.9091 Tf 5 0.188 Td [(p 1 3 =1 3 =1.Weseeinthiscasethat )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 = 3 3 = 2 6 = x .Ifwetake thetimetorewrite )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 = 3 3 = 2 withradicals,wesee x 2 = 3 3 = 2 = )]TJ/F35 5.9776 Tf 8.03 -4.528 Td [(3 p x 2 3 = 2 = q )]TJ/F35 5.9776 Tf 8.031 -4.528 Td [(3 p x 2 3 = )]TJ 5 0.437 Td [( 3 p x 3 = )]TJ/F35 5.9776 Tf 8.031 -4.528 Td [(3 p x 3 = j x j Intheplay-by-playanalysis,weseethatwhenwecanceledthe2'sinmultiplying 2 3 3 2 ,wewere, 2 Otherwisewe'drunintothesameparadoxwedidinSection3.4.

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5.3OtherAlgebraicFunctions299 infact,attemptingtocancelasquarewithasquareroot.Thefactthat p x 2 = j x j andnot simply x istheroot 3 ofthetrouble.Itmayamusethereadertoknowthat )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 3 = 2 2 = 3 = x ,and thisvericationisleftasanexercise.Themoralofthestoryisthatwhensimplifyingfractional exponents,it'susuallybesttorewritethemasradicals. 4 Thelastmajorpropertywewillstate, andleavetoCalculustoprove,isthatradicalfunctionsarecontinuousontheirdomains,sothe IntermediateValueTheorem,Theorem3.1,applies.Thismeansthatifwetakecombinationsof radicalfunctionswithpolynomialandrationalfunctionstoformwhattheauthorsconsiderthe algebraicfunctions 5 wecanmakesigndiagramsusingtheproceduresetforthinSection4.2. StepsforConstructingaSignDiagramforanAlgebraicFunction Suppose f isanalgebraicfunction. 1.Placeanyvaluesexcludedfromthedomainof f onthenumberlinewithan` 'abovethem. 2.Findthezerosof f andplacethemonthenumberlinewiththenumber0abovethem. 3.Chooseatestvalueineachoftheintervalsdeterminedinsteps1and2. 4.Determinethesignof f x foreachtestvalueinstep3,andwritethatsignabovethe correspondinginterval. OurnextexamplereviewsquiteabitofIntermediateAlgebraanddemonstratessomeofthenew featuresofthesegraphs. Example 5.3.1 Forthefollowingfunctions,statetheirdomainsandcreatesigndiagrams.Check youranswergraphicallyusingyourcalculator. 1. f x =3 x 3 p 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2. g x = p 2 )]TJ/F35 5.9776 Tf 13.94 4.523 Td [(4 p x +3 3. h x = 3 r 8 x x +1 4. k x = 2 x p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Solution. 1.Asfarasdomainisconcerned, f x hasnodenominatorsandnoevenroots,whichmeansits domainis ; 1 .Tocreatethesigndiagram,wendthezerosof f 3 Didyoulikethatpun? 4 Inmostothercases,though,rationalexponentsarepreferred. 5 AsmentionedinSection2.2, f x = p x 2 = j x j sothatabsolutevalueisalsoconsideredanalgebraicfunctions.

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300FurtherTopicsinFunctions f x =0 3 x 3 p 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x =0 3 x =0or 3 p 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x =0 x =0or )]TJ/F35 5.9776 Tf 8.03 -4.313 Td [(3 p 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 =0 3 x =0or2 )]TJ/F53 10.9091 Tf 10.91 0 Td [(x =0 x =0or x =2 Thezeros0and2dividetherealnumberlineintothreetestintervals.Thesigndiagram andaccompanyinggrapharebelow.Notethattheintervalsonwhich f is+correspond towherethegraphof f isabovethe x -axis,andwherethegraphof f isbelowthe x -axis wehavethat f is )]TJ/F15 10.9091 Tf 8.485 0 Td [(.Thecalculatorsuggestssomethingmysterioushappensnear x =2. Zoominginshowsthegraphbecomesnearlyverticalthere.You'llhavetowaituntilCalculus tofullyunderstandthisphenomenon. )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 0 + 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( y = f x y = f x near x =2. 2.In g x = p 2 )]TJ/F35 5.9776 Tf 13.94 4.523 Td [(4 p x +3,wehavetworadicalsbothofwhichareevenindexed.Tosatisfy 4 p x +3,werequire x +3 0or x )]TJ/F15 10.9091 Tf 20.167 0 Td [(3.Tosatisfy p 2 )]TJ/F35 5.9776 Tf 13.94 4.523 Td [(4 p x +3,weneed2 )]TJ/F35 5.9776 Tf 14.006 4.523 Td [(4 p x +3 0. Whileitmaybetemptingtowritethisas2 4 p x +3andtakebothsidestothefourth power,therearetimeswhenthistechniquewillproduceerroneousresults. 6 Instead,wesolve 2 )]TJ/F35 5.9776 Tf 14.119 4.524 Td [(4 p x +3 0usingasigndiagram.Ifwelet r x =2 )]TJ/F35 5.9776 Tf 14.119 4.524 Td [(4 p x +3,weknow x )]TJ/F15 10.9091 Tf 20.449 0 Td [(3,sowe concernourselveswithonlythisportionofthenumberline.Tondthezerosof r weset r x =0andsolve2 )]TJ/F35 5.9776 Tf 14.041 4.523 Td [(4 p x +3=0.Weget 4 p x +3=2sothat )]TJ/F35 5.9776 Tf 8.03 -4.313 Td [(4 p x +3 4 =2 4 fromwhich weobtain x +3=16or x =13.Sinceweraisedbothsidesofanequationtoanevenpower, weneedtochecktoseeif x =13isanextraneoussolution. 7 Wend x =13doeschecksince 2 )]TJ/F35 5.9776 Tf 13.94 4.524 Td [(4 p x +3=2 )]TJ/F35 5.9776 Tf 13.939 4.524 Td [(4 p 13+3=2 )]TJ/F35 5.9776 Tf 13.939 5.069 Td [(4 p 16=2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2=0.Belowisoursigndiagramfor r )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 + 13 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( Wend2 )]TJ/F35 5.9776 Tf 13.995 4.523 Td [(4 p x +3 0on[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 13]sothisisthedomainof g .Tondasigndiagramfor g welookforthezerosof g .Setting g x =0isequivalentto p 2 )]TJ/F35 5.9776 Tf 13.939 4.523 Td [(4 p x +3=0.Aftersquaring 6 Forinstance, )]TJ/F63 8.9664 Tf 7.167 0 Td [(2 4 p x +3,whichhasnosolutionor )]TJ/F63 8.9664 Tf 7.168 0 Td [(2 4 p x +3whosesolutionis[ )]TJ/F63 8.9664 Tf 7.167 0 Td [(3 ; 1 7 Recall,thismeanswehaveproducedacandidatewhichdoesn'tsatisfytheoriginalequation.Doyouremember howraisingbothsidesofanequationtoanevenpowercouldcausethis?

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5.3OtherAlgebraicFunctions301 bothsides,weget2 )]TJ/F35 5.9776 Tf 14.413 4.524 Td [(4 p x +3=0,whosesolutionwehavefoundtobe x =13.Sincewe squaredbothsides,wedoublecheckandnd g is,infact,0.Oursigndiagramandgraph of g arebelow.Sincethedomainof g is[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 13],whatwehavebelowisnotjusta portion ofthegraphof g ,butthe complete graph.Itisalwaysaboveoronthe x -axis,whichveries oursigndiagram. )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 + 13 Thecompletegraphof y = g x 3.Theradicalin h x isodd,soouronlyconcernisthedenominator.Setting x +1=0gives x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,soourdomainis ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 .Tondthezerosof h ,weset h x =0.To solve 3 q 8 x x +1 =0,wecubebothsidestoget 8 x x +1 =0.Weget8 x =0,or x =0.Belowis theresultingsigndiagramandcorrespondinggraph.Fromthegraph,itappearsasthough x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1isaverticalasymptote.Carryingoutananalysisas x !)]TJ/F15 10.9091 Tf 22.424 0 Td [(1asinSection4.2conrms this.Weleavethedetailstothereader.Near x =0,wehaveasituationsimilarto x =2 inthegraphof f innumber1above.Finally,itappearsasifthegraphof h hasahorizontal asymptote y =2.UsingtechniquesfromSection4.2,wendas x 8 x x +1 8.From this,itishardlysurprisingthatas x h x = 3 q 8 x x +1 3 p 8=2. + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 0 + y = h x 4.Tondthedomainof k ,wehavebothanevenrootandadenominatortoconcernourselves with.Tosatisfythesquareroot, x 2 )]TJ/F15 10.9091 Tf 10.929 0 Td [(1 0.Setting r x = x 2 )]TJ/F15 10.9091 Tf 10.929 0 Td [(1,wendthezerosof r to be x = 1,andwendthesigndiagramof r tobe + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 1 0 +

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302FurtherTopicsinFunctions Wend x 2 )]TJ/F15 10.9091 Tf 10.923 0 Td [(1 0for ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1] [ [1 ; 1 .Tokeepthedenominatorof k x awayfromzero, weset p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1=0.Weleaveittothereadertoverifythesolutionsare x = 1,bothof whichmustbeexcludedfromthedomain.Hence,thedomainof k is ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ ; 1 .To buildthesigndiagramfor k ,weneedthezerosof k .Setting k x =0resultsin 2 x p x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 =0. Weget2 x =0or x =0.However, x =0isn'tinthedomainof k ,whichmeans k hasnozeros. Weconstructoursigndiagramonthedomainof k belowalongsidethegraphof k .Itappears thatthegraphof k hastwoverticalasymptotes,oneat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1andoneat x =1.Thegap inthegraphbetweentheasymptotesisbecauseofthegapinthedomainof k .Concerning endbehavior,thereappeartobetwohorizontalasymptotes, y =2and y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Toseewhy thisisthecase,wethinkof x .Theradicandofthedenominator x 2 )]TJ/F15 10.9091 Tf 10.929 0 Td [(1 x 2 ,andas such, k x = 2 x p x 2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 2 x p x 2 = 2 x j x j .As x !1 ,wehave j x j = x so k x 2 x x =2.Ontheother hand,as x j x j = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x ,andassuch k x 2 x )]TJ/F37 7.9701 Tf 6.587 0 Td [(x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Finally,itappearsasthough thegraphof k passestheHorizontalLineTestwhichmeans k isonetooneand k )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 exists. Computing k )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 isleftasanexercise. )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 + y = k x Asthepreviousexampleillustrates,thegraphsofgeneralalgebraicfunctionscanhavefeatures we'veseenbefore,likeverticalandhorizontalasymptotes,buttheycanoccurinnewandexciting ways.Forexample, k x = 2 x p x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 hadtwodistincthorizontalasymptotes.You'llrecallthat rationalfunctionscouldhaveatmostonehorizontalasymptote.Alsosomenewcharacteristicslike `unusualsteepness' 8 andcusps 9 canappearinthegraphsofarbitraryalgebraicfunctions.Ournext examplerstdemonstrateshowwecanusesigndiagramstosolvenonlinearinequalities.Don't panic.ThetechniqueisverysimilartotheonesusedinChapters2,3and4.Wethencheckour answersgraphicallywithacalculatorandseesomeofthenewgraphicalfeaturesofthefunctions inthisextendedfamily. Example 5.3.2 Solvethefollowinginequalities.Checkyouranswersgraphicallyusingacalculator. 1. x 2 = 3
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5.3OtherAlgebraicFunctions303 Solution. 1.Tosolve x 2 = 3 0.We set r x = x 4 = 3 )]TJ/F53 10.9091 Tf 10.714 0 Td [(x 2 = 3 )]TJ/F15 10.9091 Tf 10.714 0 Td [(6andnotethatsincethedenominatorsintheexponentsare3,they correspondtocuberoots,whichmeansthedomainof r is ; 1 .Tondthezerosfor thesigndiagram,weset r x =0andattempttosolve x 4 = 3 )]TJ/F53 10.9091 Tf 11.192 0 Td [(x 2 = 3 )]TJ/F15 10.9091 Tf 11.192 0 Td [(6=0.Atthispoint, itmaybeunclearhowtoproceed.Wecouldalwaystryasalastresortconvertingbackto radicalnotation,butinthiscasewecantakeacuefromExample3.3.4.Sincethereare threeterms,andtheexponentononeofthevariableterms, x 4 = 3 ,isexactlytwicethatofthe other, x 2 = 3 ,wehaveourselvesa`quadraticindisguise'andwecanrewrite x 4 = 3 )]TJ/F53 10.9091 Tf 10.224 0 Td [(x 2 = 3 )]TJ/F15 10.9091 Tf 10.224 0 Td [(6=0 as )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 = 3 2 )]TJ/F53 10.9091 Tf 11.433 0 Td [(x 2 = 3 )]TJ/F15 10.9091 Tf 11.432 0 Td [(6=0.Ifwelet u = x 2 = 3 ,thenintermsof u ,weget u 2 )]TJ/F53 10.9091 Tf 11.432 0 Td [(u )]TJ/F15 10.9091 Tf 11.432 0 Td [(6=0. Solvingfor u ,weobtain u = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2or u =3.Replacing x 2 = 3 backinfor u ,weget x 2 = 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 or x 2 = 3 =3.ToavoidthetroubleweencounteredinthediscussionfollowingDenition5.5, wenowconvertbacktoradicalnotation.Byinterpreting x 2 = 3 as 3 p x 2 wehave 3 p x 2 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 or 3 p x 2 =3.Cubingbothsidesoftheseequationsresultsin x 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8,whichadmitsno realsolution,or x 2 =27,whichgives x = 3 p 3.Weconstructasigndiagramandnd x 4 = 3 )]TJ/F53 10.9091 Tf 11.403 0 Td [(x 2 = 3 )]TJ/F15 10.9091 Tf 11.403 0 Td [(6 > 0on )]TJ/F54 10.9091 Tf 5 -8.837 Td [( ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 p 3 [ )]TJ/F15 10.9091 Tf 5 -8.837 Td [(3 p 3 ; 1 .Tocheckouranswergraphically,weset f x = x 2 = 3 and g x = x 4 = 3 )]TJ/F15 10.9091 Tf 11.993 0 Td [(6.Thesolutionto x 2 = 3
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304FurtherTopicsinFunctions y = f x near x =0 y = g x near x =0 2.Tosolve3 )]TJ/F53 10.9091 Tf 10.96 0 Td [(x 1 = 3 x )]TJ/F53 10.9091 Tf 10.961 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 = 3 ,wegatherallthenonzerotermsononesideandobtain 3 )]TJ/F53 10.9091 Tf 11.441 0 Td [(x 1 = 3 )]TJ/F53 10.9091 Tf 11.44 0 Td [(x )]TJ/F53 10.9091 Tf 11.441 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 = 3 0.Weset r x =3 )]TJ/F53 10.9091 Tf 11.441 0 Td [(x 1 = 3 )]TJ/F53 10.9091 Tf 11.441 0 Td [(x )]TJ/F53 10.9091 Tf 11.441 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 = 3 .Asinnumber 1,thedenominatorsoftherationalexponentsareodd,whichmeanstherearenodomain concernsthere.However,thenegativeexponentonthesecondtermindicatesadenominator. Rewriting r x withpositiveexponents,weobtain r x =3 )]TJ/F53 10.9091 Tf 10.923 0 Td [(x 1 = 3 )]TJ/F37 7.9701 Tf 26.633 4.295 Td [(x )]TJ/F37 7.9701 Tf 6.586 0 Td [(x 2 = 3 .Settingthe denominatorequaltozeroweget )]TJ/F53 10.9091 Tf 10.678 0 Td [(x 2 = 3 =0,or 3 p )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 =0.Aftercubingbothsides, andsubsequentlytakingsquareroots,weget2 )]TJ/F53 10.9091 Tf 10.938 0 Td [(x =0,or x =2.Hence,thedomainof r is ; 2 [ ; 1 .Tondthezerosof r ,weset r x =0.Therearetwoschoolofthoughton howtoproceedandwedemonstrateboth. FactoringApproach. From r x =3 )]TJ/F53 10.9091 Tf 10.635 0 Td [(x 1 = 3 )]TJ/F53 10.9091 Tf 10.635 0 Td [(x )]TJ/F53 10.9091 Tf 10.635 0 Td [(x )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 = 3 ,wenotethatthequantity )]TJ/F53 10.9091 Tf 9.66 0 Td [(x iscommontobothterms.Whenwefactoroutcommonfactors,wefactoroutthe quantitywiththe smaller exponent.Inthiscase,since )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(2 3 < 1 3 ,wefactor )]TJ/F53 10.9091 Tf 11.168 0 Td [(x )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 = 3 frombothquantities.Whileitmayseemoddtodoso,weneedtofactor )]TJ/F53 10.9091 Tf 11.328 0 Td [(x )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 = 3 from )]TJ/F53 10.9091 Tf 9.901 0 Td [(x 1 = 3 ,whichresultsinsubtractingtheexponent )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(2 3 from 1 3 .Weproceedusing theusualpropertiesofexponentsandexercisespecialcarewhenreducingthe 3 3 power to1. r x =3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 1 = 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 = 3 = )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 = 3 h 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 1 3 )]TJ/F15 10.9091 Tf 6.587 -0.735 Td [( )]TJ/F35 5.9776 Tf 7.782 3.258 Td [(2 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x i = )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 = 3 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 = 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x = )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 = 3 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x since 3 p u 3 = 3 p u 3 = u = )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 = 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x = )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 = 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x Tosolve r x =0,weset )]TJ/F53 10.9091 Tf 10.948 0 Td [(x )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 = 3 )]TJ/F15 10.9091 Tf 10.91 0 Td [(4 x =0,or 6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 x )]TJ/F37 7.9701 Tf 6.586 0 Td [(x 2 = 3 =0.Wehave6 )]TJ/F15 10.9091 Tf 10.948 0 Td [(4 x =0 or x = 3 2 .

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5.3OtherAlgebraicFunctions305 CommonDenominatorApproach. Werewrite r x =3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 1 = 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 = 3 =3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 1 = 3 )]TJ/F53 10.9091 Tf 32.342 7.38 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = 3 = 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 1 = 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = 3 )]TJ/F53 10.9091 Tf 32.342 7.381 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = 3 commondenominator = 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 1 3 + 2 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = 3 )]TJ/F53 10.9091 Tf 32.341 7.381 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = 3 = 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 = 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = 3 )]TJ/F53 10.9091 Tf 32.342 7.38 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = 3 = 3 )]TJ/F53 10.9091 Tf 10.91 0 Td [(x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = 3 )]TJ/F53 10.9091 Tf 32.341 7.38 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = 3 since 3 p u 3 = 3 p u 3 = u = 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = 3 = 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 = 3 Asbefore,whenweset r x =0weobtain x = 3 2 Wenowcreateoursigndiagramandnd3 )]TJ/F53 10.9091 Tf 9.941 0 Td [(x 1 = 3 )]TJ/F53 10.9091 Tf 9.942 0 Td [(x )]TJ/F53 10.9091 Tf 9.941 0 Td [(x )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 = 3 0on 3 2 ; 2 [ ; 1 .To checkthisgraphically,weset f x =32 )]TJ/F53 10.9091 Tf 10.466 0 Td [(x 1 = 3 and g x = x )]TJ/F53 10.9091 Tf 10.466 0 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 = 3 thethickercurve. Weconrmthatthegraphsintersectat x = 3 2 andthegraphof f isbelowthegraphof g for x 3 2 ,withtheexceptionof x =2whereitappearsthegraphof g hasaverticalasymptote. + 3 2 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [( 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [( y = f x and y = g x OneapplicationofalgebraicfunctionswasgiveninExample1.7.6inSection1.1.Ourlastexample isamoresophisticatedapplicationofdistance. Example 5.3.3 CarlwishestogethighspeedinternetserviceinstalledinhisremoteSasquatch observationpostlocated30milesfromRoute117.Thenearestjunctionboxislocated50miles downroadfromthepost,asindicatedinthediagrambelow.Supposeitcosts$15permiletorun cablealongtheroadand$20permiletoruncableooftheroad.

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306FurtherTopicsinFunctions Outpost JunctionBox x y z Route117 50miles 30miles 1.Expressthetotalcost C ofconnectingtheJunctionBoxtotheOutpostasafunctionof x thenumberofmilesthecableisrunalongRoute117beforeheadingoroaddirectlytowards theOutpost.Determineareasonableapplieddomainfortheproblem. 2.Useyourcalculatortograph y = C x onitsdomain.Whatistheminimumcost?Howfar alongRoute117shouldthecableberunbeforeturningooftheroad? Solution. 1.Thecostisbrokenintotwoparts:thecosttoruncablealongRoute117at$15permile,and thecosttorunitoroadat$20permile.Since x representsthemilesofcablerunalong Route117,thecostforthatportionis15 x .Fromthediagram,weseethatthenumberof milesthecableisrunoroadis z ,sothecostofthatportionis20 z .Hence,thetotalcostis C =15 x +20 z .Ournextgoalistodetermine z asafunctionof x .Thediagramsuggestswe canusethePythagoreanTheoremtoget y 2 +30 2 = z 2 .Butwealsosee x + y =50sothat y =50 )]TJ/F53 10.9091 Tf 11.091 0 Td [(x .Hence, z 2 = )]TJ/F53 10.9091 Tf 11.09 0 Td [(x 2 +900.Solvingfor z ,weobtain z = p )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 +900. Since z representsadistance,wechoose z = p )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 +900sothatourcostasafunction of x onlyisgivenby C x =15 x +20 p )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 +900 Fromthecontextoftheproblem,wehave0 x 50. 2.Graphing y = C x onacalculatorinasuitablewindowproducesthegraphbelow.Using the`Minimum'feature,wendtherelativeminimumwhichisalsotheabsoluteminimum inthiscasetotwodecimalplacesbe : 98 ; 1146 : 86.Herethe x -coordinatetellsusthatin ordertominimizecost,weshouldrun15 : 98milesofcablealongRoute117andthenturno oftheroadandheadtowardstheoutpost.The y -coordinatetellsusthattheminimumcost, indollars,todosois$1146 : 86.TheabilitytostreamliveSasquatchCasts?Priceless.

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5.3OtherAlgebraicFunctions307 5.3.1Exercises 1.Foreachfunctionbelow Finditsdomain. Createasigndiagram. Useyourcalculatortohelpyousketchitsgraphandidentifyanyverticalorhorizontal asymptotes,`unusualsteepness'orcusps. a f x = p 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 b f x = 4 r 16 x x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 c f x = x 2 3 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(7 1 3 d f x = 5 x 3 p x 3 +8 e f x = x 3 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 1 3 f f x = p x x +5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 g f x = 3 p x 3 +3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 2.UsethetransformationspresentedinSection1.8tographthefollowingfunctions. a f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 p x +1+4 b f x =3 4 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 c f x = 5 p x +2+3 d f x = 8 p )]TJ/F53 10.9091 Tf 8.484 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 3.Solvethefollowingequationsandinequalities. a2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1= p x +3 b x 3 2 =8 c x 2 3 =4 d 3 p x x e p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2+ p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(5=3 f5 )]TJ/F15 10.9091 Tf 10.91 0 Td [( )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 3 =1 g10 )]TJ 10.909 8.569 Td [(p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 11 h 1 3 x 3 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F35 5.9776 Tf 7.782 3.259 Td [(2 3 + 3 4 x )]TJ/F35 5.9776 Tf 7.782 3.259 Td [(1 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 1 3 < 0 i 3 p x 3 +3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 >x +1 j 2 3 x +4 3 5 x )]TJ/F15 9.9626 Tf 8.787 0 Td [(2 )]TJ/F6 4.9813 Tf 7.422 2.678 Td [(1 3 + 3 5 x +4 )]TJ/F6 4.9813 Tf 7.422 2.678 Td [(2 5 x )]TJ/F15 9.9626 Tf 8.787 0 Td [(2 2 3 0 4.Showthat x 3 2 2 3 = x forall x 0. 5.VerifytheQuotientRuleforRadicalsinTheorem5.6. 6.Findtheinverseof k x = 2 x p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 7.Showthat 3 p 2isanirrationalnumberbyrstshowingthatitisazeroof p x = x 3 )]TJ/F15 10.9091 Tf 10.929 0 Td [(2and thenshowing p hasnorationalzeros.You'llneedtheRationalZerosTheorem,Theorem3.9, inordertoshowthislastpart. 8.Withthehelpofyourclassmates,generalizethepreviousexercisetoshowthat n p c isan irrationalnumberprovidedthat c n isnotarationalnumber.

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308FurtherTopicsinFunctions 9.Theperiodofapenduluminsecondsisgivenby T =2 r L g forsmalldisplacementswhere L isthelengthofthependuluminmetersand g =9 : 8meterspersecondpersecondisthe accelerationduetogravity.MySeth-Thomasantiqueschoolhouseclockneeds T = 1 2 second andIcanadjustthelengthofthependulumviaasmalldialonthebottomofthebob.At whatlengthshouldIsetthependulum? 10.TheCobb-Douglasproductionmodelstatesthattheyearlytotaldollarvalueoftheproduction output P inaneconomyisafunctionoflabor x thetotalnumberofhoursworkedinayear andcapital y thetotaldollarvalueofallofthestupurchasedinordertomakethings. Specically, P = ax b y 1 )]TJ/F37 7.9701 Tf 6.586 0 Td [(b .Byxing P ,wecreatewhat'sknownasan`isoquant'andwecan thensolvefor y asafunctionof x .Let'sassumethattheCobb-Douglasproductionmodel forthecountryofSasquatchiais P =1 : 23 x 0 : 4 y 0 : 6 aLet P =300andsolvefor y intermsof x .If x =100,whatis y ? bGraphtheisoquant300=1 : 23 x 0 : 4 y 0 : 6 .Whatinformationdoesanorderedpair x;y whichmakes P =300giveyou?Withthehelpofyourclassmates,ndseveraldierent combinationsoflaborandcapitalallofwhichyield P =300.Discussanypatternsyou maysee. 11.AccordingtoEinstein'sTheoryofSpecialRelativity,theobservedmass m ofanobjectisa functionofhowfasttheobjectistraveling.Specically, m x = m r q 1 )]TJ/F37 7.9701 Tf 12.105 4.296 Td [(x 2 c 2 where m = m r isthemassoftheobjectatrest, x isthespeedoftheobjectand c isthespeedoflight. aFindtheapplieddomainofthefunction. bCompute m : 1 c ;m : 5 c ;m : 9 c and m : 999 c cAs x c )]TJ/F15 10.9091 Tf 7.084 -3.959 Td [(,whathappensto m x ? dHowslowlymusttheobjectbetravelingsothattheobservedmassisnogreaterthan 100timesitsmassatrest? 12.SupposeFritzytheFox,positionedatapoint x;y intherstquadrant,spotsChewbacca theBunnyat ; 0.Chewbaccabeginstorunalongafencethepositive y -axistowardshis warren.Fritzy,ofcourse,takeschaseandconstantlyadjustshisdirectionsothatheisalways runningdirectlyatChewbacca.IfChewbacca'sspeedis v 1 andFrity'sspeedis v 2 ,thepath FritzywilltaketointerceptChewbacca,provided v 2 isdirectlyproportionalto,butnotequal to, v 1 ismodeledby y = 1 2 x 1+ v 1 =v 2 1+ v 1 =v 2 )]TJ/F53 10.9091 Tf 15.078 7.38 Td [(x 1 )]TJ/F37 7.9701 Tf 6.586 0 Td [(v 1 =v 2 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(v 1 =v 2 + v 1 v 2 v 2 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(v 2 1

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5.3OtherAlgebraicFunctions309 aDeterminethepathFritzywilltakeifherunsexactlytwiceasfastasChewbacca;that is, v 2 =2 v 1 .Useyourcalculatortographthispathfor x 0.Whatisthesignicance ofthe y -interceptofthegraph? bDeterminethepathFritzywilltakeifChewbaccarunsexactlytwiceasfastashedoes; thatis, v 1 =2 v 2 .Useyourcalculatortographthispathfor x> 0.Describethebehavior of y as x 0 + andinterpretthisphysically. cWiththehelpofyourclassmates,generalizepartsaandbtotwocases: v 2 >v 1 and v 2
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310FurtherTopicsinFunctions 5.3.2Answers 1.a f x = p 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 Domain:[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1] )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 + 00 1 Noasymptotes Unusualsteepnessat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and x =1 Nocusps x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(11 1 b f x = 4 r 16 x x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 Domain: )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 0] [ ; 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 0 + Verticalasymptotes: x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3and x =3 Horizontalasymptote: y =0 Unusualsteepnessat x =0 Nocusps x y )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112345678 1 2 3 4 5 c f x = x 2 3 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(7 1 3 Domain: ; 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 7 0 + Noverticalorhorizontalasymptotes 12 Unusualsteepnessat x =7 Cuspat x =0 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123456789 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 d f x = 5 x 3 p x 3 +8 Domain: ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 [ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 0 + Verticalasymptote x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 Horizontalasymptote y =5 Nounusualsteepnessorcusps x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 8 12 UsingCalculusitcanbeshownthat y = x )]TJ/F35 5.9776 Tf 10.411 3.673 Td [(7 3 isaslantasymptoteofthisgraph.

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5.3OtherAlgebraicFunctions311 e f x = x 3 2 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(7 1 3 Domain:[0 ; 1 0 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [( 7 0 + Noasymptotes Unusualsteepnessat x =7 Nocusps x y 12345678 )]TJ/F34 7.9701 Tf 6.586 0 Td [(15 )]TJ/F34 7.9701 Tf 6.586 0 Td [(10 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 5 10 15 20 25 f f x = p x x +5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 Domain:[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 ; 0] [ [4 ; 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 0 + 0 0 4 0 + Noasymptotes Unusualsteepnessat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 ;x =0and x =4 Nocusps x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 1 2 3 4 5 6 7 8 9 g f x = 3 p x 3 +3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 Domain: ; 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 2 0 + Noverticalorhorizontalasymptotes 13 Unusualsteepnessat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ;x =1and x =2 Nocusps x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 1 2 3 4 5 6 2.a f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 p x +1+4 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345678 1 2 3 4 5 6 7 b f x =3 4 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x y 7823 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 13 UsingCalculusitcanbeshownthat y = x +1isaslantasymptoteofthisgraph.

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312FurtherTopicsinFunctions c f x = 5 p x +2+3 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(34 )]TJ/F35 5.9776 Tf 5.756 0 Td [(230 1 2 3 4 5 d f x = 8 p )]TJ/F53 10.9091 Tf 8.484 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(40 )]TJ/F35 5.9776 Tf 5.756 0 Td [(30 )]TJ/F35 5.9776 Tf 5.756 0 Td [(20 )]TJ/F35 5.9776 Tf 5.756 0 Td [(10 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 3.a x = 5+ p 57 8 b x =4 c x = 8 d[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0] [ [1 ; 1 e x =6 f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 6 g[2 ; 1 h )]TJ/F15 10.9091 Tf 5 -8.836 Td [(0 ; 27 13 i ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 j ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 [ )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(22 19 [ ; 1 6. k )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = x p x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 9.9 : 8 1 4 2 0 : 062metersor6 : 2centimeters 10.aFirstrewritethemodelas P =1 : 23 x 2 5 y 3 5 .Then300=1 : 23 x 2 5 y 3 5 yields y = 300 1 : 23 x 2 5 5 3 If x =100then y 441 : 93687. 11.a[0 ;c b m : 1 c = m r p : 99 1 : 005 m r m : 5 c = m r p : 75 1 : 155 m r m : 9 c = m r p : 19 2 : 294 m r m : 999 c = m r p : 0 : 001999 22 : 366 m r cAs x c )]TJ/F53 10.9091 Tf 7.085 -3.959 Td [(;m x !1 dIftheobjectistravelingnofasterthanapproximately0 : 99995timesthespeedoflight, thenitsobservedmasswillbenogreaterthan100 m r .

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5.3OtherAlgebraicFunctions313 12.a y = 1 3 x 3 = 2 )]TJ 10.778 7.857 Td [(p x + 2 3 .Thepoint )]TJ/F15 10.9091 Tf 5 -8.836 Td [(0 ; 2 3 iswhenFritzy'spathcrossesChewbacca'spathinotherwords,whereFritzycatchesChewbacca. b y = 1 6 x 3 + 1 2 x )]TJ/F34 7.9701 Tf 12.506 4.296 Td [(2 3 .UsingthetechniquesfromChapter4,wendas x 0 + y !1 whichmeans,inthiscase,Fritzy'spursuitneverends;henevercatchesChewbacca.This makessensesinceChewbaccahasaheadstartandisrunningfasterthanFritzy. y = 1 3 x 3 = 2 )]TJ 10.909 7.858 Td [(p x + 2 3 y = 1 6 x 3 + 1 2 x )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(2 3

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314FurtherTopicsinFunctions

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Chapter6 ExponentialandLogarithmic Functions 6.1IntroductiontoExponentialandLogarithmicFunctions Ofallofthefunctionswestudyinthistext,exponentialandlogarithmicfunctionsarepossibly theoneswhichimpacteverydaylifethemost. 1 Thissectionwillintroduceustothesefunctions whiletherestofthechapterwillmorethoroughlyexploretheirproperties.Uptothispoint,we havedealtwithfunctionswhichinvolvetermslike x 2 or x 2 = 3 ,inotherwords,termsoftheform x p wherethebaseoftheterm, x ,variesbuttheexponentofeachterm, p ,remainsconstant.Inthis chapter,westudyfunctionsoftheform f x = b x wherethebase b isaconstantandtheexponent x isthevariable.Westartourexplorationofthesefunctionswith f x =2 x .Apparentlythisisa tradition.EveryCollegeAlgebrabookwehaveeverreadstartswith f x =2 x .Wemakeatable ofvalues,plotthepointsandconnecttheminapleasingfashion. x f x x;f x )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 = 1 8 )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 = 1 4 )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 2 0 2 0 =1 ; 1 1 2 1 =2 ; 2 2 2 2 =4 ; 4 3 2 3 =8 ; 8 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 1 2 3 4 5 6 7 8 y = f x =2 x Afewremarksaboutthegraphof f x =2 x whichwehaveconstructedareinorder.As x 1 TakeaclassinDierentialEquationsandyou'llseewhy.

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316ExponentialandLogarithmicFunctions andattainsvalueslike x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(100or x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1000,thefunction f x =2 x takesonvalueslike f )]TJ/F15 10.9091 Tf 8.484 0 Td [(100=2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(100 = 1 2 100 or f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1000=2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1000 = 1 2 1000 .Inotherwords,as x 2 x 1 verybig+ verysmall+ Soas x ,2 x 0 + .Thisisrepresentedgraphicallyusingthe x -axistheline y =0asa horizontalasymptote.Ontheipside,as x !1 ,wend f =2 100 f =2 1000 ,andso on,thus2 x !1 .Asaresult,ourgraphsuggeststherangeof f is ; 1 .Thegraphof f passes theHorizontalLineTestwhichmeans f isone-to-oneandhenceinvertible.Wealsonotethatwhen we`connectedthedotsinapleasingfashion',wehavemadetheimplicitassumptionthat f x =2 x iscontinuous 2 andhasadomainofallrealnumbers.Inparticular,wehavesuggestedthatthings like2 p 3 existasrealnumbers.Weshouldtakeamomenttodiscusswhatsomethinglike2 p 3 might mean,andrefertheinterestedreadertoasolidcourseinCalculusforamorerigorousexplanation. Thenumber p 3=1 : 73205 ::: isanirrationalnumber 3 andassuch,itsdecimalrepresentation neitherrepeatsnorterminates.Wecan,however,approximate p 3byterminatingdecimals,and itstandstoreason 4 wecanusethesetoapproximate2 p 3 .Forexample,ifweapproximate p 3 by1 : 73,wecanapproximate2 p 3 2 1 : 73 =2 173 100 = 100 p 2 173 .Itisnot,byanymeans,apleasant number,butitisatleastanumberthatweunderstandintermsofpowersandroots.Italsostands toreasonthatbetterandbetterapproximationsof p 3yieldbetterandbetterapproximationsof 2 p 3 ,sothevalueof2 p 3 shouldbetheresultofthissequenceofapproximations. 5 Supposewewishtostudythefamilyoffunctions f x = b x .Whichbases b makesensetostudy? Wendthatwerunintodicultyif b< 0.Forexample,if b = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,thenthefunction f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x hastrouble,forinstance,at x = 1 2 since )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 1 = 2 = p )]TJ/F15 10.9091 Tf 8.485 0 Td [(2isnotarealnumber.Ingeneral,if x isanyrationalnumberwithanevendenominator,then )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x isnotdened,sowemustrestrict ourattentiontobases b 0.Whatabout b =0?Thefunction f x =0 x isundenedfor x 0 becausewecannotdivideby0and0 0 isanindeterminantform.For x> 0,0 x =0sothefunction f x =0 x isthesameasthefunction f x =0, x> 0.Weknoweverythingwecanpossiblyknow aboutthisfunction,soweexcludeitfromourinvestigations.Theonlyotherbaseweexcludeis b =1,sincethefunction f x =1 x =1is,onceagain,afunctionwehavealreadystudied.Weare nowreadyforourdenitionofexponentialfunctions. Definition 6.1 Afunctionoftheform f x = b x where b isaxedrealnumber, b> 0, b 6 =1is calleda base b exponentialfunction Weleaveittothereadertoverify 6 thatif b> 1,thentheexponentialfunction f x = b x willshare thesamebasicshapeandcharacteristicsas f x =2 x .Whatif0
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6.1IntroductiontoExponentialandLogarithmicFunctions317 thegraphof f )]TJ/F53 10.9091 Tf 8.485 0 Td [(x isobtainedfromthegraphof f x byreectingitacrossthe y -axis.Assuch, wehave x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 1 2 3 4 5 6 7 8 y = f x =2 x reectacross y -axis )454()222()222()222()223()222()222()222()223()222()222()454(! multiplyeach x -coordinateby )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1123 1 2 3 4 5 6 7 8 y = g x =2 )]TJ/F38 5.9776 Tf 5.756 0 Td [(x = )]TJ/F35 5.9776 Tf 5.077 -3.158 Td [(1 2 x Weseethatthedomainandrangeof g matchthatof f ,namely ; 1 and ; 1 ,respectively. Like f g isalsoone-to-one.Whereas f isalwaysincreasing, g isalwaysdecreasing.Asaresult, as x g x !1 ,andontheipside,as x !1 g x 0 + .Itshouldn'tbetoosurprising thatforallchoicesofthebase0 1: { f isalwaysincreasing { As x f x 0 + { As x !1 f x !1 { Thegraphof f resembles: y = b x b> 1 If0
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318ExponentialandLogarithmicFunctions Ofallofthebasesforexponentialfunctions,twooccurthemostofteninscienticcircles.Therst, base10,isoftencalledthe commonbase .Thesecondbaseisanirrationalnumber, e 2 : 718, calledthe naturalbase .WewillmoreformallydiscusstheoriginsofthisnumberinSection6.5. Fornow,itisenoughtoknowthatsince e> 1, f x = e x isanincreasingexponentialfunction. Thefollowingexamplesoeraglimpseastothekindofreal-worldphenomenathesefunctionscan model. Example 6.1.1 Thevalueofacarcanbemodeledby V t =25 )]TJ/F34 7.9701 Tf 6.195 -4.542 Td [(4 5 x ,where x 0isageofthe carinyearsand V x isthevalueinthousandsofdollars. 1.Findandinterpret V 2.Sketchthegraphof y = V x usingtransformations. 3.Findandinterpretthehorizontalasymptoteofthegraphyoufoundin2. Solution. 1.Tond V ,wereplace x with0toobtain V =25 )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(4 5 0 =25.Since x representstheage ofthecarinyears, x =0correspondstothecarbeingbrandnew.Since V x ismeasured inthousandsofdollars, V =25correspondstoavalueof$25 ; 000.Puttingitalltogether, weinterpret V =25tomeanthepurchasepriceofthecarwas$25 ; 000. 2.Tograph y =25 )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(4 5 x ,westartwiththebasicexponentialfunction f x = )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(4 5 x .Sincethe base b = 4 5 isbetween0and1,thegraphof y = f x isdecreasing.Weplotthe y -intercept ; 1andtwootherpoints, )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 5 4 and )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 ; 4 5 ,andlabelthehorizontalasymptote y =0. Toobtain V x =25 )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(4 5 x x 0,wemultiplytheoutputfrom f by25,inotherwords, V x =25 f x .InaccordancewithTheorem1.5,thisresultsinaverticalstretchbyafactor of25.Wemultiplyallofthe y valuesinthegraphby25includingthe y valueofthe horizontalasymptoteandobtainthepoints )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 125 4 ; 25and ; 20.Thehorizontal asymptoteremains y =0.Finally,werestrictthedomainto[0 ; 1 totwiththeapplied domaingiventous.Wehavetheresultbelow. ; 1 H.A. y =0 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 2 y = f x = )]TJ/F35 5.9776 Tf 5.077 -3.158 Td [(4 5 x verticalscalebyafactorof25 )479()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()479(! multiplyeach y -coordinateby25 ; 25 H.A. y =0 x y 123456 5 10 15 20 30 y = V x =25 f x x 0 3.Weseefromthegraphof V thatitshorizontalasymptoteis y =0.Weleaveittoreaderto verifythisanalyticallybythinkingaboutwhathappensaswetakelargerandlargerpowers of 4 5 .Thismeansasthecargetsolder,itsvaluediminishesto0.

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6.1IntroductiontoExponentialandLogarithmicFunctions319 Thefunctioninthepreviousexampleisoftencalleda`decaycurve'.Increasingexponentialfunctionsareusedtomodel`growthcurves'andweshallseeseveraldierentexamplesofthosein Section6.5.Fornow,wepresentanothercommondecaycurvewhichwillserveasthebasisfor furtherstudyofexponentialfunctions.Althoughitmaylookmorecomplicatedthanthepreviousexample,itisactuallyjustabasicexponentialfunctionwhichhasbeenmodiedbyafew transformationsfromSection1.8. Example 6.1.2 AccordingtoNetwon'sLawofCooling 7 thetemperatureofcoee T indegrees Fahrenheit t minutesafteritisservedcanbemodeledby T t =70+90 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t 1.Findandinterpret T 2.Sketchthegraphof y = T t usingtransformations. 3.Findandinterpretthehorizontalasymptoteofthegraph. Solution. 1.Tond T ,wereplaceeveryoccurrenceoftheindependentvariable t with0toobtain T =70+90 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 =160.Thismeansthatthecoeewasservedat160 F. 2.Tograph y = T t usingtransformations,westartwiththebasicfunction, f t = e t .Aswe havealreadyremarked, e 2 : 718 > 1sothegraphof f isanincreasingexponentialwith y -intercept ; 1andhorizontalasymptote y =0.Thepoints )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ;e )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 : 37and ;e ; 2 : 72arealsoonthegraph.Sincetheformula T t looksrathercomplicated,we rewrite T t intheformpresentedinTheorem1.7andusethatresulttotrackthechangesto ourthreepointsandthehorizontalasymptote.Wehave T t =90 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 1 t +70=90 f )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 1 t + 70.Multiplicationoftheinputto f t ,by )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 1resultsinahorizontalexpansionbyafactor of10aswellasareectionaboutthe y -axis.Wedivideeachofthe x valuesofourpointsby )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 1whichamountstomultiplyingthemby )]TJ/F15 10.9091 Tf 8.485 0 Td [(10toobtain )]TJ/F15 10.9091 Tf 5 -8.837 Td [(10 ;e )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; 1,and )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 ;e Sincenoneofthesechangesaectedthe y values,thehorizontalasymptoteremains y =0. Next,weseethattheoutputfrom f isbeingmultipliedby90.Thisresultsinavertical stretchbyafactorof90.Wemultiplythe y -coordinatesby90toobtain )]TJ/F15 10.9091 Tf 5 -8.836 Td [(10 ; 90 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ; 90, and )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 ; 90 e .Wealsomultiplythe y valueofthehorizontalasymptote y =0by90,andit remains y =0.Finally,weadd70toallofthe y -coordinates,whichshiftsthegraphupwardsto obtain )]TJ/F15 10.9091 Tf 5 -8.837 Td [(10 ; 90 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 +70 ; 103 : 11, ; 160,and )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 ; 90 e +70 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 ; 314 : 64.Adding 70tothehorizontalasymptoteshiftsitupwardsaswellto y =70.Weconnectthesethree pointsusingthesameshapeinthesamedirectionasinthegraphof f and,lastbutnotleast, werestrictthedomaintomatchtheapplieddomain[0 ; 1 .Theresultisbelow. 7 WewilldiscussthisingreaterdetailinSection6.5.

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320ExponentialandLogarithmicFunctions ; 1 H.A. y =0 t y )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 2 3 4 5 6 7 y = f t = e t )454()222()222()222()223()222()222()222()223()222()222()454(! H.A. y =70 t y 2468101214161820 20 40 60 80 100 120 140 160 180 y = T t 3.Fromthegraph,weseethatthehorizontalasymptoteis y =70.Itisworthamomentortwo ofourtimetoseehowthishappensanalyticallyandtoreviewsomeofthe`numbersense' developedinChapter4.As t !1 ,Weget T t =70+90 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t 70+90 e verybig )]TJ/F35 5.9776 Tf 5.756 0 Td [( .Since e> 1, e verybig )]TJ/F35 5.9776 Tf 5.756 0 Td [( = 1 e verybig+ 1 verybig+ verysmall+.Thelarger t becomes,thesmaller e )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 1 t becomes,sotheterm90 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 1 t verysmall+.Hence, T t 70+verysmall+ whichmeansthegraphisapproachingthehorizontalline y =70fromabove.Thismeans thatastimegoesby,thetemperatureofthecoeeiscoolingto70 F,presumablyroom temperature. Aswehavealreadyremarked,thegraphsof f x = b x allpasstheHorizontalLineTest.Thusthe exponentialfunctionsareinvertible.Wenowturnourattentiontotheseinverses,thelogarithmic functions,whicharecalled`logs'forshort. Definition 6.2 Theinverseoftheexponentialfunction f x = b x iscalledthe base b logarithm function ,andisdenoted f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x =log b x Theexpressionlog b x isread`logbase b of x .' Wehavespecialnotationsforthecommonbase, b =10,andthenaturalbase, b = e Definition 6.3 The commonlogarithm ofarealnumber x islog 10 x andisusuallywritten log x .The naturallogarithm ofarealnumber x islog e x andisusuallywrittenln x Sincelogsaredenedastheinversesofexponentialfunctions,wecanuseTheorems5.2and5.3to tellusaboutlogarithmicfunctions.Forexample,weknowthatthedomainofalogfunctionisthe rangeofanexponentialfunction,namely ; 1 ,andthattherangeofalogfunctionisthedomain ofanexponentialfunction,namely ; 1 .Sinceweknowthebasicshapesof y = f x = b x for thedierentcasesof b ,wecanobtainthegraphof y = f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x =log b x byreectingthegraphof f acrosstheline y = x asshownbelow.The y -intercept ; 1onthegraphof f correspondsto an x -interceptof ; 0onthegraphof f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 .Thehorizontalasymptotes y =0onthegraphsofthe exponentialfunctionsbecomeverticalasymptotes x =0ontheloggraphs.

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6.1IntroductiontoExponentialandLogarithmicFunctions321 y = b x b> 1 y =log b x b> 1 y = b x ,0 0 If b> 1: { f isalwaysincreasing { As x 0 + f x { As x !1 f x !1 { Thegraphof f resembles: y =log b x b> 1 If0
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322ExponentialandLogarithmicFunctions Aswehavementioned,Theorem6.2isaconsequenceofTheorems5.2and5.3.However,itisworth thereader'stimetounderstandTheorem6.2fromanexponentialperspective.Forinstance,we knowthatthedomainof g x =log 2 x is ; 1 .Why?Becausetherangeof f x =2 x is ; 1 Inaway,thissayseverything,butatthesametime,itdoesn't.Forexample,ifwetrytond log 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1,wearetryingtondtheexponentweputon2togiveus )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Inotherwords,weare lookingfor x thatsatises2 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Thereisnosuchrealnumber,sinceallpowersof2arepositive. Whilewhatwehavesaidisexactlythesamethingassaying`thedomainof g x =log 2 x is ; 1 becausetherangeof f x =2 x is ; 1 ',wefeelitisinastudent'sbestinteresttounderstand thestatementsinTheorem6.2atthislevelinsteadofjustmerelymemorizingthefacts. Example 6.1.3 Simplifythefollowing. 1.log 3 2.log 2 1 8 3.log p 5 4.ln 3 p e 2 5.log : 001 6.2 log 2 7.117 )]TJ/F34 7.9701 Tf 7.998 0 Td [(log 117 Solution. 1.Thenumberlog 3 istheexponentweputon3toget81.Assuch,wewanttowrite81as apowerof3.Wend81=3 4 ,sothatlog 3 =4. 2.Tondlog 2 1 8 ,weneedrewrite 1 8 asapowerof2.Wend 1 8 = 1 2 3 =2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 ,solog 2 1 8 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3. 3.Todeterminelog p 5 ,weneedtoexpress25asapowerof p 5.Weknow25=5 2 ,and 5= )]TJ/F54 10.9091 Tf 5 0.187 Td [(p 5 2 ,sowehave25= )]TJ/F54 10.9091 Tf 5 0.187 Td [(p 5 2 2 = )]TJ/F54 10.9091 Tf 5 0.187 Td [(p 5 4 .Wegetlog p 5 =4. 4.First,recallthatthenotationln 3 p e 2 meanslog e 3 p e 2 ,sowearelookingfortheexponent toputon e toobtain 3 p e 2 .Rewriting 3 p e 2 = e 2 = 3 ,wendln 3 p e 2 =ln )]TJ/F53 10.9091 Tf 5 -8.836 Td [(e 2 = 3 = 2 3 5.Rewritinglog : 001aslog 10 : 001,weseethatweneedtowrite0 : 001asapowerof10.We have0 : 001= 1 1000 = 1 10 3 =10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 .Hence,log : 001=log )]TJ/F15 10.9091 Tf 5 -8.836 Td [(10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3. 6.WecanuseTheorem6.2directlytosimplify2 log 2 =8.Wecanalsounderstandthisproblem byrstndinglog 2 .Bydenition,log 2 istheexponentweputon2toget8.Since 8=2 3 ,wehavelog 2 =3.Wenowsubstitutetond2 log 2 =2 3 =8. 7.WenotethatwecannotapplyTheorem6.2directlyto117 )]TJ/F34 7.9701 Tf 7.998 0 Td [(log 117 .Whynot?Weuse apropertyofexponentstorewrite117 )]TJ/F34 7.9701 Tf 7.998 0 Td [(log 117 as 1 117 log 117 .Atthispoint,wecanapply

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6.1IntroductiontoExponentialandLogarithmicFunctions323 Theorem6.2toget117 log 117 =6andthus117 )]TJ/F34 7.9701 Tf 7.998 0 Td [(log 117 = 1 117 log 117 = 1 6 .Itisworth amomentofyourtimetothinkyourwaythroughwhy117 log 117 =6.Bydenition, log 117 istheexponentweputon117toget6.Whatarewedoingwiththisexponent? Weareputtingiton117.Bydenitionweget6.Inotherwords,theexponentialfunction f x =117 x undoesthelogarithmicfunction g x =log 117 x Upuntilthispoint,restrictionsonthedomainsoffunctionscamefromavoidingdivisionbyzero andkeepingnegativenumbersfrombeneathevenradicals.Withtheintroductionoflogs,wenow haveanotherrestriction.Sincethedomainof f x =log b x is ; 1 ,theargument 8 ofthelog mustbestrictlypositive. Example 6.1.4 Findthedomainofthefollowingfunctions.Checkyouranswersgraphicallyusing thecalculator. 1. f x =2log )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2. g x =ln x x )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 Solution. 1.Weset3 )]TJ/F53 10.9091 Tf 8.741 0 Td [(x> 0toobtain x< 3,or ; 3.Thegraphfromthecalculatorbelowveriesthis. Notethatwecouldhavegraphed f usingtransformations.TakingacuefromTheorem1.7,we rewrite f x =2log 10 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +3 )]TJ/F15 10.9091 Tf 9.852 0 Td [(1andndthemainfunctioninvolvedis y = h x =log 10 x Weselectthreepointstotrack, )]TJ/F34 7.9701 Tf 8.312 -4.541 Td [(1 10 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0and ; 1,alongwiththeverticalasymptote x =0.Since f x =2 h )]TJ/F53 10.9091 Tf 8.485 0 Td [(x +3 )]TJ/F15 10.9091 Tf 11 0 Td [(1,Theorem1.7tellsusthattoobtainthedestinationsof thesepoints,werstsubtract3fromthe x -coordinatesshiftingthegraphleft3units,then dividemultiplybythe x -coordinatesby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1causingareectionacrossthe y -axis.These transformationsapplytotheverticalasymptote x =0aswell.Subtracting3givesus x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 asourasymptote,thenmultplyingby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1givesustheverticalasymptote x =3.Next,we multiplythe y -coordinatesby2whichresultsinaverticalstretchbyafactorof2,thenwe nishbysubtracting1fromthe y -coordinateswhichshiftsthegraphdown1unit.Weleave ittothereadertoperformtheindicatedarithmeticonthepointsthemselvesandtoverify thegraphproducedbythecalculatorbelow. 2.Tondthedomainof g ,weset x x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 > 0anduseasigndiagramtosolvethisinequality.We dene r x = x x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 nditsdomaintobe r is ; 1 [ ; 1 .Setting r x =0gives x =0. + 0 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 1 + 8 Seepage41ifyou'veforgottenwhatthistermmeans.

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324ExponentialandLogarithmicFunctions Wend x x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 > 0on ; 0 [ ; 1 togetthedomainof g .Thegraphof y = g x conrms this.Wecantellfromthegraphof g thatitisnottheresultofSection1.8transformations beingappliedtothegraph y =ln x ,sobarringamoredetailedanalysisusingCalculus,the calculatorgraphisthebestwecando.Onethingworthyofnote,however,istheendbehavior of g .Thegraphsuggeststhatas x g x 0.Wecanverifythisanalytically.Using resultsfromChapter4andcontinuity,weknowthatas x x x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1.Hence,itmakes sensethat g x =ln x x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ln=0. y = f x =2log )]TJ/F53 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 y = g x =ln x x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Whilelogarithmshavesomeinterestingapplicationsoftheirownwhichyou'llexploreintheexercises,theirprimaryusetouswillbetoundoexponentialfunctions.Thisis,afterall,howthey weredened.Ourlastexamplesolidiesthisandreviewsallofthematerialinthesection. Example 6.1.5 Let f x =2 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3. 1.Graph f usingtransformationsandstatethedomainandrangeof f 2.Explainwhy f isinvertibleandndaformulafor f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x 3.Graph f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 usingtransformationsandstatethedomainandrangeof f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 4.Verify )]TJ/F53 10.9091 Tf 5 -8.836 Td [(f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 f x = x forall x inthedomainof f and )]TJ/F53 10.9091 Tf 5 -8.836 Td [(f f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = x forall x inthe domainof f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 5.Graph f and f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 onthesamesetofaxesandcheckthesymmetryabouttheline y = x Solution. 1.Ifweidentify g x =2 x ,wesee f x = g x )]TJ/F15 10.9091 Tf 11.66 0 Td [(1 )]TJ/F15 10.9091 Tf 11.66 0 Td [(3.Wepickthepoints )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 2 ; 1 and ; 2onthegraphof g alongwiththehorizontalasymptote y =0totrackthrough thetransformations.ByTheorem1.7werstadd1tothe x -coordinatesofthepointson thegraphof g shifting g totheright1unittoget )]TJ/F15 10.9091 Tf 5 -8.836 Td [(0 ; 1 2 ; 1and ; 2.Thehorizontal asymptoteremains y =0.Next,wesubtract3fromthe y -coordinates,shiftingthegraph

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6.1IntroductiontoExponentialandLogarithmicFunctions325 down3units.Wegetthepoints )]TJ/F15 10.9091 Tf 5 -8.836 Td [(0 ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(5 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1withthehorizontalasymptote nowat y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.Connectingthedotsintheorderandmannerastheywereonthegraphof g ,wegetthegraphbelow.Weseethatthedomainof f isthesameas g ,namely ; 1 butthattherangeof f is )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1 x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 8 y = h x =2 x )454()222()222()222()223()222()222()222()223()222()222()454(! x y )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 8 y = f x =2 x )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F34 7.9701 Tf 8.469 0 Td [(3 2.Thegraphof f passestheHorizontalLineTestso f isone-to-one,henceinvertible.Tond aformulafor f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x ,wenormallyset y = f x ,interchangethe x and y ,thenproceedto solvefor y .Doingsointhissituationleadsustotheequation x =2 y )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 )]TJ/F15 10.9091 Tf 11.277 0 Td [(3.Wehaveyet todiscusshowtosolvethiskindofequation,sowewillattempttondtheformulafor f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 fromaproceduralperspective.Ifwebreak f x =2 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 )]TJ/F15 10.9091 Tf 11.086 0 Td [(3intoaseriesofsteps,wend f takesaninput x andappliesthesteps asubtract1 bputasanexponenton2 csubtract3 Clearly,toundosubtracting1,wewilladd1,andsimilarlyweundosubtracting3byadding 3.Howdoweundothesecondstep?Theanswerisweusethelogarithm.Bydenition, log 2 x undoesexponentiationby2.Hence, f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 should aadd3 btakethelogarithmbase2 cadd1 Insymbols, f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x =log 2 x +3+1. 3.Tograph f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x =log 2 x +3+1usingtransformations,westartwith j x =log 2 x .We trackthepoints )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0and ; 1onthegraphof j alongwiththeverticalasymptote x =0throughthetransformationsusingTheorem1.7.Since f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = j x +3+1,werst subtract3fromeachofthe x valuesincludingtheverticalasymptotetoobtain )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(5 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ,

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326ExponentialandLogarithmicFunctions )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 0and )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1withaverticalasymptote x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.Next,weadd1tothe y valuesonthe graphandget )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(5 2 ; 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1and )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 2.Ifyouareexperiencing dejavu ,thereisagood reasonforitbutweleaveittothereadertodeterminethesourceofthisuncannyfamiliarity. Weobtainthegraphbelow.Thedomainof f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 is )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1 ,whichmatchestherangeof f andtherangeof f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 is ; 1 ,whichmatchesthedomainof f x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345678 y = j x =log 2 x )454()222()222()222()223()222()222()222()223()222()222()454(! x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345678 y = f )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 x =log 2 x +3+1 4.Wenowverifythat f x =2 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 )]TJ/F15 10.9091 Tf 11.325 0 Td [(3and f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x =log 2 x +3+1satisfythecomposition requirementforinverses.Forallrealnumbers x )]TJ/F53 10.9091 Tf 5 -8.837 Td [(f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 f x = f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 f x = f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 =log 2 \002 2 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 +3 +1 =log 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 +1 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1Sincelog 2 u = u forallrealnumbers u = x X Forallrealnumbers x> )]TJ/F15 10.9091 Tf 8.485 0 Td [(3,wehave 9 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(f f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = f )]TJ/F53 10.9091 Tf 5 -8.836 Td [(f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = f log 2 x +3+1 =2 log 2 x +3+1 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 =2 log 2 x +3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 = x +3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3Since2 log 2 u = u forallrealnumbers u> 0 = x X 5.Last,butcertainlynotleast,wegraph y = f x and y = f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x onthesamesetofaxesand seethesymmetryabouttheline y = x 9 Payattention-canyouspotinwhichstepbelowweneed x> )]TJ/F63 8.9664 Tf 7.167 0 Td [(3?

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6.1IntroductiontoExponentialandLogarithmicFunctions327 x y y = f x =2 x )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F34 7.9701 Tf 8.468 0 Td [(3 y = f )]TJ/F73 5.9776 Tf 6.498 0 Td [(1 x =log 2 x +3+1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345678 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 8

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328ExponentialandLogarithmicFunctions 6.1.1Exercises 1.Evaluatetheexpression. alog 3 blog 2 clog 8 dlog 1 5 elog )]TJ/F34 7.9701 Tf 18.898 -4.541 Td [(1 1000000 flog 4 glog 6 hlog 13 )]TJ/F54 10.9091 Tf 5 0.187 Td [(p 13 i7 log 7 3 jln )]TJ/F53 10.9091 Tf 5 -8.837 Td [(e 5 klog 9 p 10 11 llog 5 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(3 log 3 5 2.Findthedomainofthefunction. a f x =ln x 2 +1 b f x =log 7 x +8 c f x =log x +2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 d f x =ln )]TJ/F53 10.9091 Tf 10.909 0 Td [(x +ln x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 e f x = 4 p log 4 x f f x =log 9 j x +3 j)]TJ/F15 10.9091 Tf 16.363 0 Td [(4 g f x =ln p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 h f x = 1 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(log 5 x i f x = p )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x log 1 2 x j f x =ln )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 +13 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 3.Foreachfunctiongivenbelow Finditsinversefromthe`proceduralperspective'discussedinExample6.1.5. Graphthefunctionanditsinverseonthesamesetofaxes. a f x =3 x +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 b f x =log 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 c f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F37 7.9701 Tf 6.587 0 Td [(x +1 d f x =5log x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 4.TheLogarithmicScalesTherearethreewidelyusedmeasurementscaleswhichinvolve commonlogarithms:theRichterscale,thedecibelscaleandthepHscale.Thecomputations involvedinallthreescalesarenearlyidenticalsopaycloseattentiontothesubtledierences. aEarthquakesarecomplicatedeventsanditisnotourintenttoprovideacompletediscussionofthescienceinvolvedinthem.Instead,werefertheinterestedreadertoa solidcourseinGeology 10 ortheU.S.GeologicalSurvey'sEarthquakeHazardsProgram foundhere andpresentonlyasimpliedversionoftheRichterscale .TheRichterscale measuresthemagnitudeofanearthquakebycomparingtheamplitudeoftheseismic 10 Rock-solid,perhaps?

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6.1IntroductiontoExponentialandLogarithmicFunctions329 wavesofthegivenearthquaketothoseofamagnitude0event",whichwaschosento beaseismographreadingof0 : 001millimetersrecordedonaseismometer100kilometers fromtheearthquake'sepicenter.Specically,themagnitudeofanearthquakeisgivenby M x =log x 0 : 001 where x istheseismographreadinginmillimetersoftheearthquake recorded100kilometersfromtheepicenter. i.Showthat M : 001=0. ii.Compute M ; 000. iii.Showthatanearthquakewhichregistered6.7ontheRichterscalehadaseismograph readingtentimeslargerthanonewhichmeasured5.7. iv.Findtwonewsstoriesaboutrecentearthquakeswhichgivetheirmagnitudesonthe Richterscale.Howmanytimeslargerwastheseismographreadingoftheearthquake withlargermagnitude? bWhilethedecibelscalecanbeusedinmanydisciplines, 11 weshallrestrictourattention toitsuseinacoustics,specicallyitsuseinmeasuringtheintensitylevelofsound. 12 TheSoundIntensityLevel L measuredindecibelsofasoundintensity I measuredin wattspersquaremeterisgivenby L I =10log I 10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(12 .LiketheRichterscale,this scalecompares I tobaseline:10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(12 W m 2 isthethresholdofhumanhearing. i.Compute L )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 ii.Damagetoyourhearingcanstartwithshorttermexposuretosoundlevelsaround 115decibels.Whatintensity I isneededtoproducethislevel? iii.Compute L .Howdoesthiscomparewiththethresholdofpainwhichisaround 140decibels? cThepHofasolutionisameasureofitsacidityoralkalinity.Specically,pH= )]TJ/F15 10.9091 Tf 10.303 0 Td [(log[H + ] where[H + ]isthehydrogenionconcentrationinmolesperliter.AsolutionwithapH lessthan7isanacid,onewithapHgreaterthan7isabasealkalineandapHof7is regardedasneutral. i.Thehydrogenionconcentrationofpurewateris[H + ]=10 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 .FinditspH. ii.FindthepHofasolutionwith[H + ]=6 : 3 10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(13 iii.ThepHofgastricacidtheacidinyourstomachisabout0 : 7.Whatisthecorrespondinghydrogenionconcentration? 5.Showthatlog b 1=0andlog b b =1forevery b> 0 ;b 6 =1. 6.CrazybonusquestionWithoutusingyourcalculator,determinewhichislarger: e or e 11 Seethiswebpage formoreinformation. 12 Asofthewritingofthisexercise,theWikipediapagegivenhere statesthatitmaynotmeetthegeneralnotability guideline"nordoesitciteanyreferencesorsources.Indthisoddbecauseitisthisveryusageofthedecibelscale whichshowsupineveryCollegeAlgebrabookIhaveread.Perhapsthoseotherbookshavebeenwrongallalong andwe'rejustblindlyfollowingtradition.

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330ExponentialandLogarithmicFunctions 6.1.2Answers 1.alog 3 27=3 blog 2 32=5 clog 8 4= 2 3 dlog 1 5 625= )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 elog 1 1000000 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 flog 4 8= 3 2 glog 6 1=0 hlog 13 p 13= 1 2 i7 log 7 3 =3 jln e 5 =5 klog 9 p 10 11 = 11 9 llog 5 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(3 log 3 5 =1 2.a ; 1 b )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1 c )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ ; 1 d ; 7 e[1 ; 1 f ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 [ ; 1 g ; 1 h ; 125 [ ; 1 iNodomain j ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 [ )]TJ/F34 7.9701 Tf 6.195 -4.542 Td [(1 2 ; 2 3.a f x =3 x +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x =log 3 x +4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x y y = f x =3 x +2 )]TJ/F34 7.9701 Tf 8.468 0 Td [(4 y = f )]TJ/F73 5.9776 Tf 6.498 0 Td [(1 x =log 3 x +4 )]TJ/F72 7.9701 Tf 9.763 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123456 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 b f x =log 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x =4 x +1 x y y = f x =log 4 x )]TJ/F34 7.9701 Tf 8.469 0 Td [(1 y = f )]TJ/F73 5.9776 Tf 6.498 0 Td [(1 x =4 x +1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123456 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 c f x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F37 7.9701 Tf 6.587 0 Td [(x +1 f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x = )]TJ/F15 10.9091 Tf 10.303 0 Td [(log 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x x y y = f x = )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F38 5.9776 Tf 5.756 0 Td [(x +1 y = f )]TJ/F73 5.9776 Tf 6.498 0 Td [(1 x = )]TJ/F72 7.9701 Tf 9.221 0 Td [(log 2 )]TJ/F75 7.9701 Tf 9.764 0 Td [(x )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 d f x =5log x )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x =10 x +2 5 x y y = f x =5log x )]TJ/F34 7.9701 Tf 8.468 0 Td [(2 y = f )]TJ/F73 5.9776 Tf 6.498 0 Td [(1 x =10 x +2 5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(112345 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5

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6.1IntroductiontoExponentialandLogarithmicFunctions331 4.ai. M : 001=log 0 : 001 0 : 001 =log=0. ii. M ; 000=log 80 ; 000 0 : 001 =log ; 000 ; 000 7 : 9. bi. L )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 =60decibels. ii. I =10 )]TJ/F37 7.9701 Tf 6.586 0 Td [(: 5 0 : 316wattspersquaremeter. iii.Since L =120decibelsand L =140decibels,asoundwithintensitylevel 140decibelshasanintensity100timesgreaterthanasoundwithintensitylevel120 decibels. ci.ThepHofpurewateris7. ii.If[H + ]=6 : 3 10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(13 thenthesolutionhasapHof12.2. iii.[H + ]=10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 7 : 1995molesperliter.

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332ExponentialandLogarithmicFunctions 6.2PropertiesofLogarithms InSection6.1,weintroducedthelogarithmicfunctionsasinversesofexponentialfunctionsand discussedafewoftheirfunctionalpropertiesfromthatperspective.Inthissection,weexplore thealgebraicpropertiesoflogarithms.Historically,thesehaveplayedahugeruleinthescientic developmentofoursocietysince,amongotherthings,theywereusedtodevelopanalogcomputing devicescalledsliderules whichenabledscientistsandengineerstoperformaccuratecalculations leadingtosuchthingsasspacetravelandthemoonlanding .Asweshallseeshortly,logsinherit analogsofallofthepropertiesofexponentsyoulearnedinElementaryandIntermediateAlgebra. WerstextracttwopropertiesfromTheorem6.2toremindusofthedenitionofalogarithmas theinverseofanexponentialfunction. Theorem 6.3 InversePropertiesofExponentialandLogFunctions Let b> 0, b 6 =1. b a = c ifandonlyiflog b c = a log b b x = x forall x and b log b x = x forall x> 0 Next,wespelloutinmoredetailwhatitmeansforexponentialandlogarithmicfunctionstobe one-to-one. Theorem 6.4 One-to-onePropertiesofExponentialandLogFunctions Let f x = b x and g x =log b x where b> 0, b 6 =1.Then f and g areone-to-one.Inotherwords: b u = b w ifandonlyif u = w forallrealnumbers u and w log b u =log b w ifandonlyif u = w forallrealnumbers u> 0, w> 0. Wenowstatethealgebraicpropertiesofexponentialfunctionswhichwillserveasabasisforthe propertiesoflogarithms.Whilethesepropertiesmaylookidenticaltotheonesyoulearnedin ElementaryandIntermediateAlgebra,theyapplytorealnumberexponents,notjustrational exponents.Notethatinthetheoremthatfollows,weareinterestedinthepropertiesofexponential functions,sothebase b isrestrictedto b> 0, b 6 =1.Anaddedbenetofthisrestrictionisthatit eliminatesthepathologiesdiscussedinSection5.3when,forexample,wesimplied )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 = 3 3 = 2 and obtained j x j insteadofwhatwehadexpectedfromthearithmeticintheexponents, x 1 = x Theorem 6.5 AlgebraicPropertiesofExponentialFunctions Let f x = b x bean exponentialfunction b> 0, b 6 =1andlet u and w berealnumbers. ProductRule: f u + w = f u f w .Inotherwords, b u + w = b u b w QuotientRule: f u )]TJ/F53 10.9091 Tf 10.909 0 Td [(w = f u f w .Inotherwords, b u )]TJ/F37 7.9701 Tf 6.586 0 Td [(w = b u b w PowerRule: f u w = f uw .Inotherwords, b u w = b uw WhilethepropertieslistedinTheorem6.5arecertainlybelievablebasedonsimilarpropertiesof integerandrationalexponents,thefullproofsrequireCalculus.Toeachofthesepropertiesof

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6.2PropertiesofLogarithms333 exponentialfunctionscorrespondsananalogouspropertyoflogarithmicfunctions.Welistthese belowinournexttheorem. Theorem 6.6 AlgebraicPropertiesofLogarithmFunctions Let g x =log b x bea logarithmicfunction b> 0, b 6 =1andlet u> 0and w> 0berealnumbers. ProductRule: g uw = g u + g w .Inotherwords,log b uw =log b u +log b w QuotientRule: g u w = g u )]TJ/F53 10.9091 Tf 10.91 0 Td [(g w .Inotherwords,log b u w =log b u )]TJ/F15 10.9091 Tf 10.909 0 Td [(log b w PowerRule: g u w = wg u .Inotherwords,log b u w = w log b u ThereareacoupleofdierentwaystounderstandwhyTheorem6.6istrue.Considertheproduct rule:log b uw =log b u +log b w .Let a =log b uw c =log b u ,and d =log b w .Then,by denition, b a = uw b c = u and b d = w .Hence, b a = uw = b c b d = b c + d ,sothat b a = b c + d .By theone-to-onepropertyof b x ,wehave a = c + d .Inotherwords,log b uw =log b u +log b w Theremainingpropertiesareprovedsimilarly.Fromapurelyfunctionalapproach,wecansee thepropertiesinTheorem6.6asanexampleofhowinversefunctionsinterchangetherolesof inputsinoutputs.Forinstance,theProductRuleforexponentialfunctionsgiveninTheorem6.5, f u + w = f u f w ,saysthataddinginputsresultsinmultiplyingoutputs.Hence,whatever f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 is,itmusttaketheproductsofoutputsfrom f andreturnthemtothesumoftheirrespectiveinputs. Sincetheoutputsfrom f aretheinputsto f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 andvice-versa,wehavethatthat f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 musttake productsofitsinputstothesumoftheirrespectiveoutputs.ThisispreciselywhattheProductRule forLogarithmicfunctionsstatesinTheorem6.6: g uw = g u + g w .Thereaderisencouragedto viewtheremainingpropertieslistedinTheorem6.6similarly.Thefollowingexampleshelpbuild familiaritywiththeseproperties.Inourrstexample,weareaskedto`expand'thelogarithms. ThismeansthatwereadthepropertiesinTheorem6.6fromlefttorightandrewriteproducts insidethelogassumsoutsidethelog,quotientsinsidethelogasdierencesoutsidethelog,and powersinsidethelogasfactorsoutsidethelog.Whileitistheoppositeprocess,whichwewill practicelater,thatismostusefulinAlgebra,theutilityofexpandinglogarithmsbecomesapparent inCalculus. Example 6.2.1 Expandthefollowingusingthepropertiesoflogarithmsandsimplify.Assume whennecessarythatallquantitiesrepresentpositiverealnumbers. 1.log 2 8 x 2.log 0 : 1 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(10 x 2 3.ln 3 ex 2 4.log 3 s 100 x 2 yz 5 5.log 117 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 Solution. 1.Toexpandlog 2 )]TJ/F34 7.9701 Tf 6.461 -4.541 Td [(8 x ,weusetheQuotientRuleidentifying u =8and w = x andsimplify.

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334ExponentialandLogarithmicFunctions log 2 8 x =log 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(log 2 x QuotientRule =3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(log 2 x Since2 3 =8 = )]TJ/F15 10.9091 Tf 10.303 0 Td [(log 2 x +3 2.Intheexpressionlog 0 : 1 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(10 x 2 ,wehaveapowerthe x 2 andaproduct.Inordertousethe ProductRule,the entire quantityinsidethelogarithmmustberaisedtothesameexponent. Sincetheexponent2appliesonlytothe x ,werstapplytheProductRulewith u =10and w = x 2 .Oncewegetthe x 2 byitselfinsidethelog,wemayapplythePowerRulewith u = x and w =2andsimplify. log 0 : 1 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(10 x 2 =log 0 : 1 +log 0 : 1 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 ProductRule =log 0 : 1 +2log 0 : 1 x PowerRule = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+2log 0 : 1 x Since : 1 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 =10 =2log 0 : 1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 3.Wehaveapower,quotientandproductoccurringinln )]TJ/F34 7.9701 Tf 8.426 -4.541 Td [(3 ex 2 .Sincetheexponent2applies totheentirequantityinsidethelogarithm,webeginwiththePowerRulewith u = 3 ex and w =2.Next,weseetheQuotientRuleisapplicable,with u =3and w = ex ,sowereplace ln )]TJ/F34 7.9701 Tf 8.426 -4.541 Td [(3 ex withthequantityln )]TJ/F15 10.9091 Tf 11.525 0 Td [(ln ex .Sinceln )]TJ/F34 7.9701 Tf 8.426 -4.541 Td [(3 ex isbeingmultipliedby2,theentire quantityln )]TJ/F15 10.9091 Tf 10.091 0 Td [(ln ex ismultipliedby2.Finally,weapplytheProductRulewith u = e and w = x ,andreplaceln ex withthequantityln e +ln x ,andsimplify,keepinginmindthat thenaturallogislogbase e ln 3 ex 2 =2ln 3 ex PowerRule =2[ln )]TJ/F15 10.9091 Tf 10.909 0 Td [(ln ex ]QuotientRule =2ln )]TJ/F15 10.9091 Tf 10.909 0 Td [(2ln ex =2ln )]TJ/F15 10.9091 Tf 10.909 0 Td [(2[ln e +ln x ]ProductRule =2ln )]TJ/F15 10.9091 Tf 10.909 0 Td [(2ln e )]TJ/F15 10.9091 Tf 10.909 0 Td [(2ln x =2ln )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2ln x Since e 1 = e = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2ln x +2ln )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 4.InTheorem6.6,thereisnomentionofhowtodealwithradicals.However,thinkingbackto Denition5.5,wecanrewritethecuberootasa 1 3 exponent.WebeginbyusingthePower

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6.2PropertiesofLogarithms335 Rule 1 ,andwekeepinmindthatthecommonlogislogbase10. log 3 s 100 x 2 yz 5 =log 100 x 2 yz 5 1 = 3 = 1 3 log 100 x 2 yz 5 PowerRule = 1 3 log )]TJ/F15 10.9091 Tf 5 -8.837 Td [(100 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(log )]TJ/F53 10.9091 Tf 5 -8.837 Td [(yz 5 QuotientRule = 1 3 log )]TJ/F15 10.9091 Tf 5 -8.837 Td [(100 x 2 )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(1 3 log )]TJ/F53 10.9091 Tf 5 -8.837 Td [(yz 5 = 1 3 log+log )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 3 log y +log )]TJ/F53 10.9091 Tf 5 -8.836 Td [(z 5 ProductRule = 1 3 log+ 1 3 log )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 3 log y )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 3 log )]TJ/F53 10.9091 Tf 5 -8.836 Td [(z 5 = 1 3 log+ 2 3 log x )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 3 log y )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(5 3 log z PowerRule = 2 3 + 2 3 log x )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(1 3 log y )]TJ/F34 7.9701 Tf 12.104 4.296 Td [(5 3 log z Since10 2 =100 = 2 3 log x )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 3 log y )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(5 3 log z + 2 3 5.Atrstitseemsasifwehavenomeansofsimplifyinglog 117 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(4 ,sincenoneofthe propertiesoflogsaddressestheissueofexpandingadierence inside thelogarithm.However, wemayfactor x 2 )]TJ/F15 10.9091 Tf 11.064 0 Td [(4= x +2 x )]TJ/F15 10.9091 Tf 11.064 0 Td [(2therebyintroducingaproductwhichgivesuslicense tousetheProductRule. log 117 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 =log 117 [ x +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2]Factor =log 117 x +2+log 117 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2ProductRule AcoupleofremarksaboutExample6.2.1areinorder.First,whilenotexplicitlystatedintheabove example,ageneralruleofthumbtodeterminewhichlogpropertytoapplyrsttoacomplicated problemis`reverseorderofoperations.'Forexample,ifweweretosubstituteanumberfor x into theexpressionlog 0 : 1 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(10 x 2 ,wewouldrstsquarethe x ,thenmultiplyby10.Thelaststepisthe multiplication,whichtellsustherstlogpropertytoapplyistheProductRule.Inamulti-step problem,thisrulecangivetherequiredguidanceonwhichlogpropertytoapplyateachstep. ThereaderisencouragedtolookthroughthesolutionstoExample6.2.1toseethisruleinaction. Second,whilewewereinstructedtoassumewhennecessarythatallquantitiesrepresentedpositive realnumbers,theauthorswouldbecommittingasinofomissionifwefailedtopointoutthat,for instance,thefunctions f x =log 117 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 and g x =log 117 x +2+log 117 x )]TJ/F15 10.9091 Tf 9.757 0 Td [(2havedierent domains,and,hence,aredierentfunctions.Weleaveittothereadertoverifythedomainof f is ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 [ ; 1 whereasthedomainof g is ; 1 .Ingeneral,whenusinglogpropertiesto 1 Atthispointinthetext,thereaderisencouragedtocarefullyreadthrougheachstepandthinkofwhichquantity isplayingtheroleof u andwhichisplayingtheroleof w asweapplyeachproperty.

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336ExponentialandLogarithmicFunctions expandalogarithm,wemayverywellberestrictingthedomainaswedoso.Onelastcomment beforewemovetoreassemblinglogsfromtheirvariousbitsandpieces.Theauthorsarewellaware ofthepropensityforsomestudentstobecomeoverexcitedandinventtheirownpropertiesoflogs likelog 117 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 =log 117 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 )]TJ/F15 10.9091 Tf 11.265 0 Td [(log 117 ,whichsimplyisn'ttrue,ingeneral.Theunwritten 2 propertyoflogarithmsisthatifitisn'twritteninatextbook,itprobablyisn'ttrue. Example 6.2.2 Usethepropertiesoflogarithmstowritethefollowingasasinglelogarithm. 1.log 3 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(log 3 x +1 2.log x +2log y )]TJ/F15 10.9091 Tf 10.909 0 Td [(log z 3.4log 2 x +3 4. )]TJ/F15 10.9091 Tf 10.303 0 Td [(ln x )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 2 Solution. WhereasinExample6.2.1wereadthepropertiesinTheorem6.6fromlefttorightto expandlogarithms,inthisexamplewereadthemfromrighttoleft. 1.ThedierenceoflogarithmsrequirestheQuotientRule:log 3 x )]TJ/F15 10.9091 Tf 8.897 0 Td [(1 )]TJ/F15 10.9091 Tf 8.897 0 Td [(log 3 x +1=log 3 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x +1 2.Intheexpression,log x +2log y )]TJ/F15 10.9091 Tf 10.073 0 Td [(log z ,wehavebothasumanddierenceoflogarithms. However,beforeweusetheproductruletocombinelog x +2log y ,wenotethatweneed tosomehowdealwiththecoecient2onlog y .ThiscanbehandledusingthePowerRule. WecanthenapplytheProductandQuotientRulesaswemovefromlefttoright.Puttingit alltogether,wehave log x +2log y )]TJ/F15 10.9091 Tf 10.909 0 Td [(log z =log x +log )]TJ/F53 10.9091 Tf 5 -8.836 Td [(y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(log z PowerRule =log )]TJ/F53 10.9091 Tf 5 -8.836 Td [(xy 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(log z ProductRule =log xy 2 z QuotientRule 3.Wecancertainlygetstartedrewriting4log 2 x +3byapplyingthePowerRuleto4log 2 x toobtainlog 2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 4 ,butinordertousetheProductRuletohandletheaddition,weneedto rewrite3asalogarithmbase2.FromTheorem6.3,weknow3=log 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 3 ,soweget 4log 2 x +3=log 2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 4 +3PowerRule =log 2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 4 +log 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 3 Since3=log 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 3 =log 2 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 4 +log 2 =log 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(8 x 4 ProductRule 2 Theauthorsrelishtheironyinvolvedinwritingwhatfollows.

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6.2PropertiesofLogarithms337 4.Togetstartedwith )]TJ/F15 10.9091 Tf 10.303 0 Td [(ln x )]TJ/F34 7.9701 Tf 11.379 4.295 Td [(1 2 ,werewrite )]TJ/F15 10.9091 Tf 10.303 0 Td [(ln x as )]TJ/F15 10.9091 Tf 8.485 0 Td [(1ln x .WecanthenusethePower Ruletoobtain )]TJ/F15 10.9091 Tf 8.485 0 Td [(1ln x =ln )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 .InordertousetheQuotientRule,weneedtowrite 1 2 asanaturallogarithm.Theorem6.3givesus 1 2 =ln )]TJ/F53 10.9091 Tf 5 -8.836 Td [(e 1 = 2 =ln p e .Wehave )]TJ/F15 10.9091 Tf 10.303 0 Td [(ln x )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 2 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1ln x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 =ln )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 PowerRule =ln )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(ln )]TJ/F53 10.9091 Tf 5 -8.837 Td [(e 1 = 2 Since 1 2 =ln )]TJ/F53 10.9091 Tf 5 -8.837 Td [(e 1 = 2 =ln )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(ln p e =ln x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 p e QuotientRule =ln 1 x p e Aswewouldexpect,theruleofthumbforre-assemblinglogarithmsistheoppositeofwhatit wasfordismantlingthem.Thatis,ifweareinterestedinrewritinganexpressionasasingle logarithm,weapplylogpropertiesfollowingtheusualorderofoperations:dealwithmultiplesof logsrstwiththePowerRule,thendealwithadditionandsubtractionusingtheProductand QuotientRules,respectively.Additionally,wendthatusinglogpropertiesinthisfashioncan increasethedomainoftheexpression.Forexample,weleaveittothereadertoverifythedomain of f x =log 3 x )]TJ/F15 10.9091 Tf 8.696 0 Td [(1 )]TJ/F15 10.9091 Tf 8.696 0 Td [(log 3 x +1is ; 1 butthedomainof g x =log 3 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x +1 is ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ ; 1 WewillneedtokeepthisinmindwhenwesolveequationsinvolvinglogarithmsinSection6.4-it ispreciselyforthisreasonwewillhavetocheckforextraneoussolutions. Thetwologarithmbuttonscommonlyfoundoncalculatorsarethe`LOG'and`LN'buttonswhich correspondtothecommonandnaturallogs,respectively.Supposewewantedanapproximationto log 2 .Theanswershouldbealittlelessthan3,Canyouexplainwhy?buthowdowecoerce thecalculatorintotellingusamoreaccurateanswer?Weneedthefollowingtheorem. Theorem 6.7 ChangeofBase Let a;b> 0, a;b 6 =1. a x = b x log b a forallrealnumbers x log a x = log b x log b a forallrealnumbers x> 0. TheproofsoftheChangeofBaseformulasarearesultoftheotherpropertiesstudiedinthis section.Ifwestartwith b x log b a andusethePowerRuleintheexponenttorewrite x log b a as log b a x andthenapplyoneoftheInversePropertiesinTheorem6.3,weget b x log b a = b log b a x = a x ;

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338ExponentialandLogarithmicFunctions asrequired.Toverifythelogarithmicformoftheproperty,wealsousethePowerRuleandan InverseProperty.Wenotethat log a x log b a =log b a log a x =log b x ; andwegettheresultbydividingthroughbylog b a .Ofcourse,theauthorscan'thelpbutpoint outtheinverserelationshipbetweenthesetwochangeofbaseformulas.Tochangethebaseof anexponentialexpression,we multiply the input bythefactorlog b a .Tochangethebaseofa logarithmicexpression,we divide the output bythefactorlog b a .While,inthegrandscheme ofthings,bothchangeofbaseformulasarereallysayingthesamething,thelogarithmicformis theoneusuallyencounteredinAlgebrawhiletheexponentialformisn'tusuallyintroduceduntil Calculus. 3 WhatTheorem6.7reallytellsusisthatallexponentialandlogarithmicfunctionsare justscalingsofoneanother.Notonlydoesthisexplainwhytheirgraphshavesimilarshapes,but italsotellsusthatwecoulddoallofmathematicswithasinglebase-beit10, e ,42,or117.Your Calculusteacherwillhavemoretosayaboutthiswhenthetimecomes. Example 6.2.3 Useanappropriatechangeofbaseformulatoconvertthefollowingexpressionsto oneswiththeindicatedbase.Verifyyouranswersusingacalculator,asappropriate. 1.3 2 tobase10 2.2 x tobase e 3.log 4 tobase e 4.ln x tobase10 Solution. 1.WeapplytheChangeofBaseformulawith a =3and b =10toobtain3 2 =10 2log .Typing thelatterinthecalculatorproducesananswerof9asrequired. 2.Here, a =2and b = e sowehave2 x = e x ln .Toverifythisonourcalculator,wecangraph f x =2 x and g x = e x ln .Theirgraphsareindistinguishablewhichprovidesevidence thattheyarethesamefunction. y = f x =2 x and y = g x = e x ln 3 Theauthorsfeelsostronglyaboutshowingstudentsthateverypropertyoflogarithmscomesfromandcorresponds toapropertyofexponentsthatwehavebrokentraditionwiththevastmajorityofotherauthorsinthiseld.This isn'tthersttimethishappened,anditcertainlywon'tbethelast.

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6.2PropertiesofLogarithms339 3.Applyingthechangeofbasewith a =4and b = e leadsustowritelog 4 = ln ln .Evaluating thisinthecalculatorgives ln ln 1 : 16.Howdowecheckthisreallyisthevalueoflog 4 ? Bydenition,log 4 istheexponentweputon4toget5.Thecalculatorconrmsthis. 4 4.Wewriteln x =log e x = log x log e .Wegraphboth f x =ln x and g x = log x log e andnd bothgraphsappeartobeidentical. y = f x =ln x and y = g x = log x log e 4 Whichmeansifitislyingtousabouttherstansweritgaveus,atleastitisbeingconsistent.

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340ExponentialandLogarithmicFunctions 6.2.1Exercises 1.Expandthefollowingusingthepropertiesoflogarithmsandsimplify.Assumewhennecessary thatallquantitiesrepresentpositiverealnumbers. aln x 3 y 2 blog 2 128 x 2 +4 clog 5 z 25 3 dlog : 23 10 37 eln p z xy flog 5 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(25 glog p 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(4 x 3 hlog 1 3 x y 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 2.Usethepropertiesoflogarithmstowritethefollowingasasinglelogarithm. a4ln x +2ln y b3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(log x clog 2 x +log 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(log 2 z dlog 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2log 3 y e 1 2 log 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2log 3 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(log 3 z flog 2 x +log 1 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 glog 2 x +log 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 hlog 7 x +log 7 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 3.Useanappropriatechangeofbaseformulatoconvertthefollowingexpressionstooneswith theindicatedbase. a7 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 tobase e blog 3 x +2tobase10 c 2 3 x tobase e dlog x 2 +1tobase e 4.Usetheappropriatechangeofbaseformulatoapproximatethefollowinglogarithms. alog 3 blog 5 clog 4 1 10 dlog 3 5 5.Compareandcontrastthegraphsof y =ln x 2 and y =2ln x 6.ProvetheQuotientRuleandPowerRuleforLogarithms. 7.Givenumericalexamplestoshowthat,ingeneral, alog b x + y 6 =log b x +log b y blog b x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 6 =log b x )]TJ/F15 10.9091 Tf 10.909 0 Td [(log b y clog b x y 6 = log b x log b y

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6.2PropertiesofLogarithms341 8.TheHenderson-HasselbalchEquation:Suppose HA representsaweakacid.Thenwehavea reversiblechemicalreaction HA H + + A )]TJ/F53 10.9091 Tf 7.085 -4.505 Td [(: Theaciddisassociationconstant, K a ,isgivenby K = [ H + ][ A )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(] [ HA ] =[ H + ] [ A )]TJ/F15 10.9091 Tf 7.085 -3.959 Td [(] [ HA ] ; wherethesquarebracketsdenotetheconcentrationsjustastheydidinExercise4cinSection 6.1.Thesymbolp K a isdenedsimilarlytopHinthatp K a = )]TJ/F15 10.9091 Tf 10.303 0 Td [(log K a .Usingthedenition ofpHfromExercise4candthepropertiesoflogarithms,derivetheHenderson-Hasselbalch Equationwhichstates pH=p K a +log [ A )]TJ/F15 10.9091 Tf 7.085 -3.958 Td [(] [ HA ] 9.Researchthehistoryoflogarithmsincludingtheoriginoftheword`logarithm'itself.Whyis theabbreviationofnaturallog`ln'andnot`nl'? 10.Thereisasceneinthemovie`Apollo13'inwhichseveralpeopleatMissionControluseslide rulestoverifyacomputation.Wasthatsceneaccurate?Lookforotherpopculturereferences tologarithmsandsliderules.

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342ExponentialandLogarithmicFunctions 6.2.2Answers 1.a3ln x +2ln y b7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(log 2 x 2 +4 c3log 5 z )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 dlog : 23+37 e 1 2 ln z )]TJ/F15 10.9091 Tf 10.909 0 Td [(ln x )]TJ/F15 10.9091 Tf 10.909 0 Td [(ln y flog 5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(5+log 5 x +5 g3log p 2 x +8 h )]TJ/F34 7.9701 Tf 6.586 0 Td [(2+log 1 3 x +log 1 3 y )]TJ/F34 7.9701 Tf 8.469 0 Td [(2+log 1 3 y 2 +2 y +4 2.aln x 4 y 2 blog 1000 x clog 2 xy z dlog 3 x y 2 elog 3 p x y 2 z flog 2 x x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 glog 2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 hlog 7 x x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 49 3.a7 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 = e x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1ln blog 3 x +2= log x +2 log c 2 3 x = e x ln 2 3 dlog x 2 +1= ln x 2 +1 ln 4.alog 3 2 : 26186 blog 5 2 : 72271 clog 4 1 10 )]TJ/F15 10.9091 Tf 20 0 Td [(1 : 66096 dlog 3 5 )]TJ/F15 10.9091 Tf 20 0 Td [(13 : 52273

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6.3ExponentialEquationsandInequalities343 6.3ExponentialEquationsandInequalities Inthissectionwewilldeveloptechniquesforsolvingequationsinvolvingexponentialfunctions. Suppose,forinstance,wewantedtosolvetheequation2 x =128.Afteramoment'scalculation,we nd128=2 7 ,sowehave2 x =2 7 .Theone-to-onepropertyofexponentialfunctions,detailedin Theorem6.4,tellsusthat2 x =2 7 ifandonlyif x =7.Thismeansthatnotonlyis x =7asolution to2 x =2 7 ,itisthe only solution.Nowsupposewechangetheproblemeversoslightlyto2 x =129. WecoulduseoneoftheinversepropertiesofexponentialsandlogarithmslistedinTheorem6.3to write129=2 log 2 .We'dthenhave2 x =2 log 2 ,whichmeansoursolutionis x =log 2 Thismakessensebecause,afterall,thedenitionoflog 2 is`theexponentweputon2toget 129.'Indeedwecouldhaveobtainedthissolutiondirectlybyrewritingtheequation2 x =129in itslogarithmicformlog 2 = x .Eitherway,inordertogetareasonabledecimalapproximation tothisnumber,we'dusethechangeofbaseformula,Theorem6.7,togiveussomethingmore calculatorfriendly, 1 saylog 2 = ln ln .Anotherwaytoarriveatthisanswerisasfollows 2 x =129 ln x =lnTakethenaturallogofbothsides. x ln=lnPowerRule x = ln ln `Takingthenaturallog'ofbothsidesisakintosquaringbothsides:since f x =ln x isa function aslongastwoquantitiesareequal,theirnaturallogsareequal. 2 Alsonotethatwetreatlnas anyothernon-zerorealnumberanddivideitthrough 3 toisolatethevariable x .Wesummarize belowthetwocommonwaystosolveexponentialequations,motivatedbyourexamples. StepsforSolvinganEquationinvolvingExponentialFunctions 1.Isolatetheexponentialfunction. 2.aIfconvenient,expressbothsideswithacommonbaseandequatetheexponents. bOtherwise,takethenaturallogofbothsidesoftheequationandusethePowerRule. Example 6.3.1 Solvethefollowingequations.Checkyouranswergraphicallyusingacalculator. 1.2 3 x =16 1 )]TJ/F37 7.9701 Tf 6.587 0 Td [(x 2.2000=1000 3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t 3.9 3 x =7 2 x 4.75= 100 1+3 e )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 t 5.25 x =5 x +6 6. e x )]TJ/F37 7.9701 Tf 6.587 0 Td [(e )]TJ/F38 5.9776 Tf 5.757 0 Td [(x 2 =5 Solution. 1 Youcanusenaturallogsorcommonlogs.Wechoosenaturallogs.InCalculus,you'lllearnthesearethemost `mathy'ofthelogarithms. 2 Thisisalsothe`if'partofthestatementlog b u =log b w ifandonlyif u = w inTheorem6.4. 3 Pleaseresistthetemptationtodividebothsidesby`ln'insteadofln.Justlikeitwouldn'tmakesenseto dividebothsidesbythesquarerootsymbol` p 'whensolving x p 2=5,itmakesnosensetodivideby`ln'.

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344ExponentialandLogarithmicFunctions 1.Since16isapowerof2,wecanrewrite2 3 x =16 1 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x as2 3 x = )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 4 1 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x .Usingpropertiesof exponents,weget2 3 x =2 4 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x .Usingtheone-to-onepropertyofexponentialfunctions,we get3 x =4 )]TJ/F53 10.9091 Tf 9.098 0 Td [(x whichgives x = 4 7 .Tocheckgraphically,weset f x =2 3 x and g x =16 1 )]TJ/F37 7.9701 Tf 6.587 0 Td [(x andseethattheyintersectat x = 4 7 0 : 5714. 2.Webeginsolving2000=1000 3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t bydividingbothsidesby2000toisolatetheexponential whichyields3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t =2.Sinceitisinconvenienttowrite2asapowerof3,weusethenatural logtogetln )]TJ/F15 10.9091 Tf 5 -8.836 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t =ln.UsingthePowerRule,weget )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 1 t ln=ln,sowe dividebothsidesby )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 1lntoget t = )]TJ/F34 7.9701 Tf 15.797 5.375 Td [(ln 0 : 1ln = )]TJ/F34 7.9701 Tf 9.681 5.375 Td [(10ln ln .Onthecalculator,wegraph f x =2000and g x =1000 3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 x andndthattheyintersectat x = )]TJ/F34 7.9701 Tf 9.68 5.374 Td [(10ln ln )]TJ/F15 10.9091 Tf 20 0 Td [(6 : 3093. y = f x =2 3 x and y = f x =2000and y = g x =16 1 )]TJ/F75 7.9701 Tf 7.594 0 Td [(x y = g x =1000 3 )]TJ/F72 7.9701 Tf 7.594 0 Td [(0 : 1 x 3.Werstnotethatwecanrewritetheequation9 3 x =7 2 x as3 2 3 x =7 2 x toobtain3 x +2 =7 2 x Sinceitisnotconvenienttoexpressbothsidesasapowerof3or7forthatmatterweuse thenaturallog:ln )]TJ/F15 10.9091 Tf 5 -8.837 Td [(3 x +2 =ln )]TJ/F15 10.9091 Tf 5 -8.837 Td [(7 2 x .Thepowerrulegives x +2ln=2 x ln.Even thoughthisequationappearsverycomplicated,keepinmindthatlnandlnarejust constants.Theequation x +2ln=2 x lnisactuallyalinearequationandassuchwe gatherallofthetermswith x ononeside,andtheconstantsontheother.Wethendivide bothsidesbythecoecientof x ,whichweobtainbyfactoring. x +2ln=2 x ln x ln+2ln=2 x ln 2ln=2 x ln )]TJ/F53 10.9091 Tf 10.909 0 Td [(x ln 2ln= x ln )]TJ/F15 10.9091 Tf 10.909 0 Td [(lnFactor. x = 2ln 2ln )]TJ/F34 7.9701 Tf 6.586 0 Td [(ln Graphing f x =9 3 x and g x =7 2 x onthecalculator,weseethatthesetwographsintersect at x = 2ln 2ln )]TJ/F34 7.9701 Tf 6.587 0 Td [(ln 0 : 7866. 4.Ourobjectiveinsolving75= 100 1+3 e )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 t istorstisolatetheexponential.Tothatend,we cleardenominatorsandget75 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+3 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 t =100.Fromthisweget75+225 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 t =100, whichleadsto225 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 t =25,andnally, e )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 t = 1 9 .Takingthenaturallogofbothsides

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6.3ExponentialEquationsandInequalities345 givesln )]TJ/F53 10.9091 Tf 5 -8.836 Td [(e )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 t =ln )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 9 .Sincenaturallogislogbase e ,ln )]TJ/F53 10.9091 Tf 5 -8.836 Td [(e )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 t .Wecanalsouse thePowerRuletowriteln )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 9 = )]TJ/F15 10.9091 Tf 10.303 0 Td [(ln.Puttingthesetwostepstogether,wesimplify ln )]TJ/F53 10.9091 Tf 5 -8.837 Td [(e )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 t =ln )]TJ/F34 7.9701 Tf 6.196 -4.542 Td [(1 9 to )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 t = )]TJ/F15 10.9091 Tf 10.303 0 Td [(ln.Wearriveatoursolution, t = ln 2 whichsimpliesto t =ln.Canyouexplainwhy?Thecalculatorconrmsthegraphsof f x =75and g x = 100 1+3 e )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 x intersectat x =ln 1 : 099. y = f x =9 3 x and y = f x =75and y = g x =7 2 x y = g x = 100 1+3 e )]TJ/F73 5.9776 Tf 6.498 0 Td [(2 x 5.Westartsolving25 x =5 x +6byrewriting25=5 2 sothatwehave )]TJ/F15 10.9091 Tf 5 -8.836 Td [(5 2 x =5 x +6,or 5 2 x =5 x +6.Eventhoughwehaveacommonbase,havingtwotermsontherighthandside oftheequationfoilsourplanofequatingexponentsortakinglogs.Ifwestareatthislong enough,wenoticethatwehavethreetermswiththeexponentononetermexactlytwicethat ofanother.Tooursurpriseanddelight,wehavea`quadraticindisguise'.Letting u =5 x wehave u 2 = x 2 =5 2 x sotheequation5 2 x =5 x +6becomes u 2 = u +6.Solvingthisas u 2 )]TJ/F53 10.9091 Tf 11.382 0 Td [(u )]TJ/F15 10.9091 Tf 11.383 0 Td [(6=0gives u = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2or u =3.Since u =5 x ,wehave5 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2or5 x =3.Since 5 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2hasnorealsolution,Whynot?wefocuson5 x =3.Sinceitisn'tconvenientto express3asapowerof5,wetakenaturallogsandgetln x =lnsothat x ln=ln or x = ln ln .Whenwegraph f x =25 x and g x =5 x +6,weseethattheyintersectat x = ln ln 0 : 6826. 6.Atrst,it'sunclearhowtoproceedwith e x )]TJ/F37 7.9701 Tf 6.586 0 Td [(e )]TJ/F38 5.9776 Tf 5.756 0 Td [(x 2 =5,besidesclearingthedenominatorto obtain e x )]TJ/F53 10.9091 Tf 10.124 0 Td [(e )]TJ/F37 7.9701 Tf 6.587 0 Td [(x =10.Ofcourse,ifwerewrite e )]TJ/F37 7.9701 Tf 6.586 0 Td [(x = 1 e x ,weseewehaveanotherdenominator lurkingintheproblem: e x )]TJ/F34 7.9701 Tf 14.755 4.296 Td [(1 e x =10.Clearingthisdenominatorgivesus e 2 x )]TJ/F15 10.9091 Tf 11.378 0 Td [(1=10 e x andonceagain,wehaveanequationwiththreetermswheretheexponentononetermis exactlytwicethatofanother-a`quadraticindisguise.'Ifwelet u = e x ,then u 2 = e 2 x sothe equation e 2 x )]TJ/F15 10.9091 Tf 10.771 0 Td [(1=10 e x canbeviewedas u 2 )]TJ/F15 10.9091 Tf 10.772 0 Td [(1=10 u .Solving u 2 )]TJ/F15 10.9091 Tf 10.771 0 Td [(10 u )]TJ/F15 10.9091 Tf 10.771 0 Td [(1=0,weobtain bythequadraticformula u =5 p 26.Fromthis,wehave e x =5 p 26.Since5 )]TJ 10.391 9.024 Td [(p 26 < 0, wegetnorealsolutionto e x =5 )]TJ 10.438 9.025 Td [(p 26,butfor e x =5+ p 26,wetakenaturallogstoobtain x =ln )]TJ/F15 10.9091 Tf 5 -8.837 Td [(5+ p 26 .Ifwegraph f x = e x )]TJ/F37 7.9701 Tf 6.586 0 Td [(e )]TJ/F38 5.9776 Tf 5.756 0 Td [(x 2 and g x =5,weseethatthegraphsintersect at x =ln )]TJ/F15 10.9091 Tf 5 -8.836 Td [(5+ p 26 2 : 312

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346ExponentialandLogarithmicFunctions y = f x =25 x and y = f x = e x )]TJ/F37 7.9701 Tf 6.586 0 Td [(e )]TJ/F38 5.9776 Tf 5.757 0 Td [(x 2 and y = g x =5 x +6 y = g x =5 TheauthorswouldberemissnottomentionthatExample6.3.1stillholdsgreateducational value.Muchcanbelearnedaboutlogarithmsandexponentialsbyverifyingthesolutionsobtained inExample6.3.1analytically.Forexample,toverifyoursolutionto2000=1000 3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t ,we substitute t = )]TJ/F34 7.9701 Tf 9.681 5.374 Td [(10ln ln andobtain 2000 ? =1000 3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 )]TJ/F35 5.9776 Tf 7.782 4.025 Td [(10ln ln 2000 ? =1000 3 ln ln 2000 ? =1000 3 log 3 ChangeofBase 2000 ? =1000 2InverseProperty 2000 X =2000 Theothersolutionscanbeveriedbyusingacombinationoflogandinverseproperties.Somefall outquitequickly,whileothersaremoreinvolved.Weleavethemtothereader. Sinceexponentialfunctionsarecontinuousontheirdomains,theIntermediateValueTheorem3.1 applies.AswiththealgebraicfunctionsinSection5.3,thisallowsustosolveinequalitiesusing signdiagramsasdemonstratedbelow. Example 6.3.2 Solvethefollowinginequalities.Checkyouranswergraphicallyusingacalculator. 1.2 x 2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(16 0 2. e x e x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 3 3. xe 2 x < 4 x Solution. 1.Sincewealreadyhave0ononesideoftheinequality,weset r x =2 x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.327 0 Td [(16.Thedomain of r isallrealnumbers,soinordertoconstructoursigndiagram,weseedtondthezerosof r .Setting r x =0gives2 x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.478 0 Td [(16=0or2 x 2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 x =16.Since16=2 4 wehave2 x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 x =2 4 sobytheone-to-onepropertyofexponentialfunctions, x 2 )]TJ/F15 10.9091 Tf 10.658 0 Td [(3 x =4.Solving x 2 )]TJ/F15 10.9091 Tf 10.659 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.659 0 Td [(4=0 gives x =4and x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Fromthesigndiagram,wesee r x 0on ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1] [ [4 ; 1 ,which correspondstowherethegraphof y = r x =2 x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(16,isonorabovethe x -axis.

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6.3ExponentialEquationsandInequalities347 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [( 4 0 + y = r x =2 x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(16 2.Therststepweneedtotaketosolve e x e x )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 3istoget0ononesideoftheinequality.To thatend,wesubtract3frombothsidesandgetacommondenominator e x e x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 3 e x e x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 0 e x e x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(3 e x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 e x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 0Commondenomintors. 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 e x e x )]TJ/F15 10.9091 Tf 10.91 0 Td [(4 0 Weset r x = 12 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 e x e x )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 andwenotethat r isundenedwhenitsdenominator e x )]TJ/F15 10.9091 Tf 11.177 0 Td [(4=0,or when e x =4.Solvingthisgives x =ln,sothedomainof r is ; ln [ ln ; 1 .To ndthezerosof r ,wesolve r x =0andobtain12 )]TJ/F15 10.9091 Tf 10.44 0 Td [(2 e x =0.Solvingfor e x ,wend e x =6, or x =ln.Whenwebuildoursigndiagram,ndingtestvaluesmaybealittletrickysince weneedtocheckvaluesaroundlnandln.Recallthatthefunctionln x isincreasing 4 whichmeansln < ln < ln < ln < ln.Whiletheprospectofdeterminingthe signof r lnmaybeveryunsettling,rememberthat e ln =3,so r ln= 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 e ln e ln )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 = 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 Wedeterminethesignsof r lnand r lnsimilarly. 5 Fromthesigndiagram,we ndouranswertobe ; ln [ [ln ; 1 .Usingthecalculator,weseethegraphof f x = e x e x )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 isbelowthegraphof g x =3on ; ln [ ln ; 1 ,andtheyintersect at x =ln 1 : 792. 4 Thisisbecausethebaseofln x is e> 1.Ifthebase b wereintheinterval0
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348ExponentialandLogarithmicFunctions )]TJ/F15 10.9091 Tf 8.485 0 Td [( ln + ln 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( y = f x = e x e x )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 y = g x =3 3.Asbefore,westartsolving xe 2 x < 4 x bygetting0ononesideoftheinequality, xe 2 x )]TJ/F15 10.9091 Tf 9.634 0 Td [(4 x< 0. Weset r x = xe 2 x )]TJ/F15 10.9091 Tf 10.63 0 Td [(4 x andsincetherearenodenominators,even-indexedradicals,orlogs, thedomainof r isallrealnumbers.Setting r x =0produces xe 2 x )]TJ/F15 10.9091 Tf 11.046 0 Td [(4 x =0.With x both inandoutoftheexponent,thiscouldcausesomediculty.However,beforepanicsetsin, wefactoroutthe x toobtain x )]TJ/F53 10.9091 Tf 5 -8.837 Td [(e 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 =0whichgives x =0or e 2 x )]TJ/F15 10.9091 Tf 10.972 0 Td [(4=0.Tosolvethe latter,weisolatetheexponentialandtakelogstoget2 x =ln,or x = ln 2 =ln.Can youexplainthelastequalityusingpropertiesoflogs?Asinthepreviousexample,weneed tobecarefulaboutchoosingtestvalues.Sinceln=0,wechooseln )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 ,ln )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(3 2 andln. Evaluating, 6 wehave r ln )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 =ln )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 e 2ln 1 2 )]TJ/F15 10.9091 Tf 9.759 0 Td [(4ln )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 2 .ApplyingthePowerRuletothelog intheexponent,weobtainln )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 2 e ln 1 2 2 )]TJ/F15 10.9091 Tf 9.767 0 Td [(4ln )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 2 =ln )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 e ln 1 4 )]TJ/F15 10.9091 Tf 9.766 0 Td [(4ln )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 .Usingtheinverse propertiesoflogs,thisreducesto 1 4 ln )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 )]TJ/F15 10.9091 Tf 11.319 0 Td [(4ln )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 = )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(15 4 ln )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 .Since 1 2 < 1,ln )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 < 0 andweget r ln )]TJ/F34 7.9701 Tf 6.196 -4.542 Td [(1 2 is+.Continuinginthismanner,wend r x < 0on ; ln.The calculatorconrmsthatthegraphof f x = xe 2 x isbelowthegraphof g x =4 x onthis intervals. 7 + 0 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [( ln 0 + y = f x = xe 2 x and y = g x =4 x 6 Acalculatorcanbeusedatthispoint.Asusual,weproceedwithoutapologies,withtheanalyticalmethod. 7 Note:ln 0 : 693.

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6.3ExponentialEquationsandInequalities349 Example 6.3.3 RecallfromExample6.1.2thatthetemperatureofcoee T indegreesFahrenheit t minutesafteritisservedcanbemodeledby T t =70+90 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t .Whenwillthecoeebewarmer than100 F? Solution. Weneedtondwhen T t > 100,orinotherwords,weneedtosolvetheinequality 70+90 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 1 t > 100.Getting0ononesideoftheinequality,wehave90 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 1 t )]TJ/F15 10.9091 Tf 11.782 0 Td [(30 > 0,and weset r t =90 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t )]TJ/F15 10.9091 Tf 11.544 0 Td [(30.Thedomainof r isarticiallyrestrictedduetothecontextofthe problemto[0 ; 1 ,soweproceedtondthezerosof r .Solving90 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 1 t )]TJ/F15 10.9091 Tf 11.923 0 Td [(30=0resultsin e )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t = 1 3 sothat t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(10ln )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 3 which,afteraquickapplicationofthePowerRuleleavesuswith t =10ln.Ifwewishtoavoidusingthecalculatortochoosetestvalues,wenotethatsince1 < 3, 0=ln < lnsothat10ln > 0.Sowechoose t =0asatestvaluein[0 ; 10ln.Since 3 < 4,10ln < 10ln,sothelatterisourchoiceofatestvaluefortheintervalln ; 1 Oursigndiagramisbelow,andnexttoitisourgraphof t = T t fromExample6.1.2withthe horizontalline y =100. 0 + 10ln 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( H.A. y =70 y =100 t y 2468101214161820 20 40 60 80 120 140 160 180 y = T t Inordertointerpretwhatthismeansinthecontextoftherealworld,weneedareasonable approximationofthenumber10ln 10 : 986.Thismeansittakesapproximately11minutesfor thecoeetocoolto100 F.Untilthen,thecoeeiswarmerthanthat. 8 Weclosethissectionbyndingtheinverseofafunctionwhichisacompositionofarational functionwithanexponentialfunction. Example 6.3.4 Thefunction f x = 5 e x e x +1 isone-to-one.Findaformulafor f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x andcheck youranswergraphicallyusingyourcalculator. Solution. Westartbywriting y = f x ,andinterchangetherolesof x and y .Tosolvefor y ,we rstcleardenominatorsandthenisolatetheexponentialfunction. 8 Criticsmaypointoutthatsinceweneededtousethecalculatortointerpretouransweranyway,whynotuseit earliertosimplifythecomputations?Itisafairquestionwhichweanswerunfairly:it'sourbook.

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350ExponentialandLogarithmicFunctions y = 5 e x e x +1 x = 5 e y e y +1 Switch x and y x e y +1=5 e y xe y + x =5 e y x =5 e y )]TJ/F53 10.9091 Tf 10.909 0 Td [(xe y x = e y )]TJ/F53 10.9091 Tf 10.909 0 Td [(x e y = x 5 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x ln e y =ln x 5 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x y =ln x 5 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x Weclaim f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x =ln x 5 )]TJ/F37 7.9701 Tf 6.587 0 Td [(x .Toverifythisanalytically,wewouldneedtoverifythecompositions )]TJ/F53 10.9091 Tf 5 -8.836 Td [(f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 f x = x forall x inthedomainof f andthat )]TJ/F53 10.9091 Tf 5 -8.836 Td [(f f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = x forall x inthedomain of f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 .Weleavethistothereader.Toverifyoursolutiongraphically,wegraph y = f x = 5 e x e x +1 and y = g x =ln x 5 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x onthesamesetofaxesandobservethesymmetryabouttheline y = x Notethedomainof f istherangeof g andvice-versa. y = f x = 5 e x e x +1 and y = g x =ln x 5 )]TJ/F75 7.9701 Tf 7.594 0 Td [(x

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6.3ExponentialEquationsandInequalities351 6.3.1Exercises 1.Solvethefollowingequationsanalytically. a3 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 =27 b3 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 =29 c3 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 =2 x d3 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 x +5 e )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+ 0 : 06 12 12 t =3 f e )]TJ/F34 7.9701 Tf 6.587 0 Td [(5730 k = 1 2 g2 x 3 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x =1 h e 2 x =2 e x i70+90 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t =75 j 150 1+29 e )]TJ/F35 5.9776 Tf 5.756 0 Td [(0 : 8 t =75 k25 )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(4 5 x =10 2.Solvethefollowinginequalitiesanalytically. a e x > 53 b1000 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1+ 0 : 06 12 12 t 3000 c2 x 3 )]TJ/F37 7.9701 Tf 6.586 0 Td [(x < 1 d25 )]TJ/F34 7.9701 Tf 6.196 -4.542 Td [(4 5 x 10 e 150 1+29 e )]TJ/F35 5.9776 Tf 5.756 0 Td [(0 : 8 t 130 f70+90 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t 75 3.Useyourcalculatortohelpyousolvethefollowingequationsandinequalities. a e x 6withoutasigndiagram. 5.Computetheinverseof f x = e x )]TJ/F53 10.9091 Tf 10.909 0 Td [(e )]TJ/F37 7.9701 Tf 6.587 0 Td [(x 2 .Statethedomainandrangeofboth f and f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 6.InExample6.3.4,wefoundthattheinverseof f x = 5 e x e x +1 was f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x =ln x 5 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x but weleftafewlooseendsforyoutotieup. aShowthat )]TJ/F53 10.9091 Tf 5 -8.836 Td [(f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 f x = x forall x inthedomainof f andthat )]TJ/F53 10.9091 Tf 5 -8.836 Td [(f f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x = x for all x inthedomainof f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 bFindtherangeof f byndingthedomainof f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 cLet g x = 5 x x +1 and h x = e x .Showthat f = g h andthat g h )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = h )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 g )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 9 7.Withthehelpofyourclassmates,solvetheinequality e x >x n foravarietyofnatural numbers n .Whatmightyouconjectureaboutthespeed"atwhich f x = e x growsversus anypolynomial? 9 WeknowthisistrueingeneralbyExercise8inSection5.2,butit'snicetoseeaspecicexampleoftheproperty.

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352ExponentialandLogarithmicFunctions 6.3.2Answers 1.a x =4 b x = ln+ln ln c x = ln ln )]TJ/F15 10.9091 Tf 10.909 0 Td [(ln d x = ln+5ln )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 2 ln )]TJ/F15 10.9091 Tf 10.91 0 Td [(ln )]TJ/F34 7.9701 Tf 6.195 -4.542 Td [(1 2 e t = ln 12ln : 005 f k = ln )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5730 g x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; 1 h x =1 ; ln i t =10ln j t = ln )]TJ/F34 7.9701 Tf 8.313 -4.541 Td [(1 29 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 8 k x = ln )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(2 5 ln )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(4 5 2.aln ; 1 b ln 12ln : 005 ; 1 c ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 [ ; 1 d ; ln )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(2 5 ln )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(4 5 # e ; ln )]TJ/F34 7.9701 Tf 10.43 -4.541 Td [(2 377 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 8 # = )]TJ/F54 10.9091 Tf 5 -8.836 Td [( ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(5 4 ln )]TJ/F34 7.9701 Tf 10.43 -4.541 Td [(2 377 f ln )]TJ/F34 7.9701 Tf 8.313 -4.541 Td [(1 18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 1 ; 1 =[10ln ; 1 3.a ; 2 : 3217 [ : 3717 ; 1 b x )]TJ/F15 10.9091 Tf 20 0 Td [(0 : 76666 ;x =2 ;x =4 c x =0 d ; 1] e ; 2 : 7095 f x 0 : 01866 ;x 1 : 7115 4. x> 1 3 ln+1 5. f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 =ln x + p x 2 +1 .Both f and f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 havedomain ; 1 andrange ; 1 .

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6.4LogarithmicEquationsandInequalities353 6.4LogarithmicEquationsandInequalities InSection6.3wesolvedequationsandinequalitiesinvolvingexponentialfunctionsusingoneof twobasicstrategies.Wenowturnourattentiontoequationsandinequalitiesinvolvinglogarithmic functions,andnotsurprisingly,therearetwobasicstrategiestochoosefrom.Forexample,suppose wewishtosolvelog 2 x =log 2 .Theorem6.4tellsusthatthe only solutiontothisequation is x =5.Nowsupposewewishtosolvelog 2 x =3.IfwewanttouseTheorem6.4,weneedto rewrite3asalogarithmbase2.WecanuseTheorem6.3todojustthat:3=log 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 3 =log 2 Ourequationthenbecomeslog 2 x =log 2 sothat x =8.However,wecouldhavearrivedatthe sameanswer,infewersteps,byusingTheorem6.3torewritetheequationlog 2 x =3as2 3 = x or x =8.Wesummarizethetwocommonwaystosolvelogequationsbelow. StepsforSolvinganEquationinvolvingLogarithmicFuctions 1.Isolatethelogarithmicfunction. 2.aIfconvenient,expressbothsidesaslogswiththesamebaseandequatethearguments ofthelogfunctions. bOtherwise,rewritethelogequationasanexponentialequation. Example 6.4.1 Solvethefollowingequations.Checkyoursolutionsgraphicallyusingacalculator. 1.log 117 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x =log 117 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2.2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(ln x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3=1 3.log 6 x +4+log 6 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x =1 4.log 7 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 x =1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(log 7 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 5.log 2 x +3=log 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x +3 6.1+2log 4 x +1=2log 2 x Solution. 1.Sincewehavethesamebaseonbothsidesoftheequationlog 117 )]TJ/F15 10.9091 Tf 11.112 0 Td [(3 x =log 117 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 weequatewhat'sinsidethelogstoget1 )]TJ/F15 10.9091 Tf 11.694 0 Td [(3 x = x 2 )]TJ/F15 10.9091 Tf 11.694 0 Td [(3.Solving x 2 +3 x )]TJ/F15 10.9091 Tf 11.694 0 Td [(4=0gives x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4and x =1.Tochecktheseanswersusingthecalculator,wemakeuseofthechange ofbaseformulaandgraph f x = ln )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 x ln and g x = ln x 2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 ln andweseetheyintersectonly at x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4.Toseewhathappenedtothesolution x =1,wesubstituteitintoouroriginal equationtoobtainlog 117 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2=log 117 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Whiletheseexpressionslookidentical,neither isarealnumber, 1 whichmeans x =1isnotinthedomainoftheoriginalequation,andis notasolution. 2.Ourrstobjectiveinsolving2 )]TJ/F15 10.9091 Tf 9.368 0 Td [(ln x )]TJ/F15 10.9091 Tf 9.369 0 Td [(3=1istoisolatethelogarithm.Wegetln x )]TJ/F15 10.9091 Tf 9.369 0 Td [(3=1, which,asanexponentialequation,is e 1 = x )]TJ/F15 10.9091 Tf 11.409 0 Td [(3.Wegetoursolution x = e +3.Onthe calculator,weseethegraphof f x =2 )]TJ/F15 10.9091 Tf 11.73 0 Td [(ln x )]TJ/F15 10.9091 Tf 11.73 0 Td [(3intersectsthegraphof g x =1at x = e +3 5 : 718. 1 Theydo,however,representthesame family ofcomplexnumbers.Westopourselvesatthispointandreferthe readertoagoodcourseinComplexVariables.

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354ExponentialandLogarithmicFunctions y = f x =log 117 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x and y = f x =2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(ln x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3and y = g x =log 117 )]TJ/F74 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F52 10.9091 Tf 12.545 0 Td [(3 y = g x =1 3.Wecanstartsolvinglog 6 x +4+log 6 )]TJ/F53 10.9091 Tf 9.793 0 Td [(x =1byusingtheProductRuleforlogarithmsto rewritetheequationaslog 6 [ x +4 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x ]=1.Rewritingthisasanexponentialequation, weget6 1 = x +4 )]TJ/F53 10.9091 Tf 11.055 0 Td [(x .Thisreducesto x 2 )]TJ/F53 10.9091 Tf 11.055 0 Td [(x )]TJ/F15 10.9091 Tf 11.054 0 Td [(6=0,whichgives x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x =3. Graphing y = f x = ln x +4 ln + ln )]TJ/F37 7.9701 Tf 6.586 0 Td [(x ln and y = g x =1,weseetheyintersecttwice,at x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and x =3. y = f x =log 6 x +4+log 6 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x and y = g x =1 4.Takingacuefromthepreviousproblem,webeginsolvinglog 7 )]TJ/F15 10.9091 Tf 10.967 0 Td [(2 x =1 )]TJ/F15 10.9091 Tf 10.967 0 Td [(log 7 )]TJ/F53 10.9091 Tf 10.966 0 Td [(x by rstcollectingthelogarithmsonthesameside,log 7 )]TJ/F15 10.9091 Tf 9.993 0 Td [(2 x +log 7 )]TJ/F53 10.9091 Tf 9.993 0 Td [(x =1,andthenusing theProductRuletogetlog 7 [ )]TJ/F15 10.9091 Tf 10.645 0 Td [(2 x )]TJ/F53 10.9091 Tf 10.646 0 Td [(x ]=1.Rewritingthisasanexponentialequation gives7 1 = )]TJ/F15 10.9091 Tf 9.254 0 Td [(2 x )]TJ/F53 10.9091 Tf 9.254 0 Td [(x whichgivesthequadraticequation2 x 2 )]TJ/F15 10.9091 Tf 9.254 0 Td [(7 x )]TJ/F15 10.9091 Tf 9.254 0 Td [(4=0.Solving,wend x = )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 2 and x =4.Graphing,wend y = f x = ln )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x ln and y = g x =1 )]TJ/F34 7.9701 Tf 10.46 5.374 Td [(ln )]TJ/F37 7.9701 Tf 6.587 0 Td [(x ln intersect onlyat x = )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 2 .Checking x =4intheoriginalequationproduceslog 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7=1 )]TJ/F15 10.9091 Tf 10.976 0 Td [(log 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, whichisacleardomainviolation. 5.Startingwithlog 2 x +3=log 2 )]TJ/F53 10.9091 Tf 11.445 0 Td [(x +3,wegatherthelogarithmstoonesideandget log 2 x +3 )]TJ/F15 10.9091 Tf 11.547 0 Td [(log 2 )]TJ/F53 10.9091 Tf 11.547 0 Td [(x =3,andthenusetheQuotientRuletoobtainlog 2 x +3 6 )]TJ/F37 7.9701 Tf 6.587 0 Td [(x =3. Rewritingthisasanexponentialequationgives2 3 = x +3 6 )]TJ/F37 7.9701 Tf 6.587 0 Td [(x .Thisreducestothelinearequation 8 )]TJ/F53 10.9091 Tf 9.973 0 Td [(x = x +3,whichgivesus x =5.Whenwegraph f x = ln x +3 ln and g x = ln )]TJ/F37 7.9701 Tf 6.586 0 Td [(x ln +3, wendtheyintersectat x =5.

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6.4LogarithmicEquationsandInequalities355 y = f x =log 7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x and y = f x =log 2 x +3and y = g x =1 )]TJ/F52 10.9091 Tf 12.546 0 Td [(log 7 )]TJ/F74 10.9091 Tf 12.545 0 Td [(x y = g x =log 2 )]TJ/F74 10.9091 Tf 12.545 0 Td [(x +3 6.Startingwith1+2log 4 x +1=2log 2 x ,wegatherthelogstoonesidetogettheequation 1=2log 2 x )]TJ/F15 10.9091 Tf 11.576 0 Td [(2log 4 x +1.Beforewecancombinethelogarithms,however,weneeda commonbase.Since4isapowerof2,weusechangeofbasetoconvertlog 4 x +1= log 2 x +1 log 2 = 1 2 log 2 x +1.Hence,ouroriginalequationbecomes1=2log 2 x )]TJ/F15 10.9091 Tf 11.97 0 Td [(2 )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 2 log 2 x +1 or 1=2log 2 x )]TJ/F15 10.9091 Tf 10.874 0 Td [(log 2 x +1.UsingthePowerandQuotientRules,weobtain1=log 2 x 2 x +1 Rewritingthisinexponentialform,weget x 2 x +1 =2or x 2 )]TJ/F15 10.9091 Tf 10.977 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.977 0 Td [(2=0.Usingthequadratic formula,weget x =1 p 3.Graphing f x =1+ 2ln x +1 ln and g x = 2ln x ln ,weseethe graphsintersectonlyat x =1+ p 3 2 : 732.Thesolution x =1 )]TJ 11.083 9.025 Td [(p 3 < 0,whichmeansif substitutedintotheoriginalequation,theterm2log 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ 10.909 9.024 Td [(p 3 isundened. y = f x =1+2log 4 x +1and y = g x =2log 2 x Ifnothingelse,Example6.4.1demonstratestheimportanceofcheckingforextraneoussolutions 2 whensolvingequationsinvolvinglogarithms.Eventhoughwecheckedouranswersgraphically, extraneoussolutionsareeasytospot-anysupposedsolutionwhichcausesanegativenumber insidealogarithmneedstobediscarded.AswiththeequationsinExample6.3.1,muchcanbe learnedfromcheckingalloftheanswersinExample6.4.1analytically.Weleavethistothereader andturnourattentiontoinequalitiesinvolvinglogarithmicfunctions.Sincelogarithmicfunctions arecontinuousontheirdomains,wecanusesigndiagrams. Example 6.4.2 Solvethefollowinginequalities.Checkyouranswergraphicallyusingacalculator. 2 Recallthatanextraneoussolutionisananswerobtainedanalyticallywhichdoesnotsatisfytheoriginalequation.

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356ExponentialandLogarithmicFunctions 1. 1 ln x +1 1 2.log 2 x 2 < 2log 2 x +33. x log x +1 x Solution. 1.Westartsolving 1 ln x +1 1bygetting0ononesideoftheinequality: 1 ln x +1 )]TJ/F15 10.9091 Tf 11.699 0 Td [(1 0. Gettingacommondenominatoryields 1 ln x +1 )]TJ/F34 7.9701 Tf 12.704 5.374 Td [(ln x +1 ln x +1 0whichreducesto )]TJ/F34 7.9701 Tf 7.997 0 Td [(ln x ln x +1 0, or ln x ln x +1 0.Wedene r x = ln x ln x +1 andsetaboutndingthedomainandthezeros of r .Duetotheappearanceofthetermln x ,werequire x> 0.Inordertokeepthe denominatorawayfromzero,wesolveln x +1=0soln x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,so x = e )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 = 1 e .Hence, thedomainof r is )]TJ/F15 10.9091 Tf 5 -8.836 Td [(0 ; 1 e [ )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 e ; 1 .Tondthezerosof r ,weset r x = ln x ln x +1 =0sothat ln x =0,andwend x = e 0 =1.Inordertodeterminetestvaluesfor r withoutresorting tothecalculator,weneedtondnumbersbetween0, 1 e ,and1whichhaveabaseof e .Since e 2 : 718 > 1,0 < 1 e 2 < 1 e < 1 p e < 1
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6.4LogarithmicEquationsandInequalities357 0 + 1 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 8 0 + y = f x =log 2 x 2 and y = g x =2log 2 x +3 3.Webegintosolve x log x +1 x bysubtracting x frombothsidestoget x log x +1 )]TJ/F53 10.9091 Tf 9.488 0 Td [(x 0. Wedene r x = x log x +1 )]TJ/F53 10.9091 Tf 8.861 0 Td [(x andduetothepresenceofthelogarithm,werequire x +1 > 0, or x> )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Tondthezerosof r ,weset r x = x log x +1 )]TJ/F53 10.9091 Tf 11.427 0 Td [(x =0.Factoring,weget x log x +1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1=0,whichgives x =0orlog x +1 )]TJ/F15 10.9091 Tf 9.537 0 Td [(1=0.Thelattergiveslog x +1=1, or x +1=10 1 ,whichadmits x =9.Weselecttestvalues x sothat x +1isapowerof10, andweobtain )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 < )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 9 < 0 < p 10 )]TJ/F15 10.9091 Tf 11.088 0 Td [(1 < 9 < 99.Oursigndiagramgivesthesolutionto be )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 0] [ [9 ; 1 .Thecalculatorindicatesthegraphof y = f x = x log x +1isabove y = g x = x onthesolutionintervals,andthegraphsintersectat x =0and x =9. )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 + 0 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [( 9 0 + y = f x = x log x +1and y = g x = x Near x =0Near x =9 OurnextexamplerevisitstheconceptofpHasrstintroducedintheexercisesinSection6.1.

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358ExponentialandLogarithmicFunctions Example 6.4.3 InordertosuccessfullybreedIppizutishthepHofafreshwatertankmustbe atleast7.8butcanbenomorethan8.5.Determinethecorrespondingrangeofhydrogenion concentration. Solution. RecallfromExercise4cinSection6.1thatpH= )]TJ/F15 10.9091 Tf 10.303 0 Td [(log[H + ]where[H + ]isthehydrogen ionconcentrationinmolesperliter.Werequire7 : 8 )]TJ/F15 10.9091 Tf 22.114 0 Td [(log[H + ] 8 : 5or )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 : 8 log[H + ] )]TJ/F15 10.9091 Tf 20.296 0 Td [(8 : 5. Tosolvethiscompoundinequalitywesolve )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 : 8 log[H + ]andlog[H + ] )]TJ/F15 10.9091 Tf 22.166 0 Td [(8 : 5andtakethe intersectionofthesolutionsets. 3 Theformerinequalityyields0 < [H + ] 10 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 : 8 andthelatter yields[H + ] 10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(8 : 5 .Takingtheintersectiongivesusournalanswer10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(8 : 5 [H + ] 10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(7 : 8 YourChemistryprofessormaywanttheanswerwrittenas3 : 16 10 )]TJ/F34 7.9701 Tf 6.586 0 Td [(9 [H + ] 1 : 58 10 )]TJ/F34 7.9701 Tf 6.586 0 Td [(8 Aftercarefullyadjustingtheviewingwindowonthegraphingcalculatorweseethatthegraphof f x = )]TJ/F15 10.9091 Tf 10.303 0 Td [(log x liesbetweenthelines y =7 : 8and y =8 : 5ontheinterval[3 : 16 10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(9 ; 1 : 58 10 )]TJ/F34 7.9701 Tf 6.586 0 Td [(8 ]. Thegraphsof y = f x = )]TJ/F15 10.9091 Tf 10.303 0 Td [(log x y =7 : 8 and y =8 : 5 Weclosethissectionbyndinganinverseofaone-to-onefunctionwhichinvolveslogarithms. Example 6.4.4 Thefunction f x = log x 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(log x isone-to-one.Findaformulafor f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x and checkyouranswergraphicallyusingyourcalculator. Solution. Werstwrite y = f x theninterchangethe x and y andsolvefor y y = f x y = log x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(log x x = log y 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(log y Interchange x and y x )]TJ/F15 10.9091 Tf 10.909 0 Td [(log y =log y x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x log y =log y x = x log y +log y x = x +1log y x x +1 =log y y =10 x x +1 Rewriteasanexponentialequation. 3 Refertopage147foradiscussionofwhatthismeans.

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6.4LogarithmicEquationsandInequalities359 Wehave f )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 x =10 x x +1 .Graphing f and f )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 onthesameviewingwindowyields y = f x = log x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(log x and y = g x =10 x x +1

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360ExponentialandLogarithmicFunctions 6.4.1Exercises 1.Solvethefollowingequationsanalytically. alog 1 2 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 bln x 2 =ln x 2 clog 3 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(4+log 3 x +4=2 dlog 5 x +1+log 5 x +2=1 elog 2 x 3 =log 2 x flog 169 x +7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(log 169 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(9= 1 2 glog x 10 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 =4 : 7 h )]TJ/F15 10.9091 Tf 10.303 0 Td [(log x =5 : 4 i10log x 10 )]TJ/F34 7.9701 Tf 6.586 0 Td [(12 =150 jlog 3 x =log 1 3 x +8 2.Solvethefollowinginequalitiesanalytically. a x ln x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x> 0 b5 : 6 log x 10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 7 : 1 c10log x 10 )]TJ/F34 7.9701 Tf 6.586 0 Td [(12 90 d2 : 3 < )]TJ/F15 10.9091 Tf 10.303 0 Td [(log x < 5 : 4 e 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(ln x x 2 < 0 fln x 2 ln x 2 3.Useyourcalculatortohelpyousolvethefollowingequationsandinequalities. aln x = e )]TJ/F37 7.9701 Tf 6.587 0 Td [(x bln x 2 +1 5 cln x = 4 p x dln )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 +13 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(6 < 0 4.Since f x = e x isastrictlyincreasingfunction,if a
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6.4LogarithmicEquationsandInequalities361 8.ExplaintheequationinExercise1gandtheinequalityinExercise2baboveintermsofthe Richterscaleforearthquakemagnitude.SeeExercise4ainSection6.1. 9.ExplaintheequationinExercise1iandtheinequalityinExercise2caboveintermsofsound intensitylevelasmeasuredindecibels.SeeExercise4binSection6.1. 10.ExplaintheequationinExercise1handtheinequalityinExercise2daboveintermsofthe pHofasolution.SeeExercise4cinSection6.1. 11.Withthehelpofyourclassmates,solvetheinequality n p x> ln x foravarietyofnatural numbers n .Whatmightyouconjectureaboutthespeed"atwhich f x =ln x grows versusanyprincipal n th rootfunction?

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362ExponentialandLogarithmicFunctions 6.4.2Answers 1.a x =8 b x =1 ;x = e 2 c x =5 d x = 1 2 e x =1 f x =2 g x =10 1 : 7 h x =10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 : 4 i x =10 3 j x =81 2.a e; 1 b 10 2 : 6 ; 10 4 : 1 c 10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 ; 1 d )]TJ/F15 10.9091 Tf 5 -8.837 Td [(10 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 : 4 ; 10 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 : 3 e e; 1 f ; 1] [ [ e 2 ; 1 3.a x 1 : 3098 b ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 : 1414 [ : 1414 ; 1 c x 4 : 177 ;x 5503 : 665 d )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 : 0281 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 [ : 5 ; 0 : 5991 [ : 9299 ; 2 4. )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(1 2
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6.5ApplicationsofExponentialandLogarithmicFunctions363 6.5ApplicationsofExponentialandLogarithmicFunctions AswementionedinSection6.1,exponentialandlogarithmicfunctionsareusedtomodelawide varietyofbehaviorsintherealworld.Intheexamplesthatfollow,notethatwhiletheapplications aredrawnfrommanydierentdisciplines,themathematicsremainsessentiallythesame.Dueto theappliednatureoftheproblemswewillexamineinthissection,thecalculatorisoftenusedto expressouranswersasdecimalapproximations. 6.5.1ApplicationsofExponentialFunctions Perhapsthemostwell-knownapplicationofexponentialfunctionscomesfromthenancialworld. Supposeyouhave$100toinvestatyourlocalbankandtheyareoeringawhopping5%annual percentageinterestrate.Thismeansthatafteroneyear,thebankwillpay you 5%ofthat$100, or$100 : 05=$5ininterest,soyounowhave$105. 1 Thisisinaccordancewiththeformula for simpleinterest whichyouhaveundoubtedlyrunacrossatsomepointinyourmathematical upbringing. Equation 6.1 SimpleInterest Theamountofinterest I accruedatanannualrate r onan investment a P after t yearsis I = Prt Theamount A intheaccountafter t yearsisgivenby A = P + I = P + Prt = P + rt a Calledthe principal Suppose,however,thatsixmonthsintotheyear,youhearofabetterdealatarivalbank. 2 Naturally,youwithdrawyourmoneyandtrytoinvestitatthehigherratethere.Sincesixmonths isonehalfofayear,thatinitial$100yields$100 : 05 )]TJ/F34 7.9701 Tf 6.196 -4.542 Td [(1 2 =$2 : 50ininterest.Youtakeyour $102 : 50otothecompetitorandndoutthatthoserestrictionswhich may applyactuallydo applytoyou,andyoureturntoyourbankwhichhappilyacceptsyour$102 : 50fortheremaining sixmonthsoftheyear.Toyoursurpriseanddelight,attheendoftheyearyourstatementreads $105 : 06,not$105asyouhadexpected. 3 Wheredidthoseextrasixcentscomefrom?Fortherst sixmonthsoftheyear,interestwasearnedontheoriginalprincipalof$100,butforthesecond sixmonths,interestwasearnedon$102 : 50,thatis,youearnedinterestonyourinterest.Thisis thebasicconceptbehind compoundinterest .Inthepreviousdiscussion,wewouldsaythatthe interestwascompoundedtwice,orsemiannually. 4 Ifmoremoneycanbeearnedbyearninginterest oninterestalreadyearned,anaturalquestiontoaskiswhathappensiftheinterestiscompounded moreoften,say4timesayear,whichiseverythreemonths,or`quarterly.'Inthiscase,the moneyisintheaccountforthreemonths,or 1 4 ofayear,atatime.Aftertherstquarter,we 1 Howgenerousofthem! 2 Somerestrictionsmayapply. 3 Actually,thenalbalanceshouldbe$105 : 0625. 4 Usingthisconvention,simpleinterestafteroneyearisthesameascompoundingtheinterestonlyonce.

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364ExponentialandLogarithmicFunctions have A = P + rt =$100 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+0 : 05 1 4 =$101 : 25.Wenowinvestthe$101 : 25forthenextthree monthsandndthatattheendofthesecondquarter,wehave A =$101 : 25 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+0 : 05 1 4 $102 : 51. Continuinginthismanner,thebalanceattheendofthethirdquarteris$103 : 79,and,atlast,we obtain$105 : 08.Theextratwocentshardlyseemsworthit,butweseethatwedoinfactgetmore moneythemoreoftenwecompound.Inordertodevelopaformulaforthisphenomenon,weneed todosomeabstractcalculations.Supposewewishtoinvestourprincipal P atanannualrate r and compoundtheinterest n timesperyear.Thismeansthemoneysitsintheaccount 1 n th ofayear betweencompoundings.Let A k denotetheamountintheaccountafterthe k th compounding.Then A 1 = P )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1+ r )]TJ/F34 7.9701 Tf 6.647 -4.542 Td [(1 n whichsimpliesto A 1 = P )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1+ r n .Afterthesecondcompounding,weuse A 1 asournewprincipalandget A 2 = A 1 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+ r n = P )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+ r n )]TJ/F15 10.9091 Tf 16.364 -8.836 Td [(1+ r n = P )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+ r n 2 .Continuingin thisfashion,weget A 3 = P )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1+ r n 3 A 4 = P )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1+ r n 4 ,andsoon,sothat A k = P )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1+ r n k .Since wecompoundtheinterest n timesperyear,after t years,wehave nt compoundings.Wehavejust derivedthegeneralformulaforcompoundinterestbelow. Equation 6.2 CompoundedInterest: Ifaninitialprincipal P isinvestedatanannualrate r andtheinterestiscompounded n timesperyear,theamount A intheaccountafter t yearsis A = P 1+ r n nt Ifwetake P =100, r =0 : 05,and n =4,Equation6.2becomes A =100 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1+ 0 : 05 4 4 t whichreduces to A =100 : 0125 4 t .Thisequationdenestheamount A asanexponentialfunctionoftime t A t .Tocheckthisagainstourpreviouscalculations,wend A )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 4 =100 : 0125 4 1 4 =101 : 25, A )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 $102 : 51, A )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(3 4 $103 : 79,and A $105 : 08. Example 6.5.1 Suppose$2000isinvestedinanaccountwhichoers7 : 125%compoundedmonthly. 1.Expresstheamount A intheaccountasafunctionofthetermoftheinvestment t inyears. 2.Howmuchisintheaccountafter5years? 3.Howlongwillittakefortheinitialinvestmenttodouble? 4.Findandinterprettheaveragerateofchange 5 oftheamountintheaccountfromtheendof thefourthyeartotheendofthefthyear,andfromtheendofthethirty-fourthyeartothe endofthethirty-fthyear. Solution. 1.Substituting P =2000, r =0 : 07125,and n =12monthlyintoEquation6.2yields A = 2000 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1+ 0 : 07125 12 12 t .Usingfunctionnotation,weget A t =2000 : 0059375 12 t 2.Since t representsthelengthoftheinvestment,wesubstitute t =5into A t tond A = 2000 : 0059375 12 2852 : 92.After5years,wehaveapproximately$2852 : 92. 5 SeeDenition2.3inSection2.1.

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6.5ApplicationsofExponentialandLogarithmicFunctions365 3.Ourinitialinvestmentis$2000,sotondthetimeittakesthistodouble,weneedtond t when A t =4000.Weget2000 : 0059375 12 t =4000,or : 0059375 12 t =2.Takingnatural logsasinSection6.3,weget t = ln 12ln : 0059375 9 : 75.Hence,ittakesapproximately9years 9monthsfortheinvestmenttodouble. 4.Tondtheaveragerateofchangeof A fromtheendofthefourthyeartotheendofthe fthyear,wecompute A )]TJ/F37 7.9701 Tf 6.587 0 Td [(A 5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 195 : 63.Similarly,theaveragerateofchangeof A from theendofthethirty-fourthyeartotheendofthethirty-fthyearis A )]TJ/F37 7.9701 Tf 6.586 0 Td [(A 35 )]TJ/F34 7.9701 Tf 6.586 0 Td [(34 1648 : 21. Thismeansthatthevalueoftheinvestmentisincreasingatarateofapproximately$195 : 63 peryearbetweentheendofthefourthandfthyears,whilethatratejumpsto$1648 : 21per yearbetweentheendofthethirty-fourthandthirty-fthyears.So,notonlyisittruethat thelongeryouwait,themoremoneyyouhave,butalsothelongeryouwait,thefasterthe moneyincreases. 6 Wehaveobservedthatthemoretimesyoucompoundtheinterestperyear,themoremoneyyou willearninayear.Let'spushthisnotiontothelimit. 7 Consideraninvestmentof$1invested at100%interestfor1yearcompounded n timesayear.Equation6.2tellsusthattheamountof moneyintheaccountafter1yearis A = )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+ 1 n n .Belowisatableofvaluesrelating n and A n A 1 2 2 2 : 25 4 2 : 4414 12 2 : 6130 360 2 : 7145 1000 2 : 7169 10000 2 : 7181 100000 2 : 7182 Aspromised,themorecompoundingsperyear,themoremoneythereisintheaccount,butwe alsoobservethattheincreaseinmoneyisgreatlydiminishing.Wearewitnessingamathematical `tugofwar'.Whilewearecompoundingmoretimesperyear,andhencegettinginterestonour interestmoreoften,theamountoftimebetweencompoundingsisgettingsmallerandsmaller,so thereislesstimetobuildupadditionalinterest.WithCalculus,wecanshow 8 thatas n !1 A = )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1+ 1 n n e ,where e isthenaturalbaserstpresentedinSection6.1.Takingthenumber ofcompoundingsperyeartoinnityresultsinwhatiscalled continuously compoundedinterest. Theorem 6.8 Ifyouinvest$1at100%interestcompoundedcontinuously,thenyouwillhave$ e attheendofoneyear. 6 Infact,therateofincreaseoftheamountintheaccountisexponentialaswell.Thisisthequalitythatreally denesexponentialfunctionsandwereferthereadertoacourseinCalculus. 7 Onceyou'vehadasemesterofCalculus,you'llbeabletofullyappreciatethisverylamepun. 8 Ordene,dependingonyourpointofview.

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366ExponentialandLogarithmicFunctions Usingthisdenitionof e andalittleCalculus,wecantakeEquation6.2andproduceaformulafor continuouslycompoundedinterest. Equation 6.3 ContinuouslyCompoundedInterest: Ifaninitialprincipal P isinvestedat anannualrate r andtheinterestiscompoundedcontinuously,theamount A intheaccountafter t yearsis A = Pe rt IfwetakethescenarioofExample6.5.1andcomparemonthlycompoundingtocontinuouscompoundingover35years,wendthatmonthlycompoundingyields A =2000 : 0059375 12 whichisabout$24 ; 035 : 28,whereascontinuouslycompoundinggives A =2000 e 0 : 07125 which isabout$24 ; 213 : 18-adierenceoflessthan1%. Equations6.2and6.3bothuseexponentialfunctionstodescribethegrowthofaninvestment. Curiouslyenough,thesameprincipleswhichgoverncompoundinterestarealsousedtomodel shorttermgrowthofpopulations.InBiology, TheLawofUninhibitedGrowth statesas itspremisethatthe instantaneous rateatwhichapopulationincreasesatanytimeisdirectly proportionaltothepopulationatthattime. 9 Inotherwords,themoreorganismsthereareata givenmoment,thefastertheyreproduce.Formulatingthelawasstatedresultsinadierential equation,whichrequiresCalculustosolve.Itssolutionisstatedbelow. Equation 6.4 UninhibitedGrowth: IfapopulationincreasesaccordingtoTheLawofUninhibitedGrowth,thenumberoforganisms N attime t isgivenbytheformula N t = N 0 e kt ; where N = N 0 read` N nought'istheinitialnumberoforganismsand k> 0istheconstant ofproportionalitywhichsatisestheequation instantaneousrateofchangeof N t attime t = kN t ItisworthtakingsometimetocompareEquations6.3and6.4.InEquation6.3,weuse P todenote theinitialinvestment;inEquation6.4,weuse N 0 todenotetheinitialpopulation.InEquation 6.3, r denotestheannualinterestrate,andsoitshouldn'tbetoosurprisingthatthe k inEquation 6.4correspondstoagrowthrateaswell.WhileEquations6.3and6.4lookentirelydierent,they bothrepresentthesamemathematicalconcept. Example 6.5.2 Inordertoperformarthrosclerosisresearch,epithelialcellsareharvestedfrom discardedumbilicaltissueandgrowninthelaboratory.Atechnicianobservesthatacultureof twelvethousandcellsgrowstovemillioncellsinoneweek.AssumingthatthecellsfollowThe LawofUninhibitedGrowth,ndaformulaforthenumberofcells, N ,inthousands,after t days. Solution. Webeginwith N t = N 0 e kt .Since N istogivethenumberofcells inthousands wehave N 0 =12,so N t =12 e kt .Inordertocompletetheformula,weneedtodeterminethe growthrate k .Weknowthatafteroneweek,thenumberofcellshasgrowntovemillion.Since t 9 TheaveragerateofchangeofafunctionoveranintervalwasrstintroducedinSection2.1. Instantaneous rates ofchangearethebusinessofCalculus,asismentionedonPage111.

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6.5ApplicationsofExponentialandLogarithmicFunctions367 measuresdaysandtheunitsof N areinthousands,thistranslatesmathematicallyto N =5000. Wegettheequation12 e 7 k =5000whichgives k = 1 7 ln )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1250 3 .Hence, N t =12 e t 7 ln 1250 3 .Of course,inpractice,wewouldapproximate k tosomedesiredaccuracy,say k 0 : 8618,whichwe caninterpretasan86 : 18%dailygrowthrateforthecells. WhereasEquations6.3and6.4modelthegrowthofquantities,wecanuseequationslikethemto describethedeclineofquantities.Oneexamplewe'veseenalreadyisExample6.1.1inSection6.1. There,thevalueofacardeclinedfromitspurchasepriceof$25 ; 000tonothingatall.Anotherreal worldphenomenonwhichfollowssuitisradioactivedecay.Thereareelementswhichareunstable andemitenergyspontaneously.Indoingso,theamountoftheelementitselfdiminishes.The assumptionbehindthismodelisthattherateofdecayofanelementataparticulartimeisdirectly proportionaltotheamountoftheelementpresentatthattime.Inotherwords,themoreofthe elementthereis,thefastertheelementdecays.Thisispreciselythesamekindofhypothesiswhich drivesTheLawofUninhibitedGrowth,andassuch,theequationgoverningradioactivedecayis hauntinglysimilartoEquation6.4withtheexceptionthattherateconstant k isnegative. Equation 6.5 RadioactiveDecay Theamountofaradioactiveelement A attime t isgiven bytheformula A t = A 0 e kt ; where A = A 0 istheinitialamountoftheelementand k< 0istheconstantofproportionality whichsatisestheequation instantaneousrateofchangeof A t attime t = kA t Example 6.5.3 Iodine-131isacommonlyusedradioactiveisotopeusedtohelpdetecthowwell thethyroidisfunctioning.SupposethedecayofIodine-131followsthemodelgiveninEquation6.5, andthatthehalf-life 10 ofIodine-131isapproximately8days.If5gramsofIodine-131ispresent initially,ndafunctionwhichgivestheamountofIodine-131, A ,ingrams, t dayslater. Solution. Sincewestartwith5gramsinitially,Equation6.5gives A t =5 e kt .Sincethehalf-lifeis 8days,ittakes8daysforhalfoftheIodine-131todecay,leavinghalfofitbehind.Hence, A =2 : 5 whichmeans5 e 8 k =2 : 5.Solving,weget k = 1 8 ln )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 = )]TJ/F34 7.9701 Tf 9.68 5.374 Td [(ln 8 )]TJ/F15 10.9091 Tf 20 0 Td [(0 : 08664,whichwecaninterpret asalossofmaterialatarateof8 : 664%daily.Hence, A t =5 e )]TJ/F38 5.9776 Tf 7.782 4.025 Td [(t ln 8 5 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 08664 t Wenowturnourattentiontosomemoremathematicallysophisticatedmodels.Onesuchmodel isNewton'sLawofCooling,whichwerstencounteredinExample6.1.2ofSection6.1.Inthat examplewehadacupofcoeecoolingfrom160 Ftoroomtemperature70 Faccordingtothe formula T t =70+90 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t ,where t wasmeasuredinminutes.Inthissituation,weknowthe physicallimitofthetemperatureofthecoeeisroomtemperature, 11 andthedierentialequation whichgivesrisetoourformulafor T t takesthisintoaccount.Whereastheradioactivedecay 10 Thetimeittakesforhalfofthesubstancetodecay. 11 TheSecondLawofThermodynamicsstatesthatheatcanspontaneouslyowfromahotterobjecttoacolder one,butnottheotherwayaround.Thus,thecoeecouldnotcontinuetoreleaseheatintotheairsoastocoolbelow roomtemperature.

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368ExponentialandLogarithmicFunctions modelhadarateofdecayattime t directlyproportionaltotheamountoftheelementwhich remainedattime t ,Newton'sLawofCoolingstatesthattherateofcoolingofthecoeeatagiven time t isdirectlyproportionaltohowmuchofatemperaturegap existsbetweenthecoeeattime t androomtemperature,notthetemperatureofthecoeeitself.Inotherwords,thecoeecools fasterwhenitisrstserved,andasitstemperaturenearsroomtemperature,thecoeecoolsever moreslowly.Ofcourse,ifwetakeanitemfromtherefrigeratorandletitsitoutinthekitchen, theobject'stemperaturewillrisetoroomtemperature,andsincethephysicsbehindwarmingand coolingisthesame,wecombinebothcasesintheequationbelow. Equation 6.6 Newton'sLawofCoolingWarming: Thetemperature T ofanobjectat time t isgivenbytheformula T t = T a + T 0 )]TJ/F53 10.9091 Tf 10.909 0 Td [(T a e )]TJ/F37 7.9701 Tf 6.587 0 Td [(kt ; where T = T 0 istheinitialtemperatureoftheobject, T a istheambienttemperature a and k> 0istheconstantofproportionalitywhichsatisestheequation instantaneousrateofchangeof T t attime t = k T t )]TJ/F53 10.9091 Tf 10.909 0 Td [(T a a Thatis,thetemperatureofthesurroundings. Ifwere-examinethesituationinExample6.1.2with T 0 =160, T a =70,and k =0 : 1,weget, accordingtoEquation6.6, T t =160+ )]TJ/F15 10.9091 Tf 11.585 0 Td [(70 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 1 t whichreducestotheoriginalformula given.Therateconstant k =0 : 1indicatesthecoeeiscoolingatarateequalto10%ofthe dierencebetweenthetemperatureofthecoeeanditssurroundings.NoteinEquation6.6that theconstant k ispositiveforboththecoolingandwarmingscenarios.Whatdeterminesifthe function T t isincreasingordecreasingisif T 0 theinitialtemperatureoftheobjectisgreater than T a theambienttemperatureorvice-versa,asweseeinournextexample. Example 6.5.4 A40 Froastiscookedina350 Foven.After2hours,thetemperatureofthe roastis125 F. 1.AssumingthetemperatureoftheroastfollowsNewton'sLawofWarming,ndaformulafor thetemperatureoftheroast T asafunctionofitstimeintheoven, t ,inhours. 2.Theroastisdonewhentheinternaltemperaturereaches165 F.Whenwilltheroastbedone? Solution. 1.Theinitialtemperatureoftheroastis40 F,so T 0 =40.Theenvironmentinwhichwe areplacingtheroastisthe350 Foven,so T a =350.Newton'sLawofWarmingtellsus T t =350+ )]TJ/F15 10.9091 Tf 11.014 0 Td [(350 e )]TJ/F37 7.9701 Tf 6.587 0 Td [(kt ,or T t =350 )]TJ/F15 10.9091 Tf 11.014 0 Td [(310 e )]TJ/F37 7.9701 Tf 6.586 0 Td [(kt .Todetermine k ,weusethefactthat after2hours,theroastis125 F,whichmeans T =125.Thisgivesrisetotheequation 350 )]TJ/F15 10.9091 Tf 10.909 0 Td [(310 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 k =125whichyields k = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 ln )]TJ/F34 7.9701 Tf 6.196 -4.542 Td [(45 62 0 : 1602.Thetemperaturefunctionis T t =350 )]TJ/F15 10.9091 Tf 10.909 0 Td [(310 e t 2 ln 45 62 350 )]TJ/F15 10.9091 Tf 10.909 0 Td [(310 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 1602 t :

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6.5ApplicationsofExponentialandLogarithmicFunctions369 2.Todeterminewhentheroastisdone,weset T t =165.Thisgives350 )]TJ/F15 10.9091 Tf 10.841 0 Td [(310 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 1602 t =165 whosesolutionis t = )]TJ/F34 7.9701 Tf 19.325 4.295 Td [(1 0 : 1602 ln )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(37 62 3 : 22.Ittakesroughly3hoursand15minutestocook theroastcompletely. Ifwehadtakenthetimetograph y = T t inExample6.5.4,wewouldhavefoundthehorizontal asymptotetobe y =350,whichcorrespondstothetemperatureoftheoven.Wecanalsoarrive atthisconclusionbyapplyingabitof`numbersense'.As t !1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 1602 t verybig )]TJ/F15 10.9091 Tf 8.485 0 Td [(so that e )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 1602 t verysmall+.Thelargerthevalueof t ,thesmaller e )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 1602 t becomessothat T t 350 )]TJ/F15 10.9091 Tf 11.232 0 Td [(verysmall+,whichindicatesthegraphof y = T t isapproachingitshorizontal asymptote y =350frombelow.Physically,thismeanstheroastwilleventuallywarmupto350 F. 12 Thefunction T issometimescalleda limited growthmodel,sincethefunction T remainsbounded as t !1 .IfweapplytheprinciplesbehindNewton'sLawofCoolingtoabiologicalexample,it saysthegrowthrateofapopulationisdirectlyproportionaltohowmuchroomthepopulationhas togrow.Inotherwords,themoreroomforexpansion,thefasterthegrowthrate.The logistic growthmodelcombinesTheLawofUninhibitedGrowthwithlimitedgrowthandstatesthatthe rateofgrowthofapopulationvariesjointlywiththepopulationitselfaswellastheroomthe populationhastogrow. Equation 6.7 LogisticGrowth: Ifapopulationbehavesaccordingtotheassumptionsof logisticgrowth,thenumberoforganisms N attime t isgivenbytheequation N t = L 1+ Ce )]TJ/F37 7.9701 Tf 6.586 0 Td [(kLt ; where N = N 0 istheinitialpopulation, L isthelimitingpopulation a C isameasureofhow muchroomthereistogrowgivenby C = L N 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 : and k> 0istheconstantofproportionalitywhichsatisestheequation instantaneousrateofchangeof N t attime t = kN t L )]TJ/F53 10.9091 Tf 10.909 0 Td [(N t a Thatis,as t !1 N t L Thelogisticfunctionisusednotonlytomodelthegrowthoforganisms,butisalsooftenusedto modelthespreadofdiseaseandrumors. 13 Example 6.5.5 Thenumberofpeople N ,inhundreds,atalocalcommunitycollegewhohave heardtherumor`CarlisafraidofVirginiaWoolf'canbemodeledusingthelogisticequation N t = 84 1+2799 e )]TJ/F37 7.9701 Tf 6.587 0 Td [(t ; where t 0isthenumberofdaysafterApril1,2009. 12 atwhichpointitwouldbemoretoastthanroast. 13 Whichcanbejustasdamagingasdiseases.

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370ExponentialandLogarithmicFunctions 1.Findandinterpret N 2.Findandinterprettheendbehaviorof N t 3.Howlonguntil4200peoplehaveheardtherumor? 4.Checkyouranswersto2and3usingyourcalculator. Solution. 1.Wend N = 84 1+2799 e 0 = 84 2800 = 3 100 .Since N t measuresthenumberofpeoplewhohave heardtherumorinhundreds, N correspondsto3people.Since t =0correspondstoApril 1,2009,wemayconcludethatonthatday,3peoplehaveheardtherumor. 14 2.Wecouldsimplynotethat N t iswrittenintheformofEquation6.7,andidentify L =84. However,toseewhytheansweris84,weproceedanalytically.Sincethedomainof N is restrictedto t 0,theonlyendbehaviorofsignicanceis t !1 .Aswe'veseenbefore, 15 as t !1 ,have1997 e )]TJ/F37 7.9701 Tf 6.587 0 Td [(t 0 + andso N t 84 1+ verysmall+ 84.Hence,as t !1 N t 84.Thismeansthatastimegoesby,thenumberofpeoplewhowillhaveheardthe rumorapproaches8400. 3.Tondhowlongittakesuntil4200peoplehaveheardtherumor,weset N t =42.Solving 84 1+2799 e )]TJ/F38 5.9776 Tf 5.756 0 Td [(t =42gives t =ln 7 : 937.Ittakesaround8daysuntil4200peoplehave heardtherumor. 4.Wegraph y = N x usingthecalculatorandseethattheline y =84isthehorizontal asymptoteofthegraph,conrmingouranswertopart2,andthegraphintersectstheline y =42at x =ln 7 : 937,whichconrmsouranswertopart3. y = f x = 84 1+2799 e )]TJ/F38 5.9776 Tf 5.756 0 Td [(x and y = f x = 84 1+2799 e )]TJ/F38 5.9776 Tf 5.756 0 Td [(x and y =84 y =42 14 Or,morelikely,threepeoplestartedtherumor.I'dwagerJe,Jamie,andJasonstartedit.Somuchfortelling yourbestfriendssomethingincondence! 15 See,forexample,Example6.1.2.

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6.5ApplicationsofExponentialandLogarithmicFunctions371 Ifwetakethetimetoanalyzethegraphof y = N x above,wecanseegraphicallyhowlogistic growthcombinesfeaturesofuninhibitedandlimitedgrowth.Thecurveseemstorisesteeply,then atsomepoint,beginstolevelo.Thepointatwhichthishappensiscalledan inectionpoint orissometimescalledthe`pointofdiminishingreturns'.Atthispoint,eventhoughthefunctionis stillincreasing,therateatwhichitdoessobeginstodecline.Itturnsoutthepointofdiminishing returnsalwaysoccursathalfthelimitingpopulation.Inourcase,when y =42.Whilethese conceptsaremorepreciselyquantiedusingCalculus,belowaretwoviewsofthegraphof y = N x oneontheinterval[0 ; 8],theotheron[8 ; 15].Theformerlooksstrikinglylikeuninhibitedgrowth; thelatterlikelimitedgrowth. y = f x = 84 1+2799 e )]TJ/F38 5.9776 Tf 5.757 0 Td [(x for y = f x = 84 1+2799 e )]TJ/F38 5.9776 Tf 5.756 0 Td [(x for 0 x 88 x 16 6.5.2ApplicationsofLogarithms Justasmanyphysicalphenomenacanbemodeledbyexponentialfunctions,thesameistrueof logarithmicfunctions.InExercises4a,4band4cofSection6.1,weshowedthatlogarithmsare usefulinmeasuringtheintensitiesofearthquakestheRichterscale,sounddecibelsandacidsand basespH.Wenowpresentyetadierentuseoftheabasiclogarithmfunction,passwordstrength Example 6.5.6 Theinformationentropy H ,inbits,ofarandomlygeneratedpasswordconsisting of L charactersisgivenby H = L log 2 N ,where N isthenumberofpossiblesymbolsforeach characterinthepassword.Ingeneral,thehighertheentropy,thestrongerthepassword. 1.Ifa7charactercase-sensitive 16 passwordiscomprisedoflettersandnumbersonly,ndthe associatedinformationentropy. 2.Howmanypossiblesymboloptionspercharacterisrequiredtoproducea7characterpassword withaninformationentropyof50bits? Solution. 1.Thereare26lettersinthealphabet,52ifupperandlowercaselettersarecountedasdierent. Thereare10digitsthrough9foratotalof N =62symbols.Sincethepasswordistobe 7characterslong, L =7.Thus, H =7log 2 = 7ln ln 41 : 68. 16 Thatis,upperandlowercaselettersaretreatedasdierentcharacters.

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372ExponentialandLogarithmicFunctions 2.Wehave L =7and H =50andweneedtond N .Solvingtheequation50=7log 2 N gives N =2 50 = 7 141 : 323,sowewouldneed142dierentsymbolstochoosefrom. 17 Chemicalsystemsknownasbuersolutions havetheabilitytoadjusttosmallchangesinacidityto maintainarangeofpHvalues.Buersolutionshaveawidevarietyofapplicationsfrommaintaining ahealthyshtanktoregulatingthepHlevelsinblood.OurnextexampleshowshowthepHin abuersolutionisalittlemorecomplicatedthanthepHwerstencounteredinExercise4cin Section6.1. Example 6.5.7 Bloodisabuersolution.Whencarbondioxideisabsorbedintothebloodstream itproducescarbonicacidandlowersthepH.Thebodycompensatesbyproducingbicarbonate,a weakbasetopartiallyneutralizetheacid.Theequation 18 whichmodelsbloodpHinthissituation ispH=6 : 1+log )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(800 x ,where x isthepartialpressureofcarbondioxideinarterialblood,measured intorr.FindthepartialpressureofcarbondioxideinarterialbloodifthepHis7 : 4. Solution. WesetpH=7 : 4andget7 : 4=6 : 1+log )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(800 x ,orlog )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(800 x =1 : 3.Solving,wend x = 800 10 1 : 3 40 : 09.Hence,thepartialpressureofcarbondioxideinthebloodisabout40torr. Anotherplacelogarithmsareusedisindataanalysis.Suppose,forinstance,wewishtomodel thespreadofinuenzaAH1N1,theso-called`SwineFlu'.BelowisdatatakenfromtheWorld HealthOrganizationWHO where t representsthenumberofdayssinceApril28,2009,and N representsthenumberofconrmedcasesofH1N1virusworldwide. t 1 2 3 4 5 6 7 8 9 10 11 12 13 N 148 257 367 658 898 1085 1490 1893 2371 2500 3440 4379 4694 t 14 15 16 17 18 19 20 N 5251 5728 6497 7520 8451 8480 8829 Makingascatterplotofthedatatreating t astheindependentvariableand N asthedependent variablegives Whichmodelsaresuggestedbytheshapeofthedata?ThinkingbackSection2.5,wetrya QuadraticRegression,withprettygoodresults. 17 Sincethereareonly94distinctASCIIkeyboardcharacters,toachievethisstrength,thenumberofcharactersin thepasswordshouldbeincreased. 18 DerivedfromtheHenderson-HasselbalchEquation .SeeExercise8inSection6.2.Hasselbalchhimselfwas studyingcarbondioxidedissolvinginblood-aprocesscalledmetabolicacidosis .

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6.5ApplicationsofExponentialandLogarithmicFunctions373 However,isthereanyscienticreasonforthedatatobequadratic?Arethereothermodelswhich tthedataequallywell,orbetter?Scientistsoftenuselogarithmsinanattemptto`linearize'data sets-inotherwords,transformthedatasetstoproduceoneswhichresultinstraightlines.Tosee howthiscouldwork,supposeweguessedtherelationshipbetween N and t wassomekindofpower function,notnecessarilyquadratic,say N = Bt A .Totrytodeterminethe A and B ,wecantake thenaturallogofbothsidesandgetln N =ln )]TJ/F53 10.9091 Tf 5 -8.836 Td [(Bt A .Usingpropertiesoflogstoexpandtheright handsideofthisequation,wegetln N = A ln t +ln B .Ifweset X =ln t and Y =ln N ,this equationbecomes Y = AX +ln B .Inotherwords,wehavealinewithslope A and Y -intercept ln B .So,insteadofplotting N versus t ,weplotln N versusln t ln t 0 0 : 693 1 : 099 1 : 386 1 : 609 1 : 792 1 : 946 2 : 079 2 : 197 2 : 302 2 : 398 2 : 485 2 : 565 ln N 4 : 997 5 : 549 5 : 905 6 : 489 6 : 800 6 : 989 7 : 306 7 : 546 7 : 771 7 : 824 8 : 143 8 : 385 8 : 454 ln t 2 : 639 2 : 708 2 : 773 2 : 833 2 : 890 2 : 944 2 : 996 ln N 8 : 566 8 : 653 8 : 779 8 : 925 9 : 042 9 : 045 9 : 086 Runningalinearregressiononthedatagives Theslopeoftheregressionlineis a 1 : 512whichcorrespondstoourexponent A .The y -intercept b 4 : 513correspondstoln B ,sothat B 91 : 201.Inotherwords,wegetthemodel N = 91 : 201 t 1 : 512 ,somethingfromSection5.3.Ofcourse,thecalculatorhasabuilt-in`PowerRegression' feature.Ifweapplythistoouroriginaldataset,wegetthesamemodelwearrivedatbefore. 19 19 Criticsmayquestionwhytheauthorsofthebookhavechosentoevendiscusslinearizationofdatawhenthe calculatorhasaPowerRegressionbuilt-inandreadytogo.Ourresponse:talktoyoursciencefaculty.

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374ExponentialandLogarithmicFunctions Thisisallwellandgood,butthequadraticmodelappearstotthedatabetter,andwe'veyetto mentionanyscienticprinciplewhichwouldleadustobelievetheactualspreadoftheufollows anykindofpowerfunctionatall.Ifwearetoattackthisdatafromascienticperspective,itdoes seemtomakesensethat,atleastintheearlystagesoftheoutbreak,themorepeoplewhohave theu,thefasteritwillspread,whichleadsustoproposinganuninhibitedgrowthmodel.Ifwe assume N = Be At then,takinglogsasbefore,wegetln N = At +ln B .Ifweset X = t and Y =ln N ,then,onceagain,weget Y = AX +ln B ,alinewithslope A and Y -interceptln B Plottingln N versus t andgivesthefollowinglinearregression. Weseetheslopeis a 0 : 202andwhichcorrespondsto A inourmodel,andthe y -interceptis b 5 : 596whichcorrespondstoln B .Weget B 269 : 414,sothatourmodelis N =269 : 414 e 0 : 202 t Ofcourse,thecalculatorhasabuilt-in`ExponentialRegression'featurewhichproduceswhat appearstobeadierentmodel N =269 : 414 : 22333419 t .Usingpropertiesofexponents,wewrite e 0 : 202 t = )]TJ/F53 10.9091 Tf 5 -8.837 Td [(e 0 : 202 t : 223848 t ,which,hadwecarriedmoredecimalplaces,wouldhavematched thebaseofthecalculatormodelexactly. Theexponentialmodeldidn'ttthedataaswellasthequadraticorpowerfunctionmodel,but itstandstoreasonthat,perhaps,thespreadoftheuisnotunlikethatofthespreadofarumor

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6.5ApplicationsofExponentialandLogarithmicFunctions375 andthatalogisticmodelcanbeusedtomodelthedata.Thecalculatordoeshavea`Logistic Regression'feature,andusingitproducesthemodel N = 10739 : 147 1+42 : 416 e 0 : 268 t Thisappearstobeanexcellentt,butthereisnofriendlycoecientofdetermination, R 2 ,by whichtojudgethisnumerically.Therearegoodreasonsforthis,buttheyarefarbeyondthescope ofthetext.Whichofthemodels,quadratic,power,exponential,orlogisticisthe`bestmodel'? Ifby`best'wemean`tsclosesttothedata,'thenthequadraticandlogisticmodelsarearguably thewinnerswiththepowerfunctionmodelaclosesecond.However,ifwethinkaboutthescience behindthespreadoftheu,thelogisticmodelgetsanedge.Foronething,ittakesintoaccount thatonlyanitenumberofpeoplewillevergettheuaccordingtoourmodel,10 ; 739,whereas thequadraticmodelpredictsnolimittothenumberofcases.Aswehavestatedseveraltimes beforeinthetext,mathematicalmodels,regardlessoftheirsophistication,arejustthat:models, andtheyallhavetheirlimitations. 20 20 Speakingoflimitations,asofJune3,2009,therewere19,273conrmedcasesofinuenzaAH1N1.Thisis wellaboveourpredictionof10,739.Eachtimeanewreportisissued,thedatasetincreasesandthemodelmustbe recalculated.Weleavethisrecalculationtothereader.

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376ExponentialandLogarithmicFunctions 6.5.3Exercises 1.OnMay,31,2009,theAnnualPercentageRatelistedatmybankforregularsavingsaccounts was0 : 25%compoundedmonthly.UseEquation6.2toanswerthefollowing. aIf P =2000whatis A ? bSolvetheequation A t =4000for t cWhatprincipal P shouldbeinvestedsothattheaccountbalanceis $ 2000isthreeyears? 2.Mybankalsooersa36-monthCerticateofDepositCDwithanAPRof2 : 25%. aIf P =2000whatis A ? bSolvetheequation A t =4000for t cWhatprincipal P shouldbeinvestedsothattheaccountbalanceis $ 2000isthreeyears? dTheAnnualPercentageYieldisthesimple interestratethatreturnsthesameamountof interestafteroneyearasthecompoundinterestdoes.Withthehelpofyourclassmates, computetheAPYforthisinvestment. 3.UseEquation6.2toshowthatthetimeittakesforaninvestmenttodoubleinvaluedoes not dependontheprincipal P ,butrather,dependsonlyontheAPRandthenumberof compoundingsperyear.Let n =12andwiththehelpofyourclassmatescomputethe doublingtimeforavarietyofrates r .ThenlookuptheRuleof72andcompareyouranswers towhatthatrulesays.Ifyou'rereallyinterested 21 innancialmathematics,youcouldalso compareandcontrasttheRuleof72withtheRuleof70andtheRuleof69. 4.UseEquation6.5toshowthat k = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(ln h where h isthehalf-lifeoftheradioactiveisotope. 5.Thehalf-lifeoftheradioactiveisotopeCarbon-14isabout5730years. aUseEquation6.5toexpresstheamountofCarbon-14leftfromaninitial N milligrams asafunctionoftime t inyears. bWhatpercentageoftheoriginalamountofCarbon-14isleftafter20,000years? cIfanoldwoodentoolisfoundinacaveandtheamountofCarbon-14presentinitis estimatedtobeonly42%oftheoriginalamount,approximatelyhowoldisthetool? dRadiocarbondatingisnotaseasyastheseexercisesmightleadyoutobelieve.With thehelpofyourclassmates,researchradiocarbondatinganddiscusswhyourmodelis somewhatover-simplied. 21 Awesomepun!

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6.5ApplicationsofExponentialandLogarithmicFunctions377 6.Carbon-14cannotbeusedtodateinorganicmaterialsuchasrocks,buttherearemanyother methodsofradiometricdatingwhichestimatetheageofrocks.Oneofthem,RubidiumStrontiumdating,usesRubidium-87whichdecaystoStrontium-87withahalf-lifeof50 billionyears.UseEquation6.5toexpresstheamountofRubidium-87leftfromaninitial2.3 microgramsasafunctionoftime t in billions ofyears.Researchthisandotherradiometric techniquesanddiscussthemarginsoferrorforvariousmethodswithyourclassmates. 7.InExample6.1.1inSection6.1,theexponentialfunction V x =25 )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(4 5 x wasusedtomodel thevalueofacarovertime.Usethepropertiesoflogsand/orexponentstorewritethemodel intheform V t =25 e kt 8.Aporkroastwastakenoutofahardwoodsmokerwhenitsinternaltemperaturehadreached 180 Fanditwasallowedtorestina75 Fhousefor20minutesafterwhichitsinternal temperaturehaddroppedto170 F. 22 Assumingthatthetemperatureoftheroastfollows Newton'sLawofCoolingEquation6.6, aExpressthetemperature T asafunctionoftime t bFindthetimeatwhichtheroastwouldhavedroppedto140 Fhaditnotbeencarved andeaten. 9.InreferencetoExercise12inSection5.3,ifFritzytheFox'sspeedisthesameasChewbacca theBunny'sspeed,Fritzy'spursuitcurveisgivenby y x = 1 4 x 2 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 4 ln x )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 4 Useyourcalculatortographthispathfor x> 0.Describethebehaviorof y as x 0 + and interpretthisphysically. 10.Thecurrent i measuredinampsinacertainelectroniccircuitwithaconstantimpressed voltageof120voltsisgivenby i t =2 )]TJ/F15 10.9091 Tf 11.054 0 Td [(2 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(10 t where t 0isthenumberofsecondsafter thecircuitisswitchedon.Determinethevalueof i as t !1 .Thisiscalledthe steady state current. 11.IfthevoltageinthecircuitinExercise10aboveisswitchedoafter30seconds,thecurrent isgivenbythepiecewise-denedfunction i t = 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(10 t if0 t< 30 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(300 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(10 t +300 if t 30 Withthehelpofyourcalculator,graph y = i t anddiscusswithyourclassmatesthephysical signicanceofthetwopartsofthegraph0 t< 30and t 30. 22 ThisroastwasenjoyedbyJeandhisfamilyonJune10,2009.Thisisrealdata,folks!

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378ExponentialandLogarithmicFunctions 12.InExercise7inSection2.3,westatedthatthecableofasuspensionbridgeformedaparabola butthatafreehangingcabledidnot.Afreehangingcableformsacatenary anditsbasic shapeisgivenby y = e x + e )]TJ/F37 7.9701 Tf 6.587 0 Td [(x 2 .Useyourcalculatortographthisfunction.Whatareits domainandrange?Whatisitsendbehavior?Isitinvertible?Howdoyouthinkitisrelated tothefunctiongiveninExercise5inSection6.3andtheonegivenintheanswertoExercise6 inSection6.4?Whenippedupsidedown,thecatenarymakesanarch.Infact,theGateway ArchinSt.Louis,Missourihastheshape y =757 : 7 )]TJ/F15 10.9091 Tf 10.936 0 Td [(127 : 7 e x 127 : 7 + e )]TJ/F38 5.9776 Tf 14.137 3.259 Td [(x 127 : 7 2 where x and y aremeasuredinfeetand )]TJ/F15 10.9091 Tf 8.484 0 Td [(315 x 315.Whatisthehighestpointonthearch? 13.InExercise4ainSection2.5,weexaminedthedatasetgivenbelowwhichshowedhowtwo catsandtheirsurvivingospringcanproduceover80millioncatsinjusttenyears.Itis virtuallyimpossibletoseethisdataplottedonyourcalculator,soplot x versusln x aswas doneonpage374.Findalinearmodelforthisnewdataandcommentonitsgoodnessoft. Findanexponentialmodelfortheoriginaldataandcommentonitsgoodnessoft. Year x 1 2 3 4 5 6 7 8 9 10 Numberof Cats N x 12 66 382 2201 12680 73041 420715 2423316 13968290 80399780 14.EachMondayduringtheregistrationperiodbeforetheFallSemesteratLCCC,theEnrollment PlanningCouncilgetsareportpreparedbythedataanalystsinInstitutionalEectivenessand Planning. 23 Whiletheongoingenrollmentdataisanalyzedinmanydierentways,weshall focusonlyontheoverallheadcount.BelowisachartoftheenrollmentdataforFallSemester 2008.Itstarts21weeksbeforeOpeningDay"andendsonDay15"ofthesemester,but wehaverelabeledthetoprowtobe x =1through x =24sothatthemathiseasier.Thus, x =22isOpeningDay. Week x 1 2 3 4 5 6 7 8 Total Headcount 1194 1564 2001 2475 2802 3141 3527 3790 Week x 9 10 11 12 13 14 15 16 Total Headcount 4065 4371 4611 4945 5300 5657 6056 6478 Week x 17 18 19 20 21 22 23 24 Total Headcount 7161 7772 8505 9256 10201 10743 11102 11181 Withthehelpofyourclassmates,ndamodelforthisdata.Unlikemostofthephenomena wehavestudiedinthissection,thereisnosingledierentialequationwhichgovernsthe 23 TheauthorsthankDr.WendyMarleyandherstaforthisdataandDr.MarciaBallingerforthepermissionto useitinthisproblem.

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6.5ApplicationsofExponentialandLogarithmicFunctions379 enrollmentgrowth.Thusthereisnoscienticreasontorelyonalogisticfunctioneven thoughthedataplotmayleadustothatmodel.Whataresomefactorswhichinuence enrollmentatacommunitycollegeandhowcanyoutakethoseintoaccountmathematically? 15.Asofthewritingofthissetofexercises,theEnrollmentPlanningReportforFallSemester 2009hasonly10datapointsfortherst10weeksoftheregistrationperiod.Thosenumbers aregivenbelow. Week x 1 2 3 4 5 6 7 8 9 10 Total Headcount 1380 2000 2639 3153 3499 3831 4283 4742 5123 5398 Withthehelpofyourclassmates,ndamodelforthisdataandmakeapredictionfor theOpeningDayenrollmentaswellastheDay15enrollment.WARNING:Thisyear's registrationperiodisoneweekshorterthanitwasin2008soOpeningDaywouldbe x =21 andDay15is x =23. 16.Accordingtothiswebsite ,thenumberofactiveusersofFacebookhasgrownsignicantly sinceitsinitiallaunchfromaHarvarddormroominFebruary2004.Thechartbelowhas theapproximatenumber U x ofactiveusers,inmillions x monthsafterFebruary2004.For example,therstentry ; 1meansthattherewere1millionactiveusersinDecember2004 andthelastentry ; 200meansthattherewere200millionactiveusersinApril2009. Month x 10 22 34 38 44 54 59 60 62 ActiveUsersin Millions U x 1 5.5 12 20 50 100 150 175 200 Withthehelpofyourclassmates,ndamodelforthisdata.

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380ExponentialandLogarithmicFunctions 6.5.4Answers 1.a A =2000 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+ 0 : 0025 12 12 8 $2040 : 40 b t = ln 12ln )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1+ 0 : 0025 12 277 : 29years c P = 2000 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+ 0 : 0025 12 36 $1985 : 06 2.a A =2000 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1+ 0 : 0225 12 12 8 $2394 : 03 b t = ln 12ln )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+ 0 : 0225 12 30 : 83years c P = 2000 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+ 0 : 0225 12 36 $1869 : 57 d )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+ 0 : 0225 12 12 1 : 0227sotheAPYis2.27% 5.a A t = Ne )]TJ/F43 7.9701 Tf 6.587 8.807 Td [( ln 5730 t Ne )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 00012097 t b A 0 : 088978 N soabout8.9%remains c t ln : 42 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 00012097 7171yearsold 6. A t =2 : 3 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 0138629 t 7. V t =25 e ln 4 5 t 25 e )]TJ/F34 7.9701 Tf 6.586 0 Td [(0 : 22314355 t 8.a T t =75+105 e )]TJ/F34 7.9701 Tf 6.587 0 Td [(0 : 005005 t bTheroastwouldhavecooledto140 Finabout95minutes. 9.Fromthegraph,itappearsthatas x 0 + y !1 .Thisisduetothepresenceoftheln x terminthefunction.ThismeansFritzywillnevercatchChewbacca,whichmakessensesince ChewbaccahasaheadstartandFritzyonlyrunsasfastashedoes. y x = 1 4 x 2 )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 4 ln x )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 4

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6.5ApplicationsofExponentialandLogarithmicFunctions381 10.Thesteadystatecurrentis2amps. 13.Thelinearregressiononthedatabelowis y =1 : 74899 x +0 : 70739with r 2 0 : 999995.This isanexcellentt. x 1 2 3 4 5 6 7 8 9 10 ln N x 2.4849 4.1897 5.9454 7.6967 9.4478 11.1988 12.9497 14.7006 16.4523 18.2025 N x =2 : 02869 : 74879 x =2 : 02869 e 1 : 74899 x with r 2 0 : 999995.Thisisalsoanexcellentt andcorrespondstoourlinearizedmodelbecauseln : 02869 0 : 70739.

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382ExponentialandLogarithmicFunctions

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Chapter7 HookedonConics 7.1IntroductiontoConics Inthischapter,westudythe ConicSections -literally`sectionsofacone.'Imagineadoublenappedconeasseenbelowbeing`sliced'byaplane. Ifweslicethewithahorizontalplanetheresultingcurveisa circle

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384HookedonConics Tiltingtheplaneeversoslightlyproducesan ellipse Iftheplanecutsparalleltothecone,wegeta parabola Ifweslicetheconewithaverticalplane,wegeta hyperbola Forawonderfulanimationdescribingtheconicsasintersectionsofplanesandcones,seeDr.Louis Talman'sMathematicsAnimatedWebsite .

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7.1IntroductiontoConics385 Iftheslicingplanecontainsthevertexofthecone,wegettheso-called`degenerate'conics:apoint, aline,ortwointersectinglines. Wewillfocusthediscussiononthenon-degeneratecases:circles,parabolas,ellipses,andhyperbolas,inthatorder.Todetermineequationswhichdescribethesecurves,wewillmakeuseoftheir denitionsintermsofdistances.

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386HookedonConics 7.2Circles Recallfromgeometrythatacirclecanbedeterminedbyxingapointcalledthecenteranda positivenumbercalledtheradiusasfollows. Definition 7.1 A circle withcenter h;k andradius r> 0isthesetofallpoints x;y inthe planewhosedistanceto h;k is r h;k r x;y Fromthepicture,weseethatapoint x;y isonthecircleifandonlyifitsdistanceto h;k is r WeexpressthisrelationshipalgebraicallyusingtheDistanceFormula,Equation1.1,as r = p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(h 2 + y )]TJ/F53 10.9091 Tf 10.909 0 Td [(k 2 Bysquaringbothsidesofthisequation,wegetanequivalentequationsince r> 0whichgivesus thestandardequationofacircle. Equation 7.1 TheStandardEquationofaCircle: Theequationofacirclewithcenter h;k andradius r> 0is x )]TJ/F53 10.9091 Tf 10.909 0 Td [(h 2 + y )]TJ/F53 10.9091 Tf 10.909 0 Td [(k 2 = r 2 : Example 7.2.1 Writethestandardequationofthecirclewithcenter )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3andradius5. Solution. Here, h;k = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 3and r =5,soweget x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 = 2 x +2 2 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 =25 Example 7.2.2 Graph x +2 2 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 =4.Findthecenterandradius. Solution. Fromthestandardformofacircle,Equation7.1,wehavethat x +2is x )]TJ/F53 10.9091 Tf 9.702 0 Td [(h ,so h = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 and y )]TJ/F15 10.9091 Tf 10.734 0 Td [(1is y )]TJ/F53 10.9091 Tf 10.733 0 Td [(k so k =1.Thistellsusthatourcenteris )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1.Furthermore, r 2 =4,so r =2. Thuswehaveacirclecenteredat )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1witharadiusof2.Graphinggivesus

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7.2Circles 387 x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 Ifweweretoexpandtheequationinthepreviousexampleandgatherupliketerms,insteadofthe easilyrecognizable x +2 2 + y )]TJ/F15 10.9091 Tf 10.985 0 Td [(1 2 =4,we'dbecontendingwith x 2 +4 x + y 2 )]TJ/F15 10.9091 Tf 10.985 0 Td [(2 y +1=0 : If we'regivensuchanequation,wecancompletethesquareineachofthevariablestoseeifitts theformgiveninEquation7.1byfollowingthestepsgivenbelow. ToPutaCircleintoStandardForm 1.Groupthesamevariablestogetherononesideoftheequationandputtheconstantonthe otherside. 2.Completethesquareonbothvariablesasneeded. 3.Dividebothsidesbythecoecientofthesquares.Forcircles,theywillbethesame. Example 7.2.3 Completethesquaretondthecenterandradiusof3 x 2 )]TJ/F15 10.9091 Tf 10.442 0 Td [(6 x +3 y 2 +4 y )]TJ/F15 10.9091 Tf 10.442 0 Td [(4=0. Solution. 3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x +3 y 2 +4 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(4=0 3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x +3 y 2 +4 y =4add4tobothsides 3 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +3 y 2 + 4 3 y =4factoroutleadingcoecients 3 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 +3 y 2 + 4 3 y + 4 9 =4+3+3 4 9 completethesquarein x y 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 +3 y + 2 3 2 = 25 3 factor x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 + y + 2 3 2 = 25 9 dividebothsidesby3 FromEquation7.1,weidentify x )]TJ/F15 10.9091 Tf 10.377 0 Td [(1as x )]TJ/F53 10.9091 Tf 10.378 0 Td [(h ,so h =1,and y + 2 3 as y )]TJ/F53 10.9091 Tf 10.378 0 Td [(k ,so k = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(2 3 .Hence,the centeris h;k = )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(2 3 .Furthermore,weseethat r 2 = 25 9 sotheradiusis r = 5 3 Itispossibletoobtainequationslike x )]TJ/F15 10.9091 Tf 10.42 0 Td [(3 2 + y +1 2 =0or x )]TJ/F15 10.9091 Tf 10.42 0 Td [(3 2 + y +1 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,neitherof whichdescribesacircle.Doyouseewhynot?Thereaderisencouragedtothinkaboutwhat,if

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388HookedonConics any,pointslieonthegraphsofthesetwoequations.ThenextexampleusestheMidpointFormula, Equation1.2,inconjunctionwiththeideaspresentedsofarinthissection. Example 7.2.4 Writethestandardequationofthecirclewhichhas )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 3and ; 4asthe endpointsofadiameter. Solution. Werecallthatadiameterofacircleisalinesegmentcontainingthecenterandtwo pointsonthecircle.Plottingthegivendatayields x y h;k r )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123 1 2 3 4 Sincethegivenpointsareendpointsofadiameter,weknowtheirmidpoint h;k isthecenterof thecircle.Equation1.2givesus h;k = x 1 + x 2 2 ; y 1 + y 2 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+2 2 ; 3+4 2 = 1 2 ; 7 2 Thediameterofthecircleisthedistancebetweenthegivenpoints,soweknowthathalfofthe distanceistheradius.Thus, r = 1 2 q x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 1 2 + y 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 1 2 = 1 2 p )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 = 1 2 p 3 2 +1 2 = p 10 2 Finally,since p 10 2 2 = 10 4 ,ouranswerbecomes x )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 2 + y )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(7 2 2 = 10 4

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7.2Circles 389 Weclosethissectionwiththemostimportant 1 circleinallofmathematics:the UnitCircle Definition 7.2 The UnitCircle isthecirclecenteredat ; 0witharadiusof1.Thestandard equationoftheUnitCircleis x 2 + y 2 =1 : Example 7.2.5 Findthepointsontheunitcirclewith y -coordinate p 3 2 Solution. Wereplace y with p 3 2 intheequation x 2 + y 2 =1toget x 2 + y 2 =1 x 2 + p 3 2 2 =1 3 4 + x 2 =1 x 2 = 1 4 x = r 1 4 x = 1 2 Ournalanswersare 1 2 ; p 3 2 and )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(1 2 ; p 3 2 1 Whilethismayseemlikeanopinion,itisindeedafact.

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390HookedonConics 7.2.1Exercises 1.Findthestandardequationofthecirclegiventhecenterandradiusandsketchitsgraph. aCenter )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(5,radius10 bCenter )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 7 13 ,radius 1 2 cCenter ;e 2 ,radius 3 p 91 dCenter ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(9,radiusln 2.Completethesquareinordertoputtheequationintostandardform.Identifythecenterand theradiusorexplainwhytheequationdoesnotrepresentacircle. a x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x + y 2 +10 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(25 b )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(36 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(56=0 c4 x 2 +4 y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(24 y +36=0 d x 2 + x + y 2 )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(6 5 y =1 3.Findthestandardequationofthecirclewithcenter ; 5whichpassesthrough )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2. 4.Verifythatthefollowing16pointsalllieontheUnitCircle: 1 ; 0, ; 1, p 2 2 ; p 2 2 1 2 ; p 3 2 and p 3 2 ; 1 2 5.TheGiantWheelatCedarPointisacirclewithdiameter128feetwhichsitsonan8foot tallplatformmakingitsoverallheightis136feet. 2 Findanequationforthewheelassuming thatitscenterliesonthe y -axis. 6.FindanequationforthefunctionrepresentedgraphicallybythetophalfoftheUnitCircle. ExplainhowthetransformationsisSection1.8canbeusedtoproduceafunctionwhosegraph iseitherthetoporbottomofanarbitrarycircle. 7.Circlesandsemi-circleshavebeenusedmanytimesthroughoutthebooktoillustratespecic concepts.Withthehelpofyourclassmates,ndtheExamplesorExercisesinwhichacircle orsemi-circlewasusedtodiscussthefollowing. aSymmetryorlackthereof bAgraphwhichfailedtheVerticalLineTest cLocalandabsoluteextrema dTransformations eUnusualsteepness 8.Findaone-to-onefunctionwhosegraphishalfofacircle.Hint:Thinkpiecewise. 2 Source:CedarPoint'swebpage .

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7.2Circles 391 7.2.2Answers 1.a x +1 2 + y +5 2 =100 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.757 0 Td [(19 )]TJ/F35 5.9776 Tf 5.756 0 Td [(15 5 b x +3 2 + )]TJ/F53 10.9091 Tf 5 -8.836 Td [(y )]TJ/F34 7.9701 Tf 14.222 4.295 Td [(7 13 2 = 1 4 x y )]TJ/F6 4.9813 Tf 6.952 2.346 Td [(7 2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F6 4.9813 Tf 6.952 2.345 Td [(5 2 1 26 7 13 27 26 c x )]TJ/F53 10.9091 Tf 10.909 0 Td [( 2 + )]TJ/F53 10.9091 Tf 5 -8.836 Td [(y )]TJ/F53 10.9091 Tf 10.909 0 Td [(e 2 2 =91 2 3 x y )]TJ/F6 4.9813 Tf 9.575 2.9 Td [(3 p 91 + 3 p 91 e 2 )]TJ/F6 4.9813 Tf 9.575 2.9 Td [(3 p 91 e 2 e 2 + 3 p 91 d x )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 2 + y +9 2 =ln 2 x y 5 )]TJ/F35 5.9776 Tf 7.453 0 Td [(ln 5 5+ln )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 7.454 0 Td [(ln )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 5.756 0 Td [(9+ln 2.a x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 + y +5 2 =4 Center ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(5,radius r =2 b x +9 2 + y 2 =25 Center )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 ; 0,radius r =5 c x 2 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 =0 Thisisnotacircle. d )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x + 1 2 2 + )]TJ/F53 10.9091 Tf 5 -8.836 Td [(y )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(3 5 2 = 161 100 Center )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 ; 3 5 ,radius r = p 161 10 3. x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 2 =65 5. x 2 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 2 =4096

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392HookedonConics 7.3Parabolas Wehavealreadylearnedthatthegraphofaquadraticfunction f x = ax 2 + bx + c a 6 =0is calleda parabola .Tooursurpriseanddelight,wemayalsodeneparabolasintermsofdistance. Definition 7.3 Let F beapointintheplaneand D bealinenotcontaining F .A parabola is thesetofallpointsequidistantfrom F and D .Thepoint F iscalledthe focus oftheparabola andtheline D iscalledthe directrix oftheparabola. Schematically,wehavethefollowing. F D V Eachdashedlinefromthepoint F toapointonthecurvehasthesamelengthasthedashedline fromthepointonthecurvetotheline D .Thepointsuggestivelylabeled V is,asyoushould expect,the vertex .Thevertexisthepointontheparabolaclosesttothefocus. Wewanttouseonlythedistancedenitionofparabolatoderivetheequationofaparabolaand, ifallisrightwiththeuniverse,weshouldgetanexpressionmuchlikethosestudiedinSection2.3. Let p denotethedirected 1 distancefromthevertextothefocus,whichbydenitionisthesameas thedistancefromthevertextothedirectrix.Forsimplicity,assumethatthevertexis ; 0and thattheparabolaopensupwards.Hence,thefocusis ;p andthedirectrixistheline y = )]TJ/F53 10.9091 Tf 8.485 0 Td [(p Ourpicturebecomes ;p x y y = )]TJ/F53 10.9091 Tf 8.485 0 Td [(p x;y x; )]TJ/F53 10.9091 Tf 8.485 0 Td [(p ; 0 Fromthedenitionofparabola,weknowthedistancefrom ;p to x;y isthesameasthe distancefrom x; )]TJ/F53 10.9091 Tf 8.484 0 Td [(p to x;y .UsingtheDistanceFormula,Equation1.1,weget 1 We'lltalkmoreaboutwhat`directed'meanslater.

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7.3Parabolas 393 p x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 2 + y )]TJ/F53 10.9091 Tf 10.909 0 Td [(p 2 = p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F53 10.9091 Tf 8.485 0 Td [(p 2 p x 2 + y )]TJ/F53 10.9091 Tf 10.909 0 Td [(p 2 = p y + p 2 x 2 + y )]TJ/F53 10.9091 Tf 10.909 0 Td [(p 2 = y + p 2 squarebothsides x 2 + y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 py + p 2 = y 2 +2 py + p 2 expandquantities x 2 =4 py gatherliketerms Solvingfor y yields y = x 2 4 p ,whichisaquadraticfunctionoftheformfoundinEquation2.4with a = 1 4 p andvertex ; 0. Weknowfrompreviousexperiencethatifthecoecientof x 2 isnegative,theparabolaopens downwards.Intheequation y = x 2 4 p thishappenswhen p< 0.Inourformulation,wesaythat p is a`directeddistance'fromthevertextothefocus:if p> 0,thefocusisabovethevertex;if p< 0, thefocusisbelowthevertex.The focallength ofaparabolais j p j Whatifwechoosetoplacethevertexatanarbitrarypoint h;k ?Wecaneitherusetransformations verticalandhorizontalshiftsfromSection1.8orre-derivetheequationfromDenition7.3to arriveatthefollowing. Equation 7.2 TheStandardEquationofaVertical a Parabola: Theequationofavertical parabolawithvertex h;k andfocallength j p j is x )]TJ/F53 10.9091 Tf 10.91 0 Td [(h 2 =4 p y )]TJ/F53 10.9091 Tf 10.909 0 Td [(k If p> 0,theparabolaopensupwards;if p< 0,itopensdownwards. a Thatis,aparabolawhichopenseitherupwardsordownwards. Noticethatinthestandardequationoftheparabolaabove,onlyoneofthevariables, x ,issquared. Thisisaquickwaytodistinguishanequationofaparabolafromthatofacirclebecauseinthe equationofacircle,bothvariablesaresquared. Example 7.3.1 Graph x +1 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3.Findthevertex,focus,anddirectrix. Solution. WerecognizethisastheformgiveninEquation7.2.Here, x )]TJ/F53 10.9091 Tf 11.056 0 Td [(h is x +1so h = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, and y )]TJ/F53 10.9091 Tf 9.872 0 Td [(k is y )]TJ/F15 10.9091 Tf 9.871 0 Td [(3so k =3.Hence,thevertexis )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 3.Wealsoseethat4 p = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8so p = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Since p< 0,thefocuswillbebelowthevertexandtheparabolawillopendownwards.Thedistancefrom thevertextothefocusis j p j =2,whichmeansthefocusis2unitsbelowthevertex.Ifwestartat )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 3andmovedown2units,wearriveatthefocus )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1.Thedirectrix,then,is2unitsabove thevertexandifwemove2unitsupfrom )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 3,we'dbeonthehorizontalline y =5.

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394HookedonConics x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11234 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 5 Ofalloftheinformationrequestedinthepreviousexample,onlythevertexispartofthegraph oftheparabola.Soinordertogetasenseoftheactualshapeofthegraph,weneedsomemore information.Whilewecouldplotafewpointsrandomly,amoreusefulmeasureofhowwidea parabolaopensisthelengthoftheparabola'slatusrectum. 2 The latusrectum ofaparabola isthelinesegmentparalleltothedirectrixwhichcontainsthefocus.Theendpointsofthelatus rectumare,then,twopointson`opposite'sidesoftheparabola.Graphically,wehavethefollowing. F thelatusrectum D V Itturnsout 3 thatthelengthofthelatusrectumis j 4 p j ,which,inlightofEquation7.2,iseasyto nd.Inourlastexample,forinstance,whengraphing x +1 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 y )]TJ/F15 10.9091 Tf 11.152 0 Td [(3,wecanusethefact thatthelengthofthelatusrectumis j)]TJ/F15 10.9091 Tf 16.65 0 Td [(8 j =8,whichmeanstheparabolais8unitswideatthe focus,tohelpgenerateamoreaccurategraphbyplottingpoints4unitstotheleftandrightofthe focus. Example 7.3.2 Findthestandardformoftheparabolawithfocus ; 1anddirectrix y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4. Solution. Sketchingthedatayields, 2 No,I'mnotmakingthisup. 3 Considerthisanexercisetoshowwhatfollows.

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7.3Parabolas 395 x y Thevertexliesonthisverticalline midwaybetweenthefocusandthedirectrix )]TJ/F34 7.9701 Tf 6.586 0 Td [(1123 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 Fromthediagram,weseetheparabolaopensupwards.Takeamomenttothinkaboutitifyou don'tseethatimmediately.Hence,thevertexliesbelowthefocusandhasan x -coordinateof2. Tondthe y -coordinate,wenotethatthedistancefromthefocustothedirectrixis1 )]TJ/F15 10.9091 Tf 10.525 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(4=5, whichmeansthevertexlies5 = 2unitshalfwaybelowthefocus.Startingat ; 1andmoving down5 = 2unitsleavesusat ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 = 2,whichisourvertex.Sincetheparabolaopensupwards,we know p ispositive.Thus p =5 = 2.PluggingallofthisdataintoEquation7.2giveus x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 =4 5 2 y )]TJ/F55 10.9091 Tf 10.909 15.382 Td [( )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(3 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 =10 y + 3 2 Ifweinterchangetherolesof x and y ,wecanproduce`horizontal'parabolas:parabolaswhichopen totheleftortotheright.Thedirectrices 4 ofsuchanimalswouldbeverticallinesandthefocus wouldeitherlietotheleftortotherightofthevertex.Atypical`horizontal'parabolaissketched below. F D V 4 pluralof`directrix'

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396HookedonConics Equation 7.3 TheStandardEquationofaHorizontalParabola: Theequationofa horizontalparabolawithvertex h;k andfocallength j p j is y )]TJ/F53 10.9091 Tf 10.909 0 Td [(k 2 =4 p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(h If p> 0,theparabolaopenstotheright;if p< 0,itopenstotheleft. Example 7.3.3 Graph y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 =12 x +1.Findthevertex,focus,anddirectrix. Solution. WerecognizethisastheformgiveninEquation7.3.Here, x )]TJ/F53 10.9091 Tf 11.056 0 Td [(h is x +1so h = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, and y )]TJ/F53 10.9091 Tf 11.477 0 Td [(k is y )]TJ/F15 10.9091 Tf 11.476 0 Td [(2so k =2.Hence,thevertexis )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 2.Wealsoseethat4 p =12so p =3. Since p> 0,thefocuswillbetherightofthevertexandtheparabolawillopentotheright.The distancefromthevertextothefocusis j p j =3,whichmeansthefocusis3unitstotheright.If westartat )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 2andmoveright3units,wearriveatthefocus ; 2.Thedirectrix,then,is3 unitstotheleftofthevertexandifwemoveleft3unitsfrom )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 2,we'dbeontheverticalline x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4.Sincethelengthofthelatusrectumis j 4 p j =12,theparabolais12unitswideatthe focus,andthustherearepoints6unitsaboveandbelowthefocusontheparabola. x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 5 6 7 8 Aswithcircles,notallparabolaswillcometousintheformsinEquations7.2or7.3.Ifwe encounteranequationwithtwovariablesinwhichexactlyonevariableissquared,wecanattempt toputtheequationintoastandardformusingthefollowingsteps. ToPutaParabolaintoStandardForm 1.Groupthevariablewhichissquaredononesideoftheequationandputthenon-squared variableandtheconstantontheotherside. 2.Completethesquareifnecessaryanddividebythecoecientoftheperfectsquare. 3.Factoroutthecoecientofthenon-squaredvariablefromitandtheconstant.

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7.3Parabolas 397 Example 7.3.4 Considertheequation y 2 +4 y +8 x =4.Putthisequationintostandardform andgraphtheparabola.Findthevertex,focus,anddirectrix. Solution. Weneedtogetaperfectsquareinthiscase,using y ontheleft-handsideofthe equationandfactoroutthecoecientofthenon-squaredvariableinthiscase,the x onthe other. y 2 +4 y +8 x =4 y 2 +4 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 x +4 y 2 +4 y +4= )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 x +4+4completethesquarein y only y +2 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 x +8factor y +2 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 NowthattheequationisintheformgiveninEquation7.3,weseethat x )]TJ/F53 10.9091 Tf 11.168 0 Td [(h is x )]TJ/F15 10.9091 Tf 11.168 0 Td [(1so h =1, and y )]TJ/F53 10.9091 Tf 11.473 0 Td [(k is y +2so k = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Hence,thevertexis ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Wealsoseethat4 p = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8sothat p = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Since p< 0,thefocuswillbetheleftofthevertexandtheparabolawillopentotheleft. Thedistancefromthevertextothefocusis j p j =2,whichmeansthefocusis2unitstotheleft of1,soifwestartat ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2andmoveleft2units,wearriveatthefocus )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Thedirectrix, then,is2unitstotherightofthevertex,soifwemoveright2unitsfrom ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,we'dbeonthe verticalline x =3.Sincethelengthofthelatusrectumis j 4 p j is8,theparabolais8unitswideat thefocus,sotherearepoints4unitsaboveandbelowthefocusontheparabola. x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 Instudyingquadraticfunctions,wehaveseenparabolasusedtomodelphysicalphenomenasuchas thetrajectoriesofprojectiles.Otherapplicationsoftheparabolaconcernits`reectiveproperty' whichnecessitatesknowingaboutthefocusofaparabola.Forexample,manysatellitedishesare formedintheshapeofa paraboloidofrevolution asdepictedbelow.

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398HookedonConics Everycrosssectionthroughthevertexoftheparaboloidisaparabolawiththesamefocus.Tosee whythisisimportant,imaginethedashedlinesbelowaselectromagneticwavesheadingtowards aparabolicdish.Itturnsoutthatthewavesreectotheparabolaandconcentrateatthefocus whichthenbecomestheoptimalplaceforthereceiver.If,ontheotherhand,weimaginethedashed linesasemanatingfromthefocus,weseethatthewavesarereectedotheparabolainacoherent fashionasinthecaseinaashlight.Here,thebulbisplacedatthefocusandthelightraysare reectedoaparabolicmirrortogivedirectionallight. F Example 7.3.5 Asatellitedishistobeconstructedintheshapeofaparaboloidofrevolution.If thereceiverplacedatthefocusislocated2ftabovethevertexofthedish,andthedishistobe 12feetwide,howdeepwillthedishbe? Solution. Onewaytoapproachthisproblemistodeterminetheequationoftheparabolasuggestedtousbythisdata.Forsimplicity,we'llassumethevertexis ; 0andtheparabolaopens upwards.Ourstandardformforsuchaparabolais x 2 =4 py .Sincethefocusis2unitsabovethe vertex,weknow p =2,sowehave x 2 =8 y .Visually,

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7.3Parabolas 399 ? ;y y x 12unitswide )]TJ/F15 9.9626 Tf 7.749 0 Td [(66 2 Sincetheparabolais12feetwide,weknowtheedgeis6feetfromthevertex.Tondthedepth, wearelookingforthe y valuewhen x =6.Substituting x =6intotheequationoftheparabola yields6 2 =8 y or y =36 = 8=9 = 2=4 : 5.Hence,thedishwillbe9 = 2or4 : 5feetdeep.

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400HookedonConics 7.3.1Exercises 1.Sketchthegraphofthegivenparabola.Findthevertex,focusanddirectrix.Includethe endpointsofthelatusrectuminyoursketch. a y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 x +3 b y +4 2 =4 x c x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 y d )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x + 7 3 2 =2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(y + 5 2 2.Puttheequationintostandardformandidentifythevertex,focusanddirectrix. a y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(27 x +133=0 b25 x 2 +20 x +5 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(1=0 c x 2 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 y +49=0 d2 y 2 +4 y + x )]TJ/F15 10.9091 Tf 10.91 0 Td [(8=0 3.Findanequationfortheparabolawhichtsthegivencriteria. aVertex ; 0,focus ; 0 bVertex )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(9,Both ; 0, )]TJ/F15 10.9091 Tf 8.484 0 Td [(16 ; 0 arepointsonthecurve cFocus ; 1,directrix x =5 dTheendpointsoflatusrectumare )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(7and )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 4.Thegraphsofverticalparabolasrepresent y asafunctionof x ,butthoseofhorizontal parabolasdonotbecauseaparabolawhichopentotheleftorrightfailstheVerticalLine Test,Theorem1.1.However,thetophalforbottomhalfofahorizontalparaboladoesform thegraphofafunctionwhichwehavestudiedthoroughly.TowhatfunctionsamIreferring andwhereinthetexthavetheyappeared?Summarizewhatyoushouldknowaboutthese functions.HowcouldweusethetransformationsinSection1.8tobetterunderstandthe graphsofhorizontalparabolas? 5.Withthehelpofyourclassmates,researchspinningliquidmirrors.Togetyoustarted,check outthiswebsite .

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7.3Parabolas 401 7.3.2Answers 1.a Vertex )]TJ/F15 9.9626 Tf 7.749 0 Td [(3 ; 2,focus )]TJ/F15 9.9626 Tf 7.748 0 Td [(6 ; 2,directrix x =0 Endpointsoflatusrectum )]TJ/F15 9.9626 Tf 7.749 0 Td [(6 ; 8, )]TJ/F15 9.9626 Tf 7.749 0 Td [(6 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(4 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 8 bVertex ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4,focus ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 directrix x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 Endpointsoflatusrectum ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(2, ; )]TJ/F15 9.9626 Tf 7.748 0 Td [(6 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 c Vertex ; 0,focus ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(4,directrix y =4 Endpointsoflatusrectum )]TJ/F15 9.9626 Tf 7.749 0 Td [(5 ; )]TJ/F15 9.9626 Tf 7.749 0 Td [(4, ; )]TJ/F15 9.9626 Tf 7.748 0 Td [(4 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234567891011 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 dVertex )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(7 3 ; )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(5 2 ,focus )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.681 4.295 Td [(7 3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 directrix y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 Endpointsoflatusrectum )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(10 3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(4 3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 2.a y )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 2 =27 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 Vertex ; 5,focus )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(43 4 ; 5 directrix x = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(11 4 b )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x + 2 5 2 = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 5 y )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 Vertex )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(2 5 ; 1 ,focus )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(2 5 ; 19 20 directrix y = 21 20 c x +1 2 =8 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 Vertex )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 6,focus )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 8 directrix y =4 d y +1 2 = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 Vertex ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1,focus )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(79 8 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 directrix x = 81 8 3.a y 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(28 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 b x +8 2 = 64 9 y +9 c y )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 =10 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(x )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(15 2 d x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 =6 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(y + 17 2 or x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(y + 11 2

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402HookedonConics 7.4Ellipses Inthedenitionofacircle,Denition7.1,wexedapointcalledthe center andconsideredall ofthepointswhichwereaxeddistance r fromthatonepoint.Forournextconicsection,the ellipse,wextwodistinctpointsandadistance d touseinourdenition. Definition 7.4 Giventwodistinctpoints F 1 and F 2 intheplaneandaxeddistance d ,an ellipse isthesetofallpoints x;y intheplanesuchthatthesumofthedistancefrom F 1 to x;y andthedistancefrom F 2 to x;y is d .Thepoints F 1 and F 2 arecalledthe foci a ofthe ellipse. a thepluralof`focus' x;y d 1 d 2 F 1 F 2 d 1 + d 2 = d forall x;y ontheellipse Wemayimaginetakingalengthofstringandanchoringittotwopointsonapieceofpaper.The curvetracedoutbytakingapencilandmovingitsothestringisalwaystautisanellipse. The center oftheellipseisthemidpointofthelinesegmentconnectingthetwofoci.The major axis oftheellipseisthelinesegmentconnectingtwooppositeendsoftheellipsewhichalsocontains thecenterandfoci.The minoraxis oftheellipseisthelinesegmentconnectingtwoopposite endsoftheellipsewhichcontainsthecenterbutisperpendiculartothemajoraxis.The vertices ofanellipsearethepointsoftheellipsewhichlieonthemajoraxis.Noticethatthecenterisalso themidpointofthemajoraxis,henceitisthemidpointofthevertices.Inpictureswehave,

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7.4Ellipses 403 F 1 F 2 V 2 V 1 C MajorAxis MinorAxis Anellipsewithcenter C ;foci F 1 F 2 ;andvertices V 1 V 2 Notethatthemajoraxisisthelongerofthetwoaxesthroughthecenter,andlikewise,theminor axisistheshorterofthetwo.Inordertoderivethestandardequationofanellipse,weassumethat theellipsehasitscenterat ; 0,itsmajoraxisalongthe x -axis,andhasfoci c; 0and )]TJ/F53 10.9091 Tf 8.485 0 Td [(c; 0 andvertices )]TJ/F53 10.9091 Tf 8.485 0 Td [(a; 0and a; 0.Wewilllabelthe y -interceptsoftheellipseas ;b and ; )]TJ/F53 10.9091 Tf 8.485 0 Td [(b We assume a b ,and c areallpositivenumbers.Schematically, )]TJ/F53 10.9091 Tf 8.485 0 Td [(c; 0 c; 0 )]TJ/F53 10.9091 Tf 8.485 0 Td [(a; 0 a; 0 ;b ; )]TJ/F53 10.9091 Tf 8.484 0 Td [(b x;y x y Notethatsince a; 0isontheellipse,itmustsatisfytheconditionsofDenition7.4.Thatis,the distancefrom )]TJ/F53 10.9091 Tf 8.485 0 Td [(c; 0to a; 0plusthedistancefrom c; 0to a; 0mustequalthexeddistance d .Sinceallofthesepointslieonthe x -axis,weget distancefrom )]TJ/F53 10.9091 Tf 8.485 0 Td [(c; 0to a; 0+distancefrom c; 0to a; 0= d a + c + a )]TJ/F53 10.9091 Tf 10.909 0 Td [(c = d 2 a = d

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404HookedonConics Inotherwords,thexeddistance d mentionedinthedenitionoftheellipseisnoneotherthan thelengthofthemajoraxis.Wenowusethatfact ;b isontheellipse,alongwiththefactthat d =2 a toget distancefrom )]TJ/F53 10.9091 Tf 8.485 0 Td [(c; 0to ;b +distancefrom c; 0to ;b =2 a p )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F53 10.9091 Tf 8.484 0 Td [(c 2 + b )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 2 + p )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + b )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 2 =2 a p b 2 + c 2 + p b 2 + c 2 =2 a 2 p b 2 + c 2 =2 a p b 2 + c 2 = a Fromthis,weget a 2 = b 2 + c 2 ,or b 2 = a 2 )]TJ/F53 10.9091 Tf 10.547 0 Td [(c 2 ,whichwillproveusefullater.Nowconsiderapoint x;y ontheellipse.ApplyingDenition7.4,weget distancefrom )]TJ/F53 10.9091 Tf 8.485 0 Td [(c; 0to x;y +distancefrom c; 0to x;y =2 a p x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F53 10.9091 Tf 8.484 0 Td [(c 2 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 2 + p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 2 =2 a p x + c 2 + y 2 + p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y 2 =2 a Inordertomakesenseofthissituation,weneedtodosomerearranging,squaring,andmore rearranging. 1 p x + c 2 + y 2 + p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y 2 =2 a p x + c 2 + y 2 =2 a )]TJ/F55 10.9091 Tf 10.909 9.38 Td [(p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y 2 p x + c 2 + y 2 2 = 2 a )]TJ/F55 10.9091 Tf 10.909 9.38 Td [(p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y 2 2 x + c 2 + y 2 =4 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 a p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y 2 + x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y 2 4 a p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y 2 =4 a 2 + x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( x + c 2 4 a p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y 2 =4 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 cx a p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y 2 = a 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(cx a p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y 2 2 = )]TJ/F53 10.9091 Tf 5 -8.837 Td [(a 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(cx 2 a 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [( x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y 2 = a 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 a 2 cx + c 2 x 2 a 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 a 2 cx + a 2 c 2 + a 2 y 2 = a 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 a 2 cx + c 2 x 2 a 2 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 x 2 + a 2 y 2 = a 4 )]TJ/F53 10.9091 Tf 10.909 0 Td [(a 2 c 2 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(a 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 x 2 + a 2 y 2 = a 2 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(a 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 Wearenearlynished.Recallthat b 2 = a 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 sothat )]TJ/F53 10.9091 Tf 5 -8.836 Td [(a 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 x 2 + a 2 y 2 = a 2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(a 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 b 2 x 2 + a 2 y 2 = a 2 b 2 x 2 a 2 + y 2 b 2 =1 1 Inotherwords,tonsandtonsofIntermediateAlgebra.Staysharp,thisisnotforthefaintofheart.

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7.4Ellipses 405 Thisequationisforanellipsecenteredattheorigin.Iftheellipsewerecenteredatapoint h;k wewouldgetthefollowing Equation 7.4 TheStandardEquationofanEllipse: Forpositiveunequalnumbers a and b ,theequationofanellipsewithcenter h;k is x )]TJ/F53 10.9091 Tf 10.91 0 Td [(h 2 a 2 + y )]TJ/F53 10.9091 Tf 10.91 0 Td [(k 2 b 2 =1 SomeremarksaboutEquation7.4areinorder.Firstnotethatthevalues a and b determine howfarinthe x and y directions,respectively,onecountsfromthecentertoarriveatpointson theellipse.Alsotakenotethatif a>b ,thenwehaveanellipsewhosemajoraxisishorizontal, andhence,thefocilietotheleftandrightofthecenter.Inthiscase,aswe'veseeninthe derivation,thedistancefromthecentertothefocus, c ,canbefoundby c = p a 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(b 2 .If b>a therolesofthemajorandminoraxesarereversed,andthefocilieaboveandbelowthecenter. Inthiscase, c = p b 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(a 2 .Ineithercase, c isthedistancefromthecentertoeachfocus,and c = p biggerdenominator )]TJ/F15 10.9091 Tf 10.909 0 Td [(smallerdenominator.Finally,itisworthmentioningthatifwetake thestandardequationofacircle,Equation7.1,anddividebothsidesby r 2 ,weget Equation 7.5 TheAlternateStandardEquationofaCircle: Theequationofacircle withcenter h;k andradius r> 0is x )]TJ/F53 10.9091 Tf 10.909 0 Td [(h 2 r 2 + y )]TJ/F53 10.9091 Tf 10.909 0 Td [(k 2 r 2 =1 NoticethesimilaritybetweenEquation7.4andEquation7.5.Bothequationsinvolveasumof squaresequalto1;thedierenceisthatwithacircle,thedenominatorsarethesame,andwithan ellipse,theyaredierent.Ifwetakeatransformationalapproach,wecanconsiderbothequations asshiftsandstretchesoftheUnitCircle x 2 + y 2 =1inDenition7.2.Replacing x with x )]TJ/F53 10.9091 Tf 11.142 0 Td [(h and y with y )]TJ/F53 10.9091 Tf 10.146 0 Td [(k causestheusualhorizontalandverticalshifts.Replacing x with x a and y with y b causestheusualverticalandhorizontalstretches.Inotherwords,itisperfectlynetothinkofan ellipseasthedeformationofacircleinwhichthecircleisstretchedfartherinonedirectionthan theother. Example 7.4.1 Graph x +1 2 9 + y )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 2 25 =1.Findthecenter,thelineswhichcontainthemajor andminoraxes,thevertices,andthefoci. Solution. WeseethatthisequationisinthestandardformofEquation7.4.Here x )]TJ/F53 10.9091 Tf 10.747 0 Td [(h is x +1 so h = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,and y )]TJ/F53 10.9091 Tf 10.109 0 Td [(k is y )]TJ/F15 10.9091 Tf 10.109 0 Td [(2so k =2.Hence,ourellipseiscenteredat )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 2.Weseethat a 2 =9 so a =3,and b 2 =25so b =5.Thismeansthatwemove3unitsleftandrightfromthecenter and5unitsupanddownfromthecentertoarriveatpointsontheellipse.Asanaidtosketching, wedrawarectanglematchingthisdescription,calleda guiderectangle ,andsketchtheellipse insidethisrectangleasseenbelowontheleft.

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406HookedonConics x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(112 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 5 6 7 x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(112 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 5 6 7 Sincewemovedfartherinthe y directionthaninthe x direction,themajoraxiswillliealong theverticalline x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,whichmeanstheminoraxisliesalongthehorizontalline, y =2.The verticesarethepointsontheellipsewhichliealongthemajoraxissointhiscase,theyarethe points )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 7and )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.Noticethatthesearetwoofthepointsweplottedwhendrawingthe ellipse.Tondthefoci,wend c = p 25 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9= p 16=4,whichmeansthefocilie4unitsfromthe center.Sincethemajoraxisisvertical,thefocilie4unitsaboveandbelowthecenter,at )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 and )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 6.Plottingallthisinformationgivesthegraphseenaboveontheright. Example 7.4.2 Findtheequationoftheellipsewithfoci ; 1and ; 1andvertex ; 1. Solution. Plottingthedatagiventous,wehave x y 12345 1 Fromthissketch,weknowthatthemajoraxisishorizontal,meaning a>b .Sincethecenteristhe midpointofthefoci,weknowitis ; 1.Sinceonevertexis ; 1wehavethat a =3,so a 2 =9. Allthatremainsistond b 2 .Tothatend,weusethefactthat c =1toget

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7.4Ellipses 407 c = p a 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(b 2 1= p 9 )]TJ/F53 10.9091 Tf 10.909 0 Td [(b 2 1 2 = p 9 )]TJ/F53 10.9091 Tf 10.909 0 Td [(b 2 2 1=9 )]TJ/F53 10.9091 Tf 10.91 0 Td [(b 2 b 2 =8 Substitutingallofourndingsintotheequation x )]TJ/F53 10.9091 Tf 10.909 0 Td [(h 2 a 2 + y )]TJ/F53 10.9091 Tf 10.909 0 Td [(k 2 b 2 =1,wegetournalanswer tobe x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 9 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 8 =1. Aswithcirclesandparabolas,anequationmaybegivenwhichisanellipse,butisn'tinthestandard formofEquation7.4.Inthosecases,aswithcirclesandparabolasbefore,wewillneedtomassage thegivenequationintothestandardform. ToPutanEllipseintoStandardForm 1.Groupthesamevariablestogetherononesideoftheequationandputtheconstantonthe otherside. 2.Completethesquareinbothvariablesasneeded. 3.Dividebothsidesbytheconstanttermsothattheconstantontheothersideoftheequation becomes1. Example 7.4.3 Graph x 2 +4 y 2 )]TJ/F15 10.9091 Tf 10.561 0 Td [(2 x +24 y +33=0.Findthecenter,thelineswhichcontainthe majorandminoraxes,thevertices,andthefoci. Solution. Sincewehaveasumofsquaresandthesquaredtermshaveunequalcoecients,it'sa goodbetwehaveanellipseonourhands.Aparabolawouldhaveonlyonesquaredvariableand acirclewouldhavethesamecoecientonthesquaredterms.Weneedtocompletebothsquares, andthendivide,ifnecessary,togettheright-handsideequalto1. x 2 +4 y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +24 y +33=0 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +4 y 2 +24 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +4 )]TJ/F53 10.9091 Tf 5 -8.837 Td [(y 2 +6 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 +4 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(y 2 +6 y +9 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(33+1+4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 +4 y +3 2 =4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 +4 y +3 2 4 = 4 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 4 + y +3 2 =1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 4 + y +3 2 1 =1

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408HookedonConics NowthatthisequationisinthestandardformofEquation7.4,weseethat x )]TJ/F53 10.9091 Tf 10.205 0 Td [(h is x )]TJ/F15 10.9091 Tf 10.205 0 Td [(1so h =1, and y )]TJ/F53 10.9091 Tf 11.416 0 Td [(k is y +3so k = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.Hence,ourellipseiscenteredat ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.Weseethat a 2 =4so a =2,and b 2 =1so b =1.Thismeanswemove2unitsleftandrightfromthecenterand1unit upanddownfromthecentertoarriveatpointsontheellipse.Sincewemovedfartherinthe x directionthaninthe y direction,themajoraxiswillliealongthehorizontalline y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3,which meanstheminoraxisliesalongtheverticalline x =1.Theverticesarethepointsontheellipse whichliealongthemajoraxissointhiscase,theyarethepoints )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3and ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.Tond thefoci,wend c = p 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1= p 3,whichmeansthefocilie p 3unitsfromthecenter.Sincethe majoraxisishorizontal,thefocilie p 3unitstotheleftandrightofthecenter,at )]TJ 11.239 9.025 Td [(p 3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 and+ p 3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.Plottingallofthisinformationgives x y 1234 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 Asyoucomeacrossellipsesinthehomeworkexercisesandinthewild,you'llnoticetheycomein allshapesinsizes.Comparethetwoellipsesbelow. Certainly,oneellipseis`rounder'thantheother.Thisnotionofroundnessisquantiedbelow. Definition 7.5 The eccentricity ofanellipse,denoted e ,isthefollowingratio: e = distancefromthecentertoafocus distancefromthecentertoavertex Inanellipse,thefociareclosertothecenterthanthevertices,so0
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7.4Ellipses 409 Example 7.4.4 Findtheequationoftheellipsewhoseverticesare ; 5witheccentricity e = 1 4 Solution. Asbefore,weplotthedatagiventous x y Fromthissketch,weknowthatthemajoraxisisvertical,meaning b>a .Withtheverticeslocated at ; 5,weget b =5so b 2 =25.Wealsoknowthatthecenteris ; 0becausethecenteris themidpointofthevertices.Allthatremainsistond a 2 .Tothatend,weusethefactthatthe eccentricity e = 1 4 whichmeans e = distancefromthecentertoafocus distancefromthecentertoavertex = c b = c 5 = 1 4 fromwhichweget c = 5 4 .Toget a 2 ,weusethefactthat c = p b 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(a 2 c = p b 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(a 2 5 4 = p 25 )]TJ/F53 10.9091 Tf 10.909 0 Td [(a 2 5 4 2 = p 25 )]TJ/F53 10.9091 Tf 10.909 0 Td [(a 2 2 25 16 =25 )]TJ/F53 10.9091 Tf 10.909 0 Td [(a 2 a 2 =25 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(25 16 a 2 = 375 16 Substitutingallofourndingsintotheequation x )]TJ/F53 10.9091 Tf 10.91 0 Td [(h 2 a 2 + y )]TJ/F53 10.9091 Tf 10.909 0 Td [(k 2 b 2 =1,yieldsournalanswer 16 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 375 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 25 =1. Aswithparabolas,ellipseshaveareectiveproperty.Ifweimaginethedashedlinesbelowassound waves,thenthewavesemanatingfromonefocusreectothetopoftheellipseandheadtowards theotherfocus.Suchgeometryisexploitedintheconstructionofso-called`WhisperingGalleries'. Ifapersonwhispersatonefocus,apersonstandingattheotherfocuswillheartherstpersonas iftheywerestandingrightnexttothem.

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410HookedonConics F 1 F 2 Example 7.4.5 JamieandJasonwanttoexchangesecretsterriblesecretsfromacrossacrowded whisperinggallery.Recallthatawhisperinggalleryisaroomwhich,incrosssection,ishalfofan ellipse.Iftheroomis40feethighatthecenterand100feetwideattheoor,howfarfromthe outerwallshouldeachofthemstandsothattheywillbepositionedatthefocioftheellipse? Solution. Graphingthedatayields x y 100unitswide 40unitstall It'smostconvenienttoimaginethisellipsecenteredat ; 0.Sincetheellipseis100unitswide and40unitstall,weget a =50and b =40.Hence,ourellipsehastheequation x 2 50 2 + y 2 40 2 =1 : We'relookingforthefoci,andweget c = p 50 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(40 2 = p 900=30,sothatthefociare30units fromthecenter.Thatmeanstheyare50 )]TJ/F15 10.9091 Tf 11.487 0 Td [(30=20unitsfromthevertices.Hence,Jasonand Jamieshouldstand20feetfromoppositeendsofthegallery.

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7.4Ellipses 411 7.4.1Exercises 1.Graphtheellipse.Findthecenter,thelineswhichcontainthemajorandminoraxes,the vertices,thefociandtheeccentricity. a x 2 169 + y 2 25 =1 b x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 4 + y +3 2 9 =1 c x +5 2 16 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 2 1 =1 d x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 10 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 11 =1 2.Puttheequationinstandardform.Findthecenter,thelineswhichcontainthemajorand minoraxes,thevertices,thefociandtheeccentricity. a12 x 2 +3 y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(30 y +39=0 b5 x 2 +18 y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(30 x +72 y +27=0 c x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +2 y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 y +3=0 d9 x 2 +4 y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(8=0 3.Findthestandardformoftheequationoftheellipsewhichhasthegivenproperties. aCenter ; 7,Vertex ; 2,Focus ; 3 bAllpointsontheellipseareinQuadrantIVexcept ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(9and ; 0 2 cFoci ; 4,Pointoncurve 2 ; 5 p 5 3 dVertex )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 ; 5,Focus )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 5,Eccentricity 1 2 4.TheEarth'sorbitaroundthesunisanellipsewiththesunatonefocusandeccentricity e 0 : 0167.Thelengthofthesemimajoraxisthatis,halfofthemajoraxisisdened tobe1astronomicalunitAU.Theverticesoftheellipticalorbitaregivenspecialnames: `aphelion'isthevertexfartherfromthesun,and`perihelion'isthevertexclosesttothesun. FindthedistanceinAUbetweenthesunandaphelionandthedistanceinAUbetweenthe sunandperihelion. 5.ThegraphofanellipseclearlyfailstheVerticalLineTest,Theorem1.1,sotheequationof anellipsedoesnotdene y asafunctionof x .However,muchlikewithcirclesandhorizontal parabolas,wecansplitanellipseintoatophalfandabottomhalf,eachofwhichwould indeedrepresent y asafunctionof x .Withthehelpofyourclassmates,useyourcalculator tographtheellipsesgiveninExercise1above.Whatdicultiesarisewhenyouplotthem onthecalculator? 6.Withthehelpofyourclassmates,researchwhisperinggalleriesandotherwaysellipseshave beenusedinarchitectureanddesign. 7.Withthehelpofyourclassmates,researchextracorporealshock-wavelithotripsy".Ituses thereectivepropertyoftheellipsoidtodissolvekidneystones. 2 Onemightalsosaythattheellipseistangenttotheaxes"atthosetwopoints.

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412HookedonConics 7.4.2Answers 1.a x 2 169 + y 2 25 =1 Center ; 0 Majoraxisalong y =0 Minoraxisalong x =0 Vertices ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 ; 0 Foci ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 ; 0 e = 12 13 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(13 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1113 )]TJ/F35 5.9776 Tf 5.757 0 Td [(5 )]TJ/F35 5.9776 Tf 5.757 0 Td [(4 )]TJ/F35 5.9776 Tf 5.757 0 Td [(3 )]TJ/F35 5.9776 Tf 5.757 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 1 2 3 4 5 b x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 4 + y +3 2 9 =1 Center ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 Majoraxisalong x =2 Minoraxisalong y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 Vertices ; 0 ; ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 Foci ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3+ p 5 ; ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ 10.909 9.024 Td [(p 5 e = p 5 3 x y 1234 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 c x +5 2 16 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 2 1 =1 Center )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 ; 4 Majoraxisalong y =4 Minoraxisalong x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 Vertices )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 ; 4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 4 Foci )]TJ/F15 10.9091 Tf 8.485 0 Td [(5+ p 15 ; 4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ 10.909 9.024 Td [(p 15 ; 4 e = p 15 4 x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(9 )]TJ/F34 7.9701 Tf 6.586 0 Td [(8 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 d x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 10 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 11 =1 Center ; 3 Majoraxisalong x =1 Minoraxisalong y =3 Vertices ; 3+ p 11 ; ; 3 )]TJ 10.909 9.024 Td [(p 11 Foci ; 2 ; ; 4 e = 1 p 11 x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(11234 1 2 3 4 5 6

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7.4Ellipses 413 2.a x 2 3 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 2 12 =1 Center ; 5 Majoraxisalong x =0 Minoraxisalong y =5 Vertices ; 5 )]TJ 10.909 9.024 Td [(p 12 ; ; 5+ p 12 Foci ; 2 ; ; 8 e = 3 p 12 b x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 18 + y +2 2 5 =1 Center ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 Majoraxisalong y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 Minoraxisalong x =3 Vertices )]TJ 10.909 9.025 Td [(p 18 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 ; + p 18 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 Foci )]TJ 10.909 9.024 Td [(p 13 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; + p 13 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 e = p 13 p 18 c x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 16 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 8 =1 Center ; 3 MajorAxisalong y =3 MinorAxisalong x =1 Vertices ; 3, )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 3 Foci+2 p 2 ; 3, )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 p 2 ; 3 e = p 2 2 d x 2 1 + 4 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(y )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 2 2 9 =1 Center )]TJ/F15 10.9091 Tf 5 -8.837 Td [(0 ; 1 2 MajorAxisalong x =0the y -axis MinorAxisalong y = 1 2 Vertices ; 2, ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 Foci 0 ; 1+ p 5 2 0 ; 1 )]TJ 6.587 6.598 Td [(p 5 2 e = p 5 3 3.a x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 9 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 2 25 =1 b x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 2 64 + y +9 2 81 =1 c x 2 9 + y 2 25 =1 d x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 2 256 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 2 192 =1 4.Distancefromthesuntoaphelion 1 : 0167AU. Distancefromthesuntoperihelion 0 : 9833AU.

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414HookedonConics 7.5Hyperbolas Inthedenitionofanellipse,Denition7.4,wexedtwopointscalledfociandlookedatpoints whosedistancestothefocialways added toaconstantdistance d .Thosepronetosyntactical tinkeringmaywonderwhat,ifany,curvewe'dgenerateifwereplaced added with subtracted Theanswerisahyperbola. Definition 7.6 Giventwodistinctpoints F 1 and F 2 intheplaneandaxeddistance d ,a hyperbola isthesetofallpoints x;y intheplanesuchthattheabsolutevalueofthedierence ofthedistancesbetweenthefociand x;y is d .Thepoints F 1 and F 2 arecalledthe foci ofthe hyperbola. x 1 ;y 1 x 2 ;y 2 F 1 F 2 Inthegureabove: thedistancefrom F 1 to x 1 ;y 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(thedistancefrom F 2 to x 1 ;y 1 = d and thedistancefrom F 2 to x 2 ;y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(thedistancefrom F 1 to x 2 ;y 2 = d Notethatthehyperbolahastwoparts,called branches .The center ofthehyperbolaisthe midpointofthelinesegmentconnectingthetwofoci.The transverseaxis ofthehyperbolais thelinesegmentconnectingtwooppositeendsofthehyperbolawhichalsocontainsthecenterand foci.The vertices ofahyperbolaarethepointsofthehyperbolawhichlieonthetransverseaxis. Inaddition,wewillshowmomentarilythattherearelinescalled asymptotes whichthebranches ofthehyperbolaapproachforlarge x and y values.Theyserveasguidestothegraph.Inpictures,

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7.5Hyperbolas 415 V 2 V 1 F 1 F 2 TransverseAxis C Ahyperbolawithcenter C ;foci F 1 F 2 ;andvertices V 1 V 2 andasymptotesdashed Beforewederivethestandardequationofthehyperbola,weneedtodiscussonefurtherparameter, the conjugateaxis ofthehyperbola.Theconjugateaxisofahyperbolaisthelinesegment throughthecenterwhichisperpendiculartothetransverseaxisandhasthesamelengthasthe linesegmentthroughavertexwhichconnectstheasymptotes.Inpictureswehave V 2 V 1 C ConjugateAxis Notethatinthediagram,wecanconstructarectangleusinglinesegmentswithlengthsequalto thelengthsofthetransverseandconjugateaxeswhosecenteristhecenterofthehyperbolaand whosediagonalsarecontainedintheasymptotes.This guiderectangle ,whichisverysimilarto theonewecreatedintheSection7.4tohelpusgraphellipses,willaidusingraphinghyperbolas whenthetimecomes. Supposewewishtoderivetheequationofahyperbola.Forsimplicity,weshallassumethatthe centeris ; 0,theverticesare a; 0and )]TJ/F53 10.9091 Tf 8.485 0 Td [(a; 0andthefociare c; 0and )]TJ/F53 10.9091 Tf 8.485 0 Td [(c; 0.Welabelthe

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416HookedonConics endpointsoftheconjugateaxis ;b and ; )]TJ/F53 10.9091 Tf 8.485 0 Td [(b .Although b doesnotenterintoourderivation, wewillhavetojustifythischoiceasyoushallseelater.Asbefore,weassume a b ,and c areall positivenumbers.Schematicallywehave x y a; 0 )]TJ/F53 10.9091 Tf 8.485 0 Td [(a; 0 ;b ; )]TJ/F53 10.9091 Tf 8.485 0 Td [(b )]TJ/F53 10.9091 Tf 8.485 0 Td [(c; 0 c; 0 x;y Since a; 0isonthehyperbola,itmustsatisfytheconditionsofDenition7.6.Thatis,thedistance from )]TJ/F53 10.9091 Tf 8.485 0 Td [(c; 0to a; 0minusthedistancefrom c; 0to a; 0mustequalthexeddistance d .Since allthesepointslieonthe x -axis,weget distancefrom )]TJ/F53 10.9091 Tf 8.485 0 Td [(c; 0to a; 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(distancefrom c; 0to a; 0= d a + c )]TJ/F15 10.9091 Tf 10.909 0 Td [( c )]TJ/F53 10.9091 Tf 10.909 0 Td [(a = d 2 a = d Inotherwords,thexeddistance d fromthedenitionofthehyperbolaisactuallythelengthof thetransverseaxis!Wherehaveweseenthattypeofcoincidencebefore?Nowconsiderapoint x;y onthehyperbola.ApplyingDenition7.6,weget distancefrom )]TJ/F53 10.9091 Tf 8.485 0 Td [(c; 0to x;y )]TJ/F15 10.9091 Tf 10.909 0 Td [(distancefrom c; 0to x;y =2 a p x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F53 10.9091 Tf 8.484 0 Td [(c 2 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 2 )]TJ/F55 10.9091 Tf 10.909 9.381 Td [(p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 2 =2 a p x + c 2 + y 2 )]TJ/F55 10.9091 Tf 10.909 9.38 Td [(p x )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 + y 2 =2 a UsingthesamearsenalofIntermediateAlgebraweaponryweusedinderivingthestandardformula ofanellipse,Equation7.4,wearriveatthefollowing. 1 1 Itisagoodexercisetoactuallyworkthisout.

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7.5Hyperbolas 417 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(a 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 x 2 + a 2 y 2 = a 2 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(a 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(c 2 Whatremainsistodeterminetherelationshipbetween a b and c .Tothatend,wenotethatsince a and c arebothpositivenumberswith a
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418HookedonConics fromthecentertothefoci, c ,asseeninthederivation,canbefoundbytheformula c = p a 2 + b 2 Lastly,notethatwecanquicklydistinguishtheequationofahyperbolafromthatofacircleor ellipsebecausethehyperbolaformulainvolvesa dierence ofsquareswherethecircleandellipse formulasbothinvolvethe sum ofsquares. Example 7.5.1 Graphtheequation x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 4 )]TJ/F53 10.9091 Tf 12.499 7.38 Td [(y 2 25 =1 : Findthecenter,thelineswhichcontain thetransverseandconjugateaxes,thevertices,thefociandtheequationsoftheasymptotes. Solution. WerstseethatthisequationisgiventousinthestandardformofEquation7.6. Here x )]TJ/F53 10.9091 Tf 10.117 0 Td [(h is x )]TJ/F15 10.9091 Tf 10.116 0 Td [(2so h =2,and y )]TJ/F53 10.9091 Tf 10.117 0 Td [(k is y so k =0.Hence,ourhyperbolaiscenteredat ; 0.We seethat a 2 =4so a =2,and b 2 =25so b =5.Thismeanswemove2unitstotheleftandright ofthecenterand5unitsupanddownfromthecentertoarriveatpointsontheguiderectangle. Theasymptotespassthroughthecenterofthehyperbolaaswellasthecornersoftherectangle. Thisyieldsthefollowingsetup. x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123456 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 5 6 7 Sincethe y 2 termisbeingsubtractedfromthe x 2 term,weknowthatthebranchesofthehyperbola opentotheleftandright.Thismeansthatthetransverseaxisliesalongthe x -axis.Hence,the conjugateaxisliesalongtheverticalline x =2.Sincetheverticesofthehyperbolaarewherethe hyperbolaintersectsthetransverseaxis,wegetthattheverticesare2unitstotheleftandrightof ; 0at ; 0and ; 0.Tondthefoci,weneed c = p a 2 + b 2 = p 4+25= p 29.Sincethefoci lieonthetransverseaxis,wemove p 29unitstotheleftandrightof ; 0toarriveat )]TJ 10.3 9.024 Td [(p 29 ; 0 approximately )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 : 39 ; 0and+ p 29 ; 0approximately : 39 ; 0.Todeterminetheequations oftheasymptotes,recallthattheasymptotesgothroughthecenterofthehyperbola, ; 0,aswell asthecornersofguiderectangle,sotheyhaveslopesof b a = 5 2 .Usingthepoint-slopeequation ofaline,Equation2.2,yields

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7.5Hyperbolas 419 y = 5 2 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(2+0 ; soweget y = 5 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(5and y = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(5 2 x +5.Puttingitalltogether,weget x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(11234567 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 5 6 7 Example 7.5.2 Findtheequationofthehyperbolawithasymptotes y = 2 x andvertices 5 ; 0. Solution. Plottingthedatagiventous,wehave x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(55 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 5 Thisgraphnotonlytellsusthatthebranchesofthehyperbolaopentotheleftandtotheright, italsotellsusthatthecenteris ; 0.Hence,ourstandardformis x 2 a 2 )]TJ/F53 10.9091 Tf 12.104 7.38 Td [(y 2 b 2 =1 :

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420HookedonConics Sincetheverticesare 5 ; 0,wehave a =5so a 2 =25.Inordertodetermine b 2 ,werecallthat theslopesoftheasymptotesare b a .Since a =5andtheslopeoftheline y =2 x is2,wehave that b 5 =2,so b =10.Hence, b 2 =100andournalansweris x 2 25 )]TJ/F53 10.9091 Tf 15.051 7.38 Td [(y 2 100 =1 Aswiththeotherconicsections,anequationwhosegraphisahyperbolamaybegiveninaform otherthanthestandardformsinEquations7.6or7.7.Inthosecases,aswithconicsectionswhich havecomebefore,wewillneedtomassagethegivenequationintooneoftheformsinEquations 7.6or7.7. ToPutaHyperbolaintoStandardForm 1.Groupthesamevariablestogetherononesideoftheequationandputtheconstantonthe otherside 2.Completethesquareinbothvariablesasneeded 3.Dividebothsidesbytheconstanttermsothattheconstantontheothersideoftheequation becomes1 Example 7.5.3 Considertheequation9 y 2 )]TJ/F53 10.9091 Tf 10.617 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.617 0 Td [(6 x =10.Putthisequationintostandardform andgraph.Findthecenter,thelineswhichcontainthetransverseandconjugateaxes,thevertices, thefoci,andtheequationsoftheasymptotes. Solution. Weneedonlycompletethesquareonthe x ,andthendivide,ifnecessary,togetthe right-handsideequalto1. 9 y 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x =10 9 y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F53 10.9091 Tf 5 -8.836 Td [(x 2 +6 x =10 9 y 2 )]TJ/F55 10.9091 Tf 10.909 8.837 Td [()]TJ/F53 10.9091 Tf 5 -8.837 Td [(x 2 +6 x +9 =10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 9 y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( x +3 2 =1 y 2 1 9 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [( x +3 2 1 =1 NowthatthisequationisinthestandardformofEquation7.7,weseethat x )]TJ/F53 10.9091 Tf 11.659 0 Td [(h is x +3so h = )]TJ/F15 10.9091 Tf 8.484 0 Td [(3,and y )]TJ/F53 10.9091 Tf 10.912 0 Td [(k is y so k =0.Hence,ourhyperbolaiscenteredat )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 0.Wendthat a 2 =1 so a =1,and b 2 = 1 9 so b = 1 3 .Thismeansthatwemove1unittotheleftandrightofthe centerand1 = 3unitsupanddownfromthecentertoarriveatpointsontheguiderectangle.Since the x 2 termisbeingsubtractedfromthe y 2 term,weknowthebranchesofthehyperbolaopen upwardsanddownwards.Thismeansthetransverseaxisliesalongtheverticalline x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3andthe conjugateaxisliesalongthe x -axis.Sincetheverticesofthehyperbolaarewherethehyperbola intersectsthetransverseaxis,wegetthattheverticesare 1 3 ofaunitaboveandbelow )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 0at

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7.5Hyperbolas 421 )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1 3 and )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 3 .Tondthefoci,weneed c = p a 2 + b 2 = q 1 9 +1= p 10 3 .Sincethefoci lieonthetransverseaxis,wemove p 10 3 unitsaboveandbelow )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 0toarriveat )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; p 10 3 and )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ/F40 7.9701 Tf 9.68 10.993 Td [(p 10 3 .Todeterminetheasymptotes,recallthattheasymptotesgothroughthecenterof thehyperbola, )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 0,aswellasthecornersofguiderectangle,sotheyhaveslopesof b a = 1 3 Usingthepoint-slopeequationofaline,Equation2.2,weget y = 1 3 x +1and y = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 3 x )]TJ/F15 10.9091 Tf 9.874 0 Td [(1.Putting italltogether,weget x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 Hyperbolascanbeusedinpositioningproblems,asthenextexampleillustrates. Example 7.5.4 Jeisstationed10milesduewestofCarlinanotherwiseemptyforestinan attempttolocateanelusiveSasquatch.Atthestrokeofmidnight,JerecordsaSasquatchcall 9secondsearlierthanCarl.Ifthespeedofsoundthatnightis760milesperhour,determinea hyperbolicpathalongwhichSasquatchmustbelocated. Solution. SinceJehearsSasquatchsooner,itisclosertoJethanitistoCarl.Sincethe speedofsoundis760milesperhour,wecandeterminehowmuchcloserSasquatchistoJeby multiplying 760 miles hour 1hour 3600seconds 9seconds=1 : 9miles ThismeansthatSasquatchis1 : 9milesclosertoJethanitistoCarl.Inotherwords,Sasquatch mustlieonapathwhere thedistancetoCarl )]TJ/F15 10.9091 Tf 10.909 0 Td [(thedistancetoJe=1 : 9 Thisisexactlythesituationinthedenitionofahyperbola,Denition7.6.Inthiscase,Jeand Carlarelocatedatthefoci,andourxeddistance d is1.9.Forsimplicity,weassumethehyperbola iscenteredat ; 0withitsfociat )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 ; 0and ; 0.Schematically,wehave

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422HookedonConics x y JeCarl )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123456 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 2 3 4 5 6 Weareseekingacurveoftheform x 2 a 2 )]TJ/F37 7.9701 Tf 11.571 4.932 Td [(y 2 b 2 =1inwhichthedistancefromthecentertoeachfocus is c =5.Aswesawinthederivationofthestandardequationofthehyperbola,Equation7.6, d =2 a ,sothat2 a =1 : 9,or a =0 : 95and a 2 =0 : 9025.Allthatremainsistond b 2 .Tothatend, werecallthat a 2 + b 2 = c 2 so b 2 = c 2 )]TJ/F53 10.9091 Tf 11.12 0 Td [(a 2 =25 )]TJ/F15 10.9091 Tf 11.12 0 Td [(0 : 9025=24 : 0975.SinceSasquatchiscloserto JethanitistoCarl,itmustbeonthewesternlefthandbranchof x 2 0 : 9025 )]TJ/F37 7.9701 Tf 21.672 4.931 Td [(y 2 24 : 0975 =1. Inourpreviousexample,wedidnothaveenoughinformationtopindowntheexactlocationof Sasquatch.Toaccomplishthis,wewouldneedathirdobserver. Example 7.5.5 Byastrokeofluck,Kaiwasalsocampinginthewoodsduringtheeventsofthe previousexample.Hewaslocated6milesduenorthofJeandheardtheSasquatchcall18seconds afterJedid.UsethisaddedinformationtolocateSasquatch. Solution. KaiandJearenowthefociofasecondhyperbolawherethexeddistance d canbe determinedasbefore 760 miles hour 1hour 3600seconds 18seconds=3 : 8miles SinceJewaspositionedat )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 ; 0,weplaceKaiat )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 ; 6.Thisputsthecenterofthenew hyperbolaat )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 ; 3.PlottingKai'spositionandthenewcentergivesus

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7.5Hyperbolas 423 x y JeCarl Kai )]TJ/F34 7.9701 Tf 6.586 0 Td [(9 )]TJ/F34 7.9701 Tf 6.587 0 Td [(8 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123456 )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 5 6 Thesecondhyperbolaisvertical,soitmustbeoftheform y )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 2 b 2 )]TJ/F34 7.9701 Tf 12.647 5.375 Td [( x +5 2 a 2 =1.Asbefore,the distance d isthelengthofthemajoraxis,whichinthiscaseis2 b .Weget2 b =3 : 8sothat b =1 : 9 and b 2 =3 : 61.WithKai6milesdueNorthofJe,wehavethatthedistancefromthecenterto thefocusis c =3.Since a 2 + b 2 = c 2 ,weget a 2 = c 2 )]TJ/F53 10.9091 Tf 11.525 0 Td [(b 2 =9 )]TJ/F15 10.9091 Tf 11.525 0 Td [(3 : 61=5 : 39.Kaiheardthe SasquatchcallafterJe,soKaiisfartherfromSasquatchthanJe.ThusSasquatchmustlieon thesouthernbranchofthehyperbola y )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 2 3 : 61 )]TJ/F34 7.9701 Tf 12.448 5.374 Td [( x +5 2 5 : 39 =1.Lookingatthewesternbranchofthe hyperboladeterminedbyJeandCarlalongwiththesouthernbranchofthehyperboladetermined byKaiandJe,weseethatthereisexactlyonepointincommon,andthisiswhereSasquatch musthavebeenwhenitcalled. x y JeCarl Kai Sasquatch )]TJ/F34 7.9701 Tf 6.586 0 Td [(9 )]TJ/F34 7.9701 Tf 6.587 0 Td [(8 )]TJ/F34 7.9701 Tf 6.586 0 Td [(7 )]TJ/F34 7.9701 Tf 6.586 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123456 )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 5 6 Todeterminethecoordinatesofthispointofintersectionexactly,wewouldneedtechniquesfor solvingsystemsofnon-linearequations.WewillseethoselaterinSection8.7.Acalculatorcan

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424HookedonConics beofuseinapproximatingthesecoordinatesusingtheIntersectcommand.Inordertousethis command,however,werstneedtosolveeachofourhyperbolasfor y ,choosethecorrectequation toenterintothecalculator,andproceedfromthere.Weleavethisasanexercise. TheprocedureoutlinedinthetwopreviousexamplesisthebasisforLOngRangeAidtoNavigation LORAN forshort.WhileitappearstobelosingitspopularityduetoGlobalPositioningSatellites GPS,itremainsoneofmostimportantapplicationsofhyperbolas.

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7.5Hyperbolas 425 7.5.1Exercises 1.Graphthehyperbola.Findthecenter,thelineswhichcontainthetransverseandconjugate axes,thevertices,thefociandtheequationsoftheasymptotes. a y 2 9 )]TJ/F53 10.9091 Tf 12.105 7.38 Td [(x 2 16 =1 b x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 4 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [( y +3 2 9 =1 c y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 11 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 10 =1 d x +5 2 16 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 2 1 =1 2.Puttheequationinstandardform.Findthecenter,thelineswhichcontainthetransverse andconjugateaxes,thevertices,thefociandtheequationsoftheasymptotes. a12 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 y 2 +30 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(111=0b18 y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 2 +72 y +30 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(63=0 3.Findthestandardformoftheequationofthehyperbolawhichhasthegivenproperties. aCenter ; 7,Vertex ; 3,Focus ; 2 bVertex ; 1,Vertex ; 1,Focus )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 1 cFoci ; 5,Pointoncurve 2 ; 3 p 5 2 dVertex )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 ; 5,Asymptotes y = 1 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6+5 4.ThegraphofaverticalorhorizontalhyperbolaclearlyfailstheVerticalLineTest,Theorem 1.1,sotheequationofaverticalofhorizontalhyperboladoesnotdene y asafunctionof x 2 However,muchlikewithcircles,horizontalparabolasandellipses,wecansplitahyperbola intopieces,eachofwhichwouldindeedrepresent y asafunctionof x .Withthehelpofyour classmates,useyourcalculatortographthehyperbolasgiveninExercise1above.Howmany piecesdoyouneedforaverticalhyperbola?Howmanyforahorizontalhyperbola? 5.UseyourcalculatortondtheapproximatelocationofSasquatchinExample7.5.5. 6.Thelocationofanearthquake'sepicenter )]TJ/F15 10.9091 Tf 12.848 0 Td [(thepointonthesurfaceoftheEarthdirectly abovewheretheearthquakeactuallyoccurred )]TJ/F15 10.9091 Tf 12.513 0 Td [(canbedeterminedbyaprocesssimilarto howwelocatedSasquatchinExample7.5.5.AswesaidbackinExercise4ainSection6.1, earthquakesarecomplicatedeventsanditisnotourintenttoprovideacompletediscussion ofthescienceinvolvedinthem.Instead,werefertheinterestedreadertoacourseinGeology ortheU.S.GeologicalSurvey'sEarthquakeHazardsProgramfoundhere .Ourtechnique worksonlyforrelativelysmalldistancesbecauseweneedtoassumetheEarthisatinorder tousehyperbolasintheplane. 3 TheP-wavesP"standsforPrimaryofanearthquake 2 WewillseelaterinthetextthatthegraphsofcertainrotatedhyperbolaspasstheVerticalLineTest. 3 BackintheExercisesinSection1.1youwereaskedtoresearchpeoplewhobelievetheworldisat.Whatdid youdiscover?

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426HookedonConics inSasquatchiatravelat6kilometerspersecond. 4 StationArecordsthewavesrst.Then StationB,whichis100kilometersduenorthofStationA,recordsthewaves2secondslater. StationC,whichis150kilometersduewestofStationArecordsthewaves3secondsafter thatatotalof5secondsafterStationA.Whereistheepicenter? 7.Withthehelpofyourclassmates,researchtheshapeofcoolingtowersfornuclearpower plantsandotherwayshyperbolashavebeenusedinarchitectureanddesign. 8.Withthehelpofyourclassmates,researchtheCassegrainTelescope.Itusesthereective propertyofthehyperbolaaswellasthatoftheparabolatomakeaningenioustelescope. 4 Dependingonthecompositionofthecrustataspeciclocation,P-wavescantravelbetween5kpsand8kps.

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7.5Hyperbolas 427 7.5.2Answers 1.a y 2 9 )]TJ/F37 7.9701 Tf 12.104 4.295 Td [(x 2 16 =1 Center ; 0 Transverseaxison x =0 Conjugateaxison y =0 Vertices ; 3 ; ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 Foci ; 5 ; ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 Asymptotes y = 3 4 x x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.757 0 Td [(1123456 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 b x )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 2 4 )]TJ/F34 7.9701 Tf 12.105 5.375 Td [( y +3 2 9 =1 Center ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 Transverseaxison y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 Conjugateaxison x =2 Vertices ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 Foci+ p 13 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ 10.909 9.024 Td [(p 13 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 Asymptotes y = 3 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234567 )]TJ/F35 5.9776 Tf 5.757 0 Td [(10 )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 c y )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 2 11 )]TJ/F34 7.9701 Tf 12.105 5.374 Td [( x )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 2 10 =1 Center ; 3 Transverseaxison x =1 Conjugateaxison y =3 Vertices ; 3+ p 10 ; ; 3 )]TJ 10.909 9.024 Td [(p 10 Foci ; 3+ p 21 ; ; 3 )]TJ 10.909 9.024 Td [(p 21 Asymptotes y = p 10 p 11 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+3 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234567 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 7 8 9 d x +5 2 16 )]TJ/F34 7.9701 Tf 12.105 5.375 Td [( y )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 2 1 =1 Center )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ; 4 Transverseaxison y =4 Conjugateaxison x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 Vertices )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 ; 4 ; ; 4 Foci )]TJ/F15 10.9091 Tf 8.485 0 Td [(4+ p 17 ; 4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ 10.909 9.024 Td [(p 17 ; 4 Asymptotes y = 3 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(11 )]TJ/F35 5.9776 Tf 5.756 0 Td [(10 )]TJ/F35 5.9776 Tf 5.756 0 Td [(9 )]TJ/F35 5.9776 Tf 5.756 0 Td [(8 )]TJ/F35 5.9776 Tf 5.756 0 Td [(7 )]TJ/F35 5.9776 Tf 5.756 0 Td [(6 )]TJ/F35 5.9776 Tf 5.757 0 Td [(5 )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1123 1 2 3 4 5

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428HookedonConics 2.a x 2 3 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 2 12 =1 Center ; 5 Transverseaxison y =5 Conjugateaxison x =0 Vertices p 3 ; 5 ; )]TJ 8.485 9.025 Td [(p 3 ; 5 Foci p 15 ; 5 ; )]TJ 8.485 9.024 Td [(p 15 ; 5 Asymptotes y = 2 x +5 b y +2 2 5 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 18 =1 Center ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 Transverseaxison x =3 Conjugateaxison y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 Vertices ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+ p 5 ; ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ 10.909 9.024 Td [(p 5 Foci ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+ p 5 ; ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ 10.909 9.025 Td [(p 5 Asymptotes y = p 5 p 18 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 3.a y )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 2 16 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 9 =1 b x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 2 16 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 33 =1 c y 2 9 )]TJ/F53 10.9091 Tf 12.105 7.38 Td [(x 2 16 =1 d x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 2 256 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 2 64 =1 5.Sasquatchislocatedatthepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 9629 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 8113. 6.ByplacingStationAat ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(50andStationBat ; 50,thetwosecondtimedierence yieldsthehyperbola y 2 36 )]TJ/F53 10.9091 Tf 17.698 7.38 Td [(x 2 2464 =1withfociAandBandcenter ; 0.PlacingStationC at )]TJ/F15 10.9091 Tf 8.485 0 Td [(150 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(50andusingfociAandCgivesusacenterof )]TJ/F15 10.9091 Tf 8.485 0 Td [(75 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(50andthehyperbola x +75 2 225 )]TJ/F15 10.9091 Tf 12.134 7.381 Td [( y +50 2 5400 =1.Thepointofintersectionofthesetwohyperbolaswhichiscloser toAthanBandclosertoAthanCis )]TJ/F15 10.9091 Tf 8.485 0 Td [(57 : 8444 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 : 21336sothatistheepicenter.

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Chapter8 SystemsofEquationsandMatrices 8.1SystemsofLinearEquations:GaussianElimination Upuntilnow,whenweconcernedourselveswithsolvingdierenttypesofequationstherewasonly oneequationtosolveatatime.Givenanequation f x = g x ,wecouldcheckoursolutions geometricallybyndingwherethegraphsof y = f x and y = g x intersect.The x -coordinates oftheseintersectionpointscorrespondtothesolutionstotheequation f x = g x ,andthe y coordinateswerelargelyignored.Ifwemodifytheproblemandaskfortheintersectionpointsof thegraphsof y = f x and y = g x ,whereboththesolutionto x and y areofinterest,wehave whatisknownasa systemofequations ,usuallywrittenas y = f x y = g x The`curlybracket'notationmeanswearetondall pairs ofpoints x;y whichsatisfy both equations.Webeginourstudyofsystemsofequationsbyreviewingsomebasicnotionsfrom IntermediateAlgebra. Definition 8.1 A linearequationintwovariables isanequationoftheform a 1 x + a 2 y = c where a 1 a 2 and c arerealnumbersandatleastoneof a 1 and a 2 isnonzero. Forreasonswhichwillbecomeclearlaterinthesection,weareusingsubscriptsinDenition8.1 toindicatedierent,butxed,realnumbersandthosesubscriptshavenomathematicalmeaning beyondthat.Forexample,3 x )]TJ/F37 7.9701 Tf 11.913 4.931 Td [(y 2 =0 : 1isalinearequationintwovariableswith a 1 =3, a 2 = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 and c =0 : 1.Wecanalsoconsider x =5tobealinearequationintwovariablesbyidentifying a 1 =1, a 2 =0,and c =5. 1 If a 1 and a 2 areboth0,thendependingon c ,wegeteitheran equationwhichis always true,calledan identity ,oranequationwhichis never true,calleda contradiction .If c =0,thenweget0=0,whichisalwaystrue.If c 6 =0,thenwe'dhave 0 6 =0,whichisnevertrue.Eventhoughidentitiesandcontradictionshavealargeroletoplay 1 Criticsmayarguethat x =5isclearlyanequationinonevariable.Itcanalsobeconsideredanequationin117 variableswiththecoecientsof116variablessetto0.AswithmanyconventionsinMathematics,thecontextwill clarifythesituation.

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430SystemsofEquationsandMatrices intheupcomingsections,wedonotconsiderthemlinearequations.Thekeytoidentifyinglinear equationsistonotethatthevariablesinvolvedaretotherstpowerandthatthecoecientsofthe variablesarenumbers.Someexamplesofequationswhicharenon-linearare x 2 + y =1, xy =5and e 2 x +ln y =1.WeleaveittothereadertoexplainwhythesedonotsatisfyDenition8.1.From whatweknowfromSections1.2and2.1,thegraphsoflinearequationsarelines.Ifwecoupletwo ormorelinearequationstogether,ineecttondthepointsofintersectionoftwoormorelines, weobtaina systemoflinearequationsintwovariables .Ourrstexamplereviewssomeof thebasictechniquesrstlearnedinIntermediateAlgebra. Example 8.1.1 Solvethefollowingsystemsofequations.Checkyouransweralgebraicallyand graphically. 1. 2 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y =1 y =3 2. 3 x +4 y = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y =5 3. x 3 )]TJ/F34 7.9701 Tf 12.105 4.931 Td [(4 y 5 = 7 5 2 x 9 + y 3 = 1 2 4. 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 y =6 3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 y =9 5. 6 x +3 y =9 4 x +2 y =12 6. 8 < : x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y =0 x + y =2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x + y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 Solution. 1.Ourrstsystemisnearlysolvedforus.Thesecondequationtellsusthat y =3.Tondthe correspondingvalueof x ,we substitute thisvaluefor y intothetherstequationtoobtain 2 x )]TJ/F15 10.9091 Tf 11.131 0 Td [(3=1,sothat x =2.Oursolutiontothesystemis ; 3.Tocheckthisalgebraically, wesubstitute x =2and y =3intoeachequationandseethattheyaresatised.Wesee 2 )]TJ/F15 10.9091 Tf 11.353 0 Td [(3=1,and3=3,asrequired.Tocheckouranswergraphically,wegraphthelines 2 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y =1and y =3andverifythattheyintersectat ; 3. 2.Tosolvethesecondsystem,weusethe addition methodto eliminate thevariable x .We takethetwoequationsasgivenand`addequalstoequals'toobtain 3 x +4 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y =5 3 y =3 Thisgivesus y =1.Wenowsubstitute y =1intoeitherofthetwoequations,say )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x )]TJ/F53 10.9091 Tf 8.887 0 Td [(y =5, toget )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.991 0 Td [(1=5sothat x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Oursolutionis )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1.Substituting x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and y =1 intotherstequationgives3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+4= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,whichistrue,and,likewise,whenwecheck )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1inthesecondequation,weget )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.306 0 Td [(1=5,whichisalsotrue.Geometrically,the lines3 x +4 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y =5intersectat )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1.

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8.1SystemsofLinearEquations:GaussianElimination431 ; 3 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(11234 1 2 4 2 x )]TJ/F37 7.9701 Tf 8.468 0 Td [(y =1 y =3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 ; 1 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(4 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 x +4 y = )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F72 7.9701 Tf 7.594 0 Td [(3 x )]TJ/F75 7.9701 Tf 9.764 0 Td [(y =5 3.Theequationsinthethirdsystemaremoreapproachableifwecleardenominators.We multiplybothsidesoftherstequationby15andbothsidesofthesecondequationby18 toobtainthekinder,gentlersystem 5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 y =21 4 x +6 y =9 Addingthesetwoequationsdirectlyfailstoeliminateeitherofthevariables,butwenote thatifwemultiplytherstequationby4andthesecondby )]TJ/F15 10.9091 Tf 8.485 0 Td [(5,wewillbeinapositionto eliminatethe x term 20 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(48 y =84 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(30 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(45 )]TJ/F15 10.9091 Tf 8.485 0 Td [(78 y =39 Fromthisweget y = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 .Wecantemporarilyavoidtoomuchunpleasantnessbychoosingto substitute y = )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 intooneoftheequivalentequationswefoundbyclearingdenominators, sayinto5 x )]TJ/F15 10.9091 Tf 11.564 0 Td [(12 y =21.Weget5 x +6=21whichgives x =3.Ouransweris )]TJ/F15 10.9091 Tf 5 -8.837 Td [(3 ; )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 2 Atthispoint,wehavenochoice )]TJ/F15 10.9091 Tf 12.887 0 Td [(inordertocheckanansweralgebraically,wemustsee iftheanswersatisesbothofthe original equations,sowesubstitute x =3and y = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 intoboth x 3 )]TJ/F34 7.9701 Tf 12.325 4.932 Td [(4 y 5 = 7 5 and 2 x 9 + y 3 = 1 2 .Weleaveittothereadertoverifythatthesolution iscorrect.Graphingbothofthelinesinvolvedwithconsiderablecareyieldsanintersection pointof )]TJ/F15 10.9091 Tf 5 -8.836 Td [(3 ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 4.Aneeriecalmsettlesoverusaswecautiouslyapproachourfourthsystem.Doitsfriendly integercoecientsbeliesomethingmoresinister?Wenotethatifwemultiplybothsidesof therstequationby3andthebothsidesofthesecondequationby )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,wearereadyto eliminatethe x

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432SystemsofEquationsandMatrices 6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 y =18 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x +12 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(18 0=0 Weeliminatednotonlythe x ,butthe y aswellandweareleftwiththeidentity0=0.This meansthatthesetwodierentlinearequationsare,infact,equivalent.Inotherwords,ifan orderedpair x;y satisestheequation2 x )]TJ/F15 10.9091 Tf 11.099 0 Td [(4 y =6,it automatically satisestheequation 3 x )]TJ/F15 10.9091 Tf 10.319 0 Td [(6 y =9.Onewaytodescribethesolutionsettothissystemistousetherostermethod 2 andwrite f x;y :2 x )]TJ/F15 10.9091 Tf 11.37 0 Td [(4 y =6 g .Whilethisiscorrectandcorrespondsexactlytowhat's happeninggraphically,asweshallseeshortly,wetakethisopportunitytointroducethe notionofa parametricsolution .Ourrststepistosolve2 x )]TJ/F15 10.9091 Tf 11.736 0 Td [(4 y =6foroneofthe variables,say y = 1 2 x )]TJ/F34 7.9701 Tf 12.917 4.295 Td [(3 2 .Foreachvalueof x ,theformula y = 1 2 x )]TJ/F34 7.9701 Tf 12.917 4.295 Td [(3 2 determinesthe corresponding y -valueofasolution.Sincewehavenorestrictionon x ,itiscalleda free variable .Welet x = t ,aso-called`parameter',andget y = 1 2 t )]TJ/F34 7.9701 Tf 11.778 4.295 Td [(3 2 .Oursetofsolutionscan thenbedescribedas )]TJ/F53 10.9091 Tf 11.364 -8.836 Td [(t; 1 2 t )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(3 2 :
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8.1SystemsofLinearEquations:GaussianElimination433 5.Multiplyingbothsidesoftherstequationby2andthebothsidesofthesecondequation by )]TJ/F15 10.9091 Tf 8.485 0 Td [(3,wesetthestagetoeliminate x 12 x +6 y =18 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 y = )]TJ/F15 10.9091 Tf 8.484 0 Td [(36 0= )]TJ/F15 10.9091 Tf 8.485 0 Td [(18 Asinthepreviousexample,both x and y droppedoutoftheequation,butweareleftwith anirrevocablecontradiction,0= )]TJ/F15 10.9091 Tf 8.485 0 Td [(18.Thistellsusthatitisimpossibletondapair x;y whichsatisesbothequations;inotherwords,thesystemhasnosolution.Graphically,we seethatthelines6 x +3 y =9and4 x +2 y =12aredistinctandparallel,andassuchdonot intersect. 6.Wecanbegintosolveourlastsystembyaddingthersttwoequations x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y =0 + x + y =2 2 x =2 whichgives x =1.Substitutingthisintotherstequationgives1 )]TJ/F53 10.9091 Tf 11.32 0 Td [(y =0sothat y =1. Weseemtohavedeterminedasolutiontooursystem, ; 1.Whilethischecksinthe rsttwoequations,whenwesubstitute x =1and y =1intothethirdequation,weget )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2whichsimpliestothecontradiction )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Graphingthelines x )]TJ/F53 10.9091 Tf 9.225 0 Td [(y =0, x + y =2,and )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x + y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,weseethatthersttwolinesdo,infact,intersectat ; 1, however,allthreelinesneverintersectatthesamepointsimultaneously,whichiswhatis requiredifasolutiontothesystemistobefound. x y 12 )]TJ/F35 5.9776 Tf 5.756 0 Td [(3 )]TJ/F35 5.9776 Tf 5.756 0 Td [(2 )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 2 3 4 5 6 6 x +3 y =9 4 x +2 y =12 x y )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 1 y )]TJ/F37 7.9701 Tf 8.468 0 Td [(x =0 y + x =2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 x + y = )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 AfewremarksaboutExample8.1.1areinorder.Itisclearthatsomesystemsofequationshave solutions,andsomedonot.Thosewhichhavesolutionsarecalled consistent ,thosewithno

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434SystemsofEquationsandMatrices solutionarecalled inconsistent .Wealsodistinguishthetwodierenttypesofbehavioramong consistentsystems.Thosewhichadmitfreevariablesarecalled dependent ;thosewithnofree variablesarecalled independent 4 Usingthisnewvocabulary,weclassifynumbers1,2and3in Example8.1.1asconsistentindependentsystems,number4isconsistentdependent,andnumbers 5and6areinconsistent. 5 Thesystemin6aboveiscalled overdetermined ,sincewehavemore equationsthanvariables. 6 Notsurprisingly,asystemwithmorevariablesthanequationsiscalled underdetermined .Whilethesysteminnumber6aboveisoverdeterminedandinconsistent, thereexistoverdeterminedconsistentsystemsbothdependentandindependentandweleaveit tothereadertothinkaboutwhatishappeningalgebraicallyandgeometricallyinthesecases. Likewise,therearebothconsistentandinconsistentunderdeterminedsystems, 7 butaconsistent underdeterminedsystemoflinearequationsisnecessarilydependent. 8 InordertomovethissectionbeyondareviewofIntermediateAlgebra,wenowdenewhatismeant byalinearequationin n variables. Definition 8.2 A linearequationin n variables x 1 x 2 ,..., x n isanequationoftheform a 1 x 1 + a 2 x 2 + ::: + a n x n = c where a 1 a 2 ,... a n and c arerealnumbersandatleastoneof a 1 a 2 ..., a n isnonzero. Insteadofusingmorefamiliarvariableslike x y ,andeven z and/or w inDenition8.2,weuse subscriptstodistinguishthedierentvariables.Wehavenoideahowmanyvariablesmaybe involved,soweusenumberstodistinguishtheminsteadofletters.Thereisanendlesssupplyof distinctnumbers.Asanexample,thelinearequation3 x 1 )]TJ/F53 10.9091 Tf 9.425 0 Td [(x 2 =4representsthesamerelationship betweenthevariables x 1 and x 2 astheequation3 x )]TJ/F53 10.9091 Tf 11.259 0 Td [(y =4doesbetweenthevariables x and y Inaddition,justaswecannotcombinethetermsintheexpression3 x )]TJ/F53 10.9091 Tf 10.745 0 Td [(y ,wecannotcombinethe termsintheexpression3 x 1 )]TJ/F53 10.9091 Tf 11.392 0 Td [(x 2 .Couplingmorethanonelinearequationin n variablesresults ina systemoflinearequationsin n variables .Whensolvingthesesystems,itbecomes increasinglyimportanttokeeptrackofwhatoperationsareperformedtowhichequationsandto developastrategybasedonthekindofmanipulationswe'vealreadyemployed.Tothisend,we rstremindourselvesofthemaneuverswhichcanbeappliedtoasystemoflinearequationsthat resultinanequivalentsystem. 9 4 Inthecaseofsystemsoflinearequations,regardlessofthenumberofequationsorvariables,consistentindependentsystemshaveexactlyonesolution.Thereaderisencouragedtothinkaboutwhythisisthecaseforlinear equationsintwovariables.Hint:thinkgeometrically. 5 Theadjectives`dependent'and`independent'applyonlyto consistent systems-theydescribethetypeofsolutions. 6 Ifwethinkifeachvariablebeinganunknownquantity,thenostensibly,torecovertwounknownquantities, weneedtwopiecesofinformation-i.e.,twoequations.Havingmorethantwoequationssuggestswehavemore informationthannecessarytodeterminethevaluesoftheunknowns.Whilethisisnotnecessarilythecase,itdoes explainthechoiceofterminology`overdetermined'. 7 Weneedmorethantwovariablestogiveanexampleofthelatter. 8 Again,experiencewithsystemswithmorevariableshelpstoseethishere,asdoesasolidcourseinLinearAlgebra. 9 Thatis,asystemwiththesamesolutionset.

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8.1SystemsofLinearEquations:GaussianElimination435 Theorem 8.1 Givenasystemofequations,thefollowingmoveswillresultinanequivalent systemofequations. Interchangethepositionofanytwoequations. Replaceanequationwithanonzeromultipleofitself. a Replaceanequationwithitselfplusanonzeromultipleofanotherequation. a Thatis,anequationwhichresultsfrommultiplyingbothsidesoftheequationbythesamenonzeronumber. WehaveseenplentyofinstancesofthesecondandthirdmovesinTheorem8.1whenwesolved thesystemsExample8.1.1.Therstmove,whileitobviouslyadmitsanequivalentsystem,seems silly.Ourperceptionwillchangeasweconsidermoreequationsandmorevariablesinthis,and latersections. Considerthesystemofequations 8 > < > : x )]TJ/F34 7.9701 Tf 12.104 4.296 Td [(1 3 y + 1 2 z =1 y )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(1 2 z =4 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 Clearly z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1,andwesubstitutethisintothesecondequation y )]TJ/F34 7.9701 Tf 12.381 4.295 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=4toobtain y = 7 2 Finally,wesubstitute y = 7 2 and z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1intotherstequationtoget x )]TJ/F34 7.9701 Tf 12.712 4.295 Td [(1 3 )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(7 2 + 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=1, sothat x = 8 3 .Thereadercanverifythatthesevaluesof x y and z satisfyallthreeoriginal equations.Itistemptingforustowritethesolutiontothissystembyextendingtheusual x;y notationto x;y;z andlistoursolutionas )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(8 3 ; 7 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 .Thequestionquicklybecomeswhatdoes an`orderedtriple'like )]TJ/F34 7.9701 Tf 6.195 -4.542 Td [(8 3 ; 7 2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 represent?Justasorderedpairsareusedtolocatepointsonthe two-dimensionalplane,orderedtriplescanbeusedtolocatepointsinspace. 10 Moreover,justas equationsinvolvingthevariables x and y describegraphsofone-dimensionallinesandcurvesinthe two-dimensionalplane,equationsinvolvingvariables x y ,and z describeobjectscalled surfaces inthree-dimensionalspace.Eachoftheequationsintheabovesystemcanbevisualizedasaplane situatedinthree-space.Geometrically,thesystemistryingtondtheintersection,orcommon point,ofallthreeplanes.Ifyouimaginethreesheetsofnotebookpapereachrepresentingaportion oftheseplanes,youwillstarttoseethecomplexitiesinvolvedinhowthreesuchplanescanintersect. Belowisasketchofthethreeplanes.Itturnsoutthatanytwooftheseplanesintersectinaline, 11 soourintersectionpointiswhereallthreeoftheselinesmeet. 10 YouwereaskedtothinkaboutthisinExercise13inSection1.1. 11 Infact,theselinesaredescribedbytheparametricsolutionstothesystemsformedbytakinganytwoofthese equationsbythemselves.

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436SystemsofEquationsandMatrices Sincethegeometryforequationsinvolvingmorethantwovariablesiscomplicated,wewillfocus oureortsonthealgebra.Returningtothesystem 8 > < > : x )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 3 y + 1 2 z =1 y )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 z =4 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 wenotethereasonitwassoeasytosolveisthatthethirdequationissolvedfor z ,thesecond equationinvolvesonly y and z ,andsincethecoecientof y is1,itmakesiteasytosolvefor y usingourknownvaluefor z .Lastly,thecoecientof x intherstequationis1makingiteasyto substitutetheknownvaluesof y and z andthensolvefor x .Weformalizethispatternbelowfor themostgeneralsystemsoflinearequations.Again,weusesubscriptedvariablestodescribethe generalcase.Thevariablewiththesmallestsubscriptinagivenequationistypicallycalledthe leadingvariable ofthatequation. Definition 8.3 Asystemoflinearequationswithvariables x 1 x 2 ,... x n issaidtobein triangularform providedallofthefollowingconditionshold: 1.Thesubscriptsofthevariablesineachequationarealwaysincreasingfromlefttoright. 2.Theleadingvariableineachequationhascoecient1. 3.Thesubscriptontheleadingvariableinagivenequationisgreaterthanthesubscripton theleadingvariableintheequationaboveit. 4.Anyequationwithoutvariables a cannotbeplacedaboveanequationwithvariables. a necessarilyanidentityorcontradiction Inourprevioussystem,ifmaketheobviouschoices x = x 1 y = x 2 ,and z = x 3 ,weseethatthe

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8.1SystemsofLinearEquations:GaussianElimination437 systemisintriangularform. 12 Anexampleofamorecomplicatedsystemintriangularformis 8 > > < > > : x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 + x 4 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 6 =6 x 2 +2 x 3 =1 x 4 +3 x 5 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 6 =8 x 5 +9 x 6 =10 Ourgoalhenceforthwillbetotransformagivensystemoflinearequationsintotriangularform usingthemovesinTheorem8.1. Example 8.1.2 UseTheorem8.1toputthefollowingsystemsintotriangularformandthensolve thesystemifpossible.Classifyeachsystemasconsistentindependent,consistentdependent,or inconsistent. 1. 8 < : 3 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =3 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 y +3 z =16 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 2. 8 < : 2 x +3 y )]TJ/F53 10.9091 Tf 10.909 0 Td [(z =1 10 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(z =2 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 y +2 z =5 3. 8 < : 3 x 1 + x 2 + x 4 =6 2 x 1 + x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 =4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 =0 Solution. 1.Fordenitiveness,welabelthetopmostequationinthesystem E 1,theequationbeneaththat E 2,andsoforth.Wenowattempttoputthesystemintriangularformusinganalgorithm knownas GaussianElimination .Whatthismeansisthat,startingwith x ,wetransform thesystemsothatconditions2and3inDenition8.3aresatised.Thenwemoveonto thenextvariable,inthiscase y ,andrepeat.Sincethevariablesinalloftheequationshave aconsistentorderingfromlefttoright,ourrstmoveistogetan x in E 1'sspotwitha coecientof1.Whiletherearemanywaystodothis,theeasiestistoapplytherstmove listedinTheorem8.1andinterchange E 1and E 3. 8 < : E 13 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =3 E 22 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 y +3 z =16 E 3 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 Switch E 1and E 3 )332()223()222()222()222()223()222()222()222()223()332(! 8 < : E 1 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 E 22 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 y +3 z =16 E 33 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =3 TosatisfyDenition8.3,weneedtoeliminatethe x 'sfrom E 2and E 3.Weaccomplishthis byreplacingeachofthemwithasumofthemselvesandamultipleof E 1.Toeliminatethe x from E 2,weneedtomultiply E 1by )]TJ/F15 10.9091 Tf 8.485 0 Td [(2thenadd;toeliminatethe x from E 3,weneedto multiply E 1by )]TJ/F15 10.9091 Tf 8.485 0 Td [(3thenadd.ApplyingthethirdmovelistedinTheorem8.1twice,weget 8 < : E 1 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 E 22 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 y +3 z =16 E 33 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =3 Replace E 2with )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 E 1+ E 2 )490()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()489(! Replace E 3with )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 E 1+ E 3 8 < : E 1 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 E 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 y + z =6 E 32 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 12 Iflettersareusedinsteadofsubscriptedvariables,Denition8.3canbesuitablymodiedusingalphabetical orderofthevariablesinsteadofnumericalorderonthesubscriptsofthevariables.

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438SystemsofEquationsandMatrices NowweenforcetheconditionsstatedinDenition8.3forthevariable y .Tothatendwe needtogetthecoecientof y in E 2equalto1.WeapplythesecondmovelistedinTheorem 8.1andreplace E 2withitselftimes )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 2 8 < : E 1 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 E 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 y + z =6 E 32 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 Replace E 2with )]TJ/F35 5.9776 Tf 7.782 3.258 Td [(1 2 E 2 )269()222()223()222()222()222()222()223()222()222()222()223()222()269(! 8 < : E 1 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 E 2 y )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(1 2 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 E 32 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 z = )]TJ/F15 10.9091 Tf 8.484 0 Td [(12 Toeliminatethe y in E 3,weadd )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 E 2toit. 8 < : E 1 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 E 2 y )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(1 2 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 E 32 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 z = )]TJ/F15 10.9091 Tf 8.484 0 Td [(12 Replace E 3with )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 E 2+ E 3 )490()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()489(! 8 < : E 1 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 E 2 y )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(1 2 z = )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 E 3 )]TJ/F53 10.9091 Tf 8.485 0 Td [(z = )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 Finally,weapplythesecondmovefromTheorem8.1onelasttimeandmultiply E 3by )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 tosatisfytheconditionsofDenition8.3forthevariable z 8 < : E 1 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 E 2 y )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 z = )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 E 3 )]TJ/F53 10.9091 Tf 8.485 0 Td [(z = )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 Replace E 3with )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 E 3 )352()222()222()223()222()222()222()223()222()222()222()223()222()352(! 8 < : E 1 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 E 2 y )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 E 3 z =6 Nowweproceedtosubstitute.Pluggingin z =6into E 2gives y )]TJ/F15 10.9091 Tf 11.258 0 Td [(3= )]TJ/F15 10.9091 Tf 8.485 0 Td [(3sothat y =0. With y =0and z =6, E 1becomes x )]TJ/F15 10.9091 Tf 11.377 0 Td [(0+6=5,or x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Oursolutionis )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; 6. Weleaveittothereadertocheckthatsubstitutingtherespectivevaluesfor x y ,and z into theoriginalsystemresultsinthreeidentities.Sincewehavefoundasolution,thesystemis consistent;sincetherearenofreevariables,itisindependent. 2.Proceedingaswedidin1,ourrststepistogetanequationwith x inthe E 1positionwith 1asitscoecient.Sincethereisnoeasyx,wemultiply E 1by 1 2 8 < : E 12 x +3 y )]TJ/F53 10.9091 Tf 10.909 0 Td [(z =1 E 210 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(z =2 E 34 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 y +2 z =5 Replace E 1with 1 2 E 1 )293()222()223()222()222()222()223()222()222()222()222()223()293(! 8 < : E 1 x + 3 2 y )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 z = 1 2 E 210 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(z =2 E 34 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 y +2 z =5 Nowit'stimetotakecareofthe x 'sin E 2and E 3. 8 < : E 1 x + 3 2 y )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(1 2 z = 1 2 E 210 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(z =2 E 34 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 y +2 z =5 Replace E 2with )]TJ/F34 7.9701 Tf 6.587 0 Td [(10 E 1+ E 2 )296()222()222()222()223()222()222()222()223()222()222()222()222()223()222()222()222()296(! Replace E 3with )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 E 1+ E 3 8 < : E 1 x + 3 2 y )]TJ/F34 7.9701 Tf 12.104 4.296 Td [(1 2 z = 1 2 E 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 y +4 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 E 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 y +4 z =3

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8.1SystemsofLinearEquations:GaussianElimination439 Ournextstepistogetthecoecientof y in E 2equalto1.Tothatend,wehave 8 > < > : E 1 x + 3 2 y )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 2 z = 1 2 E 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 y +4 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 E 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 y +4 z =3 Replace E 2with )]TJ/F35 5.9776 Tf 9.609 3.258 Td [(1 15 E 2 )379()223()222()222()222()223()222()222()222()222()223()222()222()222()380(! 8 > < > : E 1 x + 3 2 y )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 z = 1 2 E 2 y )]TJ/F34 7.9701 Tf 14.222 4.295 Td [(4 15 z = 1 5 E 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(15 y +4 z =3 Finally,werid E 3of y 8 > < > : E 1 x + 3 2 y )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 z = 1 2 E 2 y )]TJ/F34 7.9701 Tf 14.221 4.295 Td [(4 15 z = 1 5 E 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 y +4 z =3 Replace E 3with15 E 2+ E 3 )320()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()320(! 8 > < > : E 1 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 E 2 y )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 2 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 E 30=6 Thelastequation,0=6,isacontradictionsothesystemhasnosolution.Accordingto Theorem8.1,sincethissystemhasnosolutions,neitherdoestheoriginal,thuswehavean inconsistentsystem. 3.Forourlastsystem,webeginbymultiplying E 1by 1 3 togetacoecientof1on x 1 8 < : E 13 x 1 + x 2 + x 4 =6 E 22 x 1 + x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 =4 E 3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 =0 Replace E 1with 1 3 E 1 )293()222()223()222()222()222()223()222()222()222()222()223()293(! 8 < : E 1 x 1 + 1 3 x 2 + 1 3 x 4 =2 E 22 x 1 + x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 =4 E 3 x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 =0 Nextweeliminate x 1 from E 2 8 > < > : E 1 x 1 + 1 3 x 2 + 1 3 x 4 =2 E 22 x 1 + x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 =4 E 3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 =0 Replace E 2 )346()222()222()222()223()222()222()222()223()345(! with )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 E 1+ E 2 8 > < > : E 1 x 1 + 1 3 x 2 + 1 3 x 4 =2 E 2 1 3 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(2 3 x 4 =0 E 3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 =0 Weswitch E 2and E 3togetacoecientof1for x 2 8 > < > : E 1 x 1 + 1 3 x 2 + 1 3 x 4 =2 E 2 1 3 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(2 3 x 4 =0 E 3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 =0 Switch E 2and E 3 )332()223()222()222()222()223()222()222()222()223()332(! 8 > < > : E 1 x 1 + 1 3 x 2 + 1 3 x 4 =2 E 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 =0 E 3 1 3 x 2 )]TJ/F53 10.9091 Tf 10.91 0 Td [(x 3 )]TJ/F34 7.9701 Tf 12.104 4.296 Td [(2 3 x 4 =0 Finally,weeliminate x 2 in E 3.

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440SystemsofEquationsandMatrices 8 > < > : E 1 x 1 + 1 3 x 2 + 1 3 x 4 =2 E 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 x 4 =0 E 3 1 3 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(2 3 x 4 =0 Replace E 3 )263()222()222()223()222()222()222()222()223()263(! with )]TJ/F35 5.9776 Tf 7.782 3.258 Td [(1 3 E 2+ E 3 8 > < > : E 1 x 1 + 1 3 x 2 + 1 3 x 4 =2 E 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 =0 E 30=0 Equation E 3reducesto0=0,whichisalwaystrue.Sincewehavenoequationswith x 3 or x 4 asleadingvariables,theyarebothfree,whichmeanswehaveaconsistentdependent system.Weparametrizethesolutionsetbyletting x 3 = s and x 4 = t andobtainfrom E 2 that x 2 =3 s +2 t .Substitutingthisand x 4 = t into E 1,wehave x 1 + 1 3 s +2 t + 1 3 t =2 whichgives x 1 =2 )]TJ/F53 10.9091 Tf 9.917 0 Td [(s )]TJ/F53 10.9091 Tf 9.917 0 Td [(t .Oursolutionistheset f )]TJ/F53 10.9091 Tf 9.917 0 Td [(s )]TJ/F53 10.9091 Tf 9.917 0 Td [(t; 2 s +3 t;s;t : < > : E 1 x + 3 2 y )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 z = 1 2 E 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 y +4 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 E 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 y +4 z =3 thatequations E 2and E 3whentakentogetherformacontradictionsincewehaveidenticallefthand sidesanddierentrighthandsides.Thealgorithmtakestwomorestepstoreachthiscontradiction. WealsonotethatsubstitutioninGaussianEliminationisdelayeduntilalltheeliminationisdone, thusitgetscalled back-substitution .Thismayalsobeinecientinmanycases.Restassured, thetechniqueofsubstitutionasyoumayhavelearneditinIntermediateAlgebrawillonceagain takecenterstageinSection8.7.Lastly,wenotethatthesystemin3aboveisunderdetermined, andasitisconsistent,wehavefreevariablesinouranswer.Weclosethissectionwithastandard `mixture'typeapplicationofsystemsoflinearequations. Example 8.1.3 Lucasneedstocreatea500millilitersmLofa40%acidsolution.Hehasstock solutionsof30%and90%acidaswellasallofthedistilledwaterhewants.Set-upandsolvea systemoflinearequationswhichdeterminesallofthepossiblecombinationsofthestocksolutions andwaterwhichwouldproducetherequiredsolution. Solution. Weareafterthreeunknowns,theamountinmLofthe30%stocksolutionwhich we'llcall x ,theamountinmLofthe90%stocksolutionwhichwe'llcall y andtheamount inmLofwaterwhichwe'llcall w .Wenowneedtodeterminesomerelationshipsbetweenthese variables.Ourgoalistoproduce500millilitersofa40%acidsolution.Thisproducthastwo deningcharacteristics.First,itmustbe500mL;second,itmustbe40%acid.Wetakeeach 13 Here,anychoiceof s and t willdetermineasolutionwhichisapointin4-dimensionalspace.Yeah,wehave troublevisualizingthat,too.

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8.1SystemsofLinearEquations:GaussianElimination441 ofthesequalitiesinturn.First,thetotalvolumeof500mLmustbethesumofthecontributed volumesofthetwostocksolutionsandthewater.Thatis amountof30%stocksolution+amountof90%stocksolution+amountofwater=500mL Usingourdenedvariables,thisreducesto x + y + w =500.Next,weneedtomakesurethenal solutionis40%acid.Sincewatercontainsnoacid,theacidwillcomefromthestocksolutionsonly. Wend40%of500mLtobe200mLwhichmeansthenalsolutionmustcontain200mLofacid. Wehave amountofacidin30%stocksolution+amountofacid90%stocksolution=200mL Theamountofacidin x mLof30%stockis0 : 30 x andtheamountofacidin y mLof90%solution is0 : 90 y .Wehave0 : 30 x +0 : 90 y =200.Convertingtofractions, 14 oursystemofequationsbecomes x + y + w =500 3 10 x + 9 10 y =200 Wersteliminatethe x fromthesecondequation E 1 x + y + w =500 E 2 3 10 x + 9 10 y =200 Replace E 2with )]TJ/F35 5.9776 Tf 9.609 3.258 Td [(3 10 E 1+ E 2 )239()223()222()222()222()223()222()222()222()222()223()222()222()222()223()222()222()240(! E 1 x + y + w =500 E 2 3 5 y )]TJ/F34 7.9701 Tf 14.222 4.295 Td [(3 10 w =50 Next,wegetacoecientof1ontheleadingvariablein E 2 E 1 x + y + w =500 E 2 3 5 y )]TJ/F34 7.9701 Tf 14.222 4.295 Td [(3 10 w =50 Replace E 2with 5 3 E 2 )293()222()223()222()222()222()223()222()222()222()222()223()293(! E 1 x + y + w =500 E 2 y )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 w = 250 3 Noticethatwehavenoequationtodetermine w ,andassuch, w isfree.Weset w = t andfrom E 2 get y = 1 2 t + 250 3 .Substitutinginto E 1gives x + )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 2 t + 250 3 + t =500sothat x = )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(3 2 t + 1250 3 .This systemisconsistent,dependentanditssolutionsetis f )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F34 7.9701 Tf 9.681 4.295 Td [(3 2 t + 1250 3 ; 1 2 t + 250 3 ;t :
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442SystemsofEquationsandMatrices 8.1.1Exercises 1.Putthefollowingsystemsoflinearequationsintotriangularformandthensolvethesystemifpossible.Classifyeachsystemasconsistentindependent,consistentdependent,or inconsistent. a )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x + y =17 x + y =5 b 8 < : x + y + z =3 2 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x +5 y +7 z =7 c 8 < : 4 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 2 y +6 z =30 x + z =5 d 8 < : 4 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 2 y +6 z =30 x + z =6 e x + y + z = )]TJ/F15 10.9091 Tf 8.484 0 Td [(17 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 z =0 f 8 > > < > > : 2 x 1 + x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 4 =16 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 1 + x 2 +12 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 x 1 +2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(16 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 =25 x 1 +2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 4 =11 g 8 > > < > > : x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 4 =0 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 + x 3 =0 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 3 + x 4 =1 h 8 > > < > > : x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 3 +3 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x 1 + x 2 +5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 =0 x 2 +5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 =1 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 x 3 +6 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2.FindtwootherformsoftheparametricsolutiontoExercise1cabovebyreorganizingthe equationssothat x or y canbethefreevariable. 3.AtTheOldHomeFill'erUpandKeepona-Truckin'Cafe,Mavismixestwodierenttypes ofcoeebeanstoproduceahouseblend.Thersttypecosts $ 3perpoundandthesecond costs $ 8perpound.HowmuchofeachtypedoesMavisusetomake50poundsofablend whichcosts $ 6perpound? 4.AtTheCrispyCritter'sHeadShopandPatchouliEmporiumalongwiththeirdriedupweeds, sunowerseedsandastrologicalpostcardstheysellanherbalteablend.Byweight,TypeI herbalteais30%peppermint,40%rosehipsand30%chamomile,TypeIIhaspercents40%, 20%and40%,respectively,andTypeIIIhaspercents35%,30%and35%,respectively.How muchofeachTypeofteaisneededtomake2poundsofanewblendofteathatisequal partspeppermint,rosehipsandchamomile? 5.DiscusswithyourclassmateshowyouwouldapproachExercise4aboveiftheyneededtouse upapoundofTypeIteatomakeroomontheshelfforanewcanister. 6.Discusswithyourclassmateswhyitisimpossibletomixa20%acidsolutionwitha40%acid solutiontoproducea60%acidsolution.Ifyouweretotrytomake100mLofa60%acid solutionusingstocksolutionsat20%and40%,respectively,whatwouldthetriangularform oftheresultingsystemlooklike?

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8.1SystemsofLinearEquations:GaussianElimination443 8.1.2Answers 1.Becausetriangularformisnotunique,wegiveonlyonepossibleanswertothatpartofthe question.Yoursmaybedierentandstillbecorrect. a x + y =5 y =7 Consistentindependent Solution )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 ; 7 b 8 > < > : x )]TJ/F34 7.9701 Tf 12.105 4.296 Td [(5 3 y )]TJ/F34 7.9701 Tf 12.104 4.296 Td [(7 3 z = )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(7 3 y + 5 4 z =2 z =0 Consistentindependent Solution ; 2 ; 0 c 8 > < > : x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 4 y + 1 4 z = 5 4 y +3 z =15 0=0 Consistentdependent Solution )]TJ/F53 10.9091 Tf 8.484 0 Td [(t +5 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 t +15 ;t forallrealnumbers t d 8 > < > : x )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 4 y + 1 4 z = 5 4 y +3 z =15 0=1 Inconsistent Nosolution e x + y + z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(17 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 z =0 Consistentdependent Solution )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 t )]TJ/F15 10.9091 Tf 10.909 0 Td [(17 ; 3 t;t forallrealnumbers t f 8 > > < > > : x 1 + 2 3 x 2 )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(16 3 x 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 4 = 25 3 x 2 +4 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 =2 0=0 0=0 Consistentdependent Solution s )]TJ/F53 10.9091 Tf 10.909 0 Td [(t +7 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 s +3 t +2 ;s;t forallrealnumbers s and t g 8 > > > < > > > : x 1 )]TJ/F53 10.9091 Tf 10.91 0 Td [(x 3 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 x 2 )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 x 4 =0 x 3 )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 x 4 =1 x 4 =4 Consistentindependent Solution ; 2 ; 3 ; 4 h 8 > > < > > : x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 3 +3 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x 2 +5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 = 1 2 0=1 0=0 Inconsistent Nosolution 2.If x isthefreevariablethenthesolutionis t; 3 t; )]TJ/F53 10.9091 Tf 8.485 0 Td [(t +5andif y isthefreevariablethenthe solutionis )]TJ/F34 7.9701 Tf 6.196 -4.542 Td [(1 3 t;t; )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 3 t +5 3.Mavisneeds20poundsof $ 3perpoundcoeeand30poundsof $ 8perpoundcoee. 4. 4 3 )]TJ/F34 7.9701 Tf 11.29 4.296 Td [(1 2 t poundsofTypeI, 2 3 )]TJ/F34 7.9701 Tf 11.29 4.296 Td [(1 2 t poundsofTypeIIand t poundsofTypeIIIwhere0 t 4 3 .

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444SystemsofEquationsandMatrices 8.2SystemsofLinearEquations:AugmentedMatrices InSection8.1weintroducedGaussianEliminationasameansoftransformingasystemoflinear equationsintotriangularformwiththeultimategoalofproducinganequivalentsystemoflinear equationswhichiseasiertosolve.Iftakeastepbackandstudytheprocess,weseethatallofour movesaredeterminedentirelybythe coecients ofthevariablesinvolved,andnotthevariables themselves.MuchthesamethinghappenedwhenwestudiedlongdivisioninSection3.2.Justas wedevelopedsyntheticdivisiontostreamlinethatprocess,inthissection,weintroduceasimilar bookkeepingdevicetohelpussolvesystemsoflinearequations.Tothatend,wedenea matrix asarectangulararrayofrealnumbers.Wetypicallyenclosematriceswithsquarebrackets,`['and `]',andwesizematricesbythenumberofrowsandcolumnstheyhave.Forexample,the size sometimescalledthe dimension of 30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(510 is2 3becauseithas2rowsand3columns.Theindividualnumbersinamatrixarecalledits entries andareusuallylabeledwithdoublesubscripts:thersttellswhichrowtheelementisin andthesecondtellswhichcolumnitisin.Therowsarenumberedfromtoptobottomandthe columnsarenumberedfromlefttoright.Matricesthemselvesareusuallydenotedbyuppercase letters A B C ,etc.whiletheirentriesareusuallydenotedbythecorrespondingletter.So,for instance,ifwehave A = 30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(510 then a 11 =3, a 12 =0, a 13 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, a 21 =2, a 22 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5,and a 23 =10.Weshallexplorematricesas mathematicalobjectswiththeirownalgebrainSection8.3andintroducethemheresolelyasa bookkeepingdevice.Considerthesystemoflinearequationsfromnumber2inExample8.1.2 8 < : E 12 x +3 y )]TJ/F53 10.9091 Tf 10.909 0 Td [(z =1 E 210 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(z =2 E 34 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 y +2 z =5 Weencodethissystemintoamatrixbyassigningeachequationtoacorrespondingrow.Within thatrow,eachvariableandtheconstantgetsitsowncolumn,andtoseparatethevariablesonthe lefthandsideoftheequationfromtheconstantsontherighthandside,weuseaverticalbar, j Notethatin E 2,since y isnotpresent,werecorditscoecientas0.Thematrixassociatedwith thissystemis xyzc E 1 E 2 E 3 2 4 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 100 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(92 5 3 5

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8.2SystemsofLinearEquations:AugmentedMatrices445 Thismatrixiscalledan augmentedmatrix becausethecolumncontainingtheconstantsis appendedtothematrixcontainingthecoecients. 1 Tosolvethissystem,wecanusethesame kindoperationsonthe rows ofthematrixthatweperformedonthe equations ofthesystem.More specically,wehavethefollowinganalogofTheorem8.1below. Theorem 8.2 RowOperations: Givenanaugmentedmatrixforasystemoflinearequations, thefollowingrowoperationsproduceanaugmentedmatrixwhichcorrespondstoanequivalent systemoflinearequations. Interchangeanytworows. Replacearowwithanonzeromultipleofitself. a Replacearowwithitselfplusanonzeromultipleofanotherrow. b a Thatis,therowobtainedbymultiplyingeachentryintherowbythesamenonzeronumber. b Whereweaddentriesincorrespondingcolumns. AsademonstrationofthemovesinTheorem8.2,werevisitsomeofthestepsthatwereusedin solvingthesystemsoflinearequationsinExample8.1.2ofSection8.1.Thereaderisencouragedto performtheindicatedoperationsontherowsoftheaugmentedmatrixtoseethatthemachinations areidenticaltowhatisdonetothecoecientsofthevariablesintheequations.Werstseea demonstrationofswitchingtworowsusingtherststepofpart1inExample8.1.2. 8 < : E 13 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =3 E 22 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 y +3 z =16 E 3 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 Switch E 1and E 3 )332()223()222()222()222()223()222()222()222()223()332(! 8 < : E 1 x )]TJ/F53 10.9091 Tf 10.91 0 Td [(y + z =5 E 22 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 y +3 z =16 E 33 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =3 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(43 16 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 5 3 5 Switch R 1and R 3 )353()222()223()222()222()222()223()222()222()222()353(! 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 5 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(43 16 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 3 5 Next,wehaveademonstrationofreplacingarowwithanonzeromultipleofitselfusingtherst stepofpart3inExample8.1.2. 8 < : E 13 x 1 + x 2 + x 4 =6 E 22 x 1 + x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 =4 E 3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 =0 Replace E 1with 1 3 E 1 )293()222()223()222()222()222()223()222()222()222()223()222()293(! 8 < : E 1 x 1 + 1 3 x 2 + 1 3 x 4 =2 E 22 x 1 + x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 =4 E 3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 =0 2 4 3101 6 21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 4 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 3 5 Replace R 1with 1 3 R 1 )314()222()222()223()222()222()222()222()223()222()222()222()314(! 2 4 1 1 3 0 1 3 2 21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 4 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 3 5 Finally,wehaveanexampleofreplacingarowwithitselfplusamultipleofanotherrowusingthe secondstepfrompart2inExample8.1.2. 1 WeshallstudythecoecientandconstantmatricesseparatelyinSection8.3.

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446SystemsofEquationsandMatrices 8 < : E 1 x + 3 2 y )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 2 z = 1 2 E 210 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(z =2 E 34 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 y +2 z =5 Replace E 2with )]TJ/F34 7.9701 Tf 6.586 0 Td [(10 E 1+ E 2 )296()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()222()296(! Replace E 3with )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 E 1+ E 3 8 < : E 1 x + 3 2 y )]TJ/F34 7.9701 Tf 12.105 4.295 Td [(1 2 z = 1 2 E 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 y +4 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 E 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 y +4 z =3 2 4 1 3 2 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 2 1 2 100 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(92 5 3 5 Replace R 2with )]TJ/F34 7.9701 Tf 6.587 0 Td [(10 R 1+ R 2 )327()222()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()326(! Replace R 3with )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 R 1+ R 3 2 4 1 3 2 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 1 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(154 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(154 3 3 5 Thematrixequivalentof`triangularform'is rowechelonform .Thereaderisencouragedto refertoDenition8.3forcomparison.Notethattheanalogof`leadingvariable'ofanequation is`leadingentry'ofarow.Specically,therstnonzeroentryifitexistsinarowiscalledthe leadingentry ofthatrow. Definition 8.4 Amatrixissaidtobein rowechelonform providedallofthefollowing conditionshold: 1.Therstnonzeroentryineachrowis1. 2.Theleading1ofagivenrowmustbetotherightoftheleading1oftherowaboveit. 3.Anyrowofallzeroscannotbeplacedabovearowwithnonzeroentries. Tosolveasystemofalinearequationsusinganaugmentedmatrix,weencodethesystemintoan augmentedmatrixandapplyGaussianEliminationtotherowstogetthematrixintorow-echelon form.Wethendecodethematrixandbacksubstitute.Thenextexampleillustratesthisnicely. Example 8.2.1 Useanaugmentedmatrixtotransformthefollowingsystemoflinearequations intotriangularform.Solvethesystem. 8 < : 3 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =8 x +2 y )]TJ/F53 10.9091 Tf 10.909 0 Td [(z =4 2 x +3 y )]TJ/F15 10.9091 Tf 10.91 0 Td [(4 z =10 Solution. Werstencodethesystemintoanaugmentedmatrix. 8 < : 3 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =8 x +2 y )]TJ/F53 10.9091 Tf 10.909 0 Td [(z =4 2 x +3 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 z =10 Encodeintothematrix )347()222()222()223()222()222()222()223()222()222()222()223()222()347(! 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 8 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 10 3 5 ThinkingbacktoGaussianEliminationatanequationslevel,ourrstorderofbusinessistoget x in E 1withacoecientof1.Atthematrixlevel,thismeansgettingaleading1in R 1.Thisisin accordancewiththerstcriteriainDenition8.4.Tothatend,weinterchange R 1and R 2. 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 8 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 10 3 5 Switch R 1and R 2 )353()222()223()222()222()222()223()222()222()222()353(! 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 8 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 10 3 5

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8.2SystemsofLinearEquations:AugmentedMatrices447 Ournextstepistoeliminatethe x 'sfrom E 2and E 3.Fromamatrixstandpoint,thismeanswe need0'sbelowtheleading1in R 1.Thisguaranteestheleading1in R 2willbetotherightofthe leading1in R 1inaccordancewiththesecondrequirementofDenition8.4. 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 8 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 10 3 5 Replace R 2with )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 R 1+ R 2 )243()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()243(! Replace R 3with )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 R 1+ R 3 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(74 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 3 5 Nowwerepeattheaboveprocessforthevariable y whichmeansweneedtogettheleadingentry in R 2tobe1. 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(74 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 3 5 Replace R 2with )]TJ/F35 5.9776 Tf 7.782 3.258 Td [(1 7 R 2 )290()222()222()222()223()222()222()222()223()222()222()222()223()289(! 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 01 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(4 7 4 7 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 3 5 Toguaranteetheleading1in R 3istotherightoftheleading1in R 2,wegeta0inthesecond columnof R 3. 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 01 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(4 7 4 7 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 3 5 Replace R 3with R 2+ R 3 )461()222()223()222()222()222()223()222()222()222()223()222()222()222()222()462(! 2 6 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 01 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(4 7 4 7 00 )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(18 7 18 7 3 7 5 Finally,wegettheleadingentryin R 3tobe1. 2 6 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 01 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(4 7 4 7 00 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(18 7 18 7 3 7 5 Replace R 3with )]TJ/F35 5.9776 Tf 9.608 3.259 Td [(7 18 R 3 )400()222()222()223()222()222()222()223()222()222()222()223()222()222()400(! 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 01 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(4 7 4 7 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 Decodingfromthematrixgivesasystemintriangularform 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 01 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(4 7 4 7 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 Decodefromthematrix )228()222()222()223()222()222()222()223()222()222()222()223()222()228(! 8 < : x +2 y )]TJ/F53 10.9091 Tf 10.909 0 Td [(z =4 y )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(4 7 z = 4 7 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 Weget z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, y = 4 7 z + 4 7 = 4 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+ 4 7 =0and x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 y + z +4= )]TJ/F15 10.9091 Tf 8.484 0 Td [(2+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+4=3fora nalanswerof ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Weleaveittothereadertocheck. AspartofGaussianElimination,weusedrowoperationstoobtain0'sbeneatheachleading1to putthematrixintorowechelonform.Ifwealsorequirethat0'saretheonlynumbersabovea leading1,wehavewhatisknownasthe reducedrowechelonform ofthematrix. Definition 8.5 Amatrixissaidtobein reducedrowechelonform providedbothofthe followingconditionshold: 1.Thematrixisinrowechelonform. 2.Theleading1saretheonlynonzeroentryintheirrespectivecolumns.

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448SystemsofEquationsandMatrices Ofwhatsignicanceisthereducedrowechelonformofamatrix?Toillustrate,let'staketherow echelonformfromExample8.2.1andperformthenecessarystepstoputintoreducedrowechelon form.Westartbyusingtheleading1in R 3tozerooutthenumbersintherowsaboveit. 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 01 )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(4 7 4 7 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 Replace R 1with R 3+ R 1 )462()222()222()223()222()222()222()223()222()222()222()222()223()222()222()222()462(! Replace R 2with 4 7 R 3+ R 2 2 4 120 3 010 0 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 Finally,wetakecareofthe2in R 1abovetheleading1in R 2. 2 4 120 3 010 0 001 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 5 Replace R 1with )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 R 2+ R 1 )243()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()243(! 2 4 100 3 010 0 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 Tooursurpriseanddelight,whenwedecodethismatrix,weobtainthesolutioninstantlywithout havingtodealwithanyback-substitutionatall. 2 4 100 3 010 0 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 Decodefromthematrix )228()222()223()222()222()222()222()223()222()222()222()223()222()228(! 8 < : x =3 y =0 z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 Notethatinthepreviousdiscussion,wecouldhavestartedwith R 2andusedittogetazeroabove itsleading1andthendonethesamefortheleading1in R 3.Bystartingwith R 3,however,weget morezerosrst,andthemorezerosthereare,thefastertheremainingcalculationswillbe. 2 Itis alsoworthnotingthatwhileamatrixhasseveral 3 rowechelonforms,ithasonlyonereducedrow echelonform.Theprocessbywhichwehaveputamatrixintoreducedrowechelonformiscalled Gauss-JordanElimination Example 8.2.2 Solvethefollowingsystemusinganaugmentedmatrix.UseGauss-JordanEliminationtoputtheaugmentedmatrixintoreducedrowechelonform. 8 < : x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 1 + x 4 =2 2 x 1 +4 x 3 =5 4 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 4 =3 Solution. Werstencodethesystemintoamatrix.Payattentiontothesubscripts! 8 < : x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 1 + x 4 =2 2 x 1 +4 x 3 =5 4 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 4 =3 Encodeintothematrix )347()222()222()223()222()222()222()223()222()222()222()223()222()347(! 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3101 2 2040 5 040 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 3 5 Next,wegetaleading1intherstcolumnof R 1. 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3101 2 2040 5 040 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 3 5 Replace R 1with )]TJ/F35 5.9776 Tf 7.782 3.259 Td [(1 3 R 1 )290()222()222()222()223()222()222()222()223()222()222()222()223()289(! 2 4 1 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 3 0 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(2 3 2040 5 040 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 3 5 2 Carlalsondsstartingwith R 3tobemoresymmetric,inapurelypoeticway. 3 innite,infact

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8.2SystemsofLinearEquations:AugmentedMatrices449 Nowweeliminatethenonzeroentrybelowourleading1. 2 4 1 )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 3 0 )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 3 )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(2 3 2040 5 040 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 3 5 Replace R 2with )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 R 1+ R 2 )243()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()243(! 2 6 4 1 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 3 0 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(2 3 0 2 3 4 2 3 19 3 040 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 3 7 5 Weproceedtogetaleading1in R 2. 2 6 4 1 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 3 0 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(2 3 0 2 3 4 2 3 19 3 040 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 3 7 5 Replace R 2with 3 2 R 2 )314()222()222()223()222()222()222()222()223()222()222()222()314(! 2 6 4 1 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 0 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(2 3 0161 19 2 040 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 3 7 5 Wenowzeroouttheentrybelowtheleading1in R 2. 2 6 4 1 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 0 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 3 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(2 3 0161 19 2 040 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 3 7 5 Replace R 3with )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 R 2+ R 3 )243()222()222()223()222()222()222()223()222()222()222()223()222()222()222()222()243(! 2 6 4 1 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 0 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 3 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(2 3 0161 19 2 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(35 3 7 5 Next,it'stimeforaleading1in R 3. 2 6 4 1 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 3 0 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(2 3 0161 19 2 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(35 3 7 5 Replace R 3with )]TJ/F35 5.9776 Tf 9.609 3.258 Td [(1 24 R 3 )400()222()223()222()222()222()223()222()222()222()222()223()222()222()400(! 2 6 4 1 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 0 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(2 3 0161 19 2 001 5 24 35 24 3 7 5 Thematrixisnowinrowechelonform.Togetthereducedrowechelonform,westartwiththe lastleading1weproducedandworktoget0'saboveit. 2 6 4 1 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 0 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(2 3 0161 19 2 001 5 24 35 24 3 7 5 Replace R 2with )]TJ/F34 7.9701 Tf 6.587 0 Td [(6 R 3+ R 2 )243()222()222()223()222()222()222()223()222()222()222()223()222()222()222()223()243(! 2 6 4 1 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 3 0 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 3 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(2 3 010 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 4 3 4 001 5 24 35 24 3 7 5 Lastly,wegeta0abovetheleading1of R 2. 2 6 4 1 )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(1 3 0 )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(1 3 )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(2 3 010 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 4 3 4 001 5 24 35 24 3 7 5 Replace R 1with 1 3 R 2+ R 1 )462()222()222()223()222()222()222()223()222()222()222()222()223()222()222()222()462(! 2 6 4 100 )]TJ/F34 7.9701 Tf 11.798 4.296 Td [(5 12 )]TJ/F34 7.9701 Tf 11.798 4.296 Td [(5 12 010 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(1 4 3 4 001 5 24 35 24 3 7 5 Atlast,wedecodetoget 2 6 4 100 )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(5 12 )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(5 12 010 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 4 3 4 001 5 24 35 24 3 7 5 Decodefromthematrix )228()222()223()222()222()222()222()223()222()222()222()223()222()228(! 8 > < > : x 1 )]TJ/F34 7.9701 Tf 14.222 4.295 Td [(5 12 x 4 = )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(5 12 x 2 )]TJ/F34 7.9701 Tf 12.104 4.295 Td [(1 4 x 4 = 3 4 x 3 + 5 24 x 4 = 35 24 Wehavethat x 4 isfreeandweassignittheparameter t .Weobtain x 3 = )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(5 24 t + 35 24 x 2 = 1 4 t + 3 4 and x 1 = 5 12 t )]TJ/F34 7.9701 Tf 13.773 4.295 Td [(5 12 .Oursolutionis )]TJ/F34 7.9701 Tf 14.677 -4.541 Td [(5 12 t )]TJ/F34 7.9701 Tf 14.222 4.295 Td [(5 12 ; 1 4 t + 3 4 ; )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(5 24 t + 35 24 ;t :
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450SystemsofEquationsandMatrices Likeallgoodalgorithms,puttingamatrixinrowechelonorreducedrowechelonformcaneasily beprogrammedintoacalculator,and,doubtless,yourgraphingcalculatorhassuchafeature.We usethisinournextexample. Example 8.2.3 Findthequadraticfunctionwhichpassesthroughthepoints )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 3, ; 4, ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2. Solution. AccordingtoDenition2.5,aquadraticfunctionhastheform f x = ax 2 + bx + c where a 6 =0.Ourgoalistond a b and c sothatthethreegivenpointsareonthegraphof f .If )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 3isonthegraphof f ,then f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=3,or a )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 + b )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+ c =3whichreducesto a )]TJ/F53 10.9091 Tf 10.888 0 Td [(b + c =3,anhonest-to-goodnesslinearequationwiththevariables a b and c .Sincethepoint ; 4isalsoonthegraphof f ,then f =4whichgivesustheequation4 a +2 b + c =4.Lastly, thepoint ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2isonthegraphof f givesus25 a +5 b + c = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Puttingthesetogether,weobtain asystemofthreelinearequations.Encodingthisintoanaugmentedmatrixproduces 8 < : a )]TJ/F53 10.9091 Tf 10.909 0 Td [(b + c =3 4 a +2 b + c =4 25 a +5 b + c = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 Encodeintothematrix )347()222()222()223()222()222()222()223()222()222()222()223()222()347(! 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 421 4 2551 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 Usingacalculator, 4 wend a = )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(7 18 b = 13 18 and c = 37 9 .Hence,theoneandonlyquadraticwhich tsthebillis f x = )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(7 18 x 2 + 13 18 x + 37 9 .Toverifythisanalytically,weseethat f )]TJ/F15 10.9091 Tf 8.484 0 Td [(1=3, f =4, and f = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Wecanusethecalculatortocheckoursolutionaswellbyplottingthethreedata pointsandthefunction f Thegraphof f x = )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(7 18 x 2 + 13 18 x + 37 9 withthepoints )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 3, ; 4and ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 4 We'vetorturedyouenoughalreadywithfractionsinthisexposition!

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8.2SystemsofLinearEquations:AugmentedMatrices451 8.2.1Exercises 1.Statewhetherthegivenmatrixisinreducedrowechelonform,rowechelonformonlyorin neitherofthoseforms. a 10 3 01 3 b 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(43 16 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 5 3 5 c 2 4 114 3 013 6 000 1 3 5 d 2 4 100 0 010 0 000 1 3 5 e 2 4 1043 0 0136 0 0000 0 3 5 f 114 3 013 6 2.Thefollowingmatricesareinreducedrowechelonform.Decodefromeachmatrixthesolution ofthecorrespondingsystemoflinearequationsorstatethatthesystemisinconsistent. a 10 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 01 7 b 2 4 100 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 010 20 001 19 3 5 c 2 4 1003 4 0106 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 0010 2 3 5 d 2 4 1003 0 0126 0 0000 1 3 5 e 2 6 6 4 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(81 7 014 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 0000 0 0000 0 3 7 7 5 f 2 4 109 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 20 000 0 3 5 g 2 6 6 4 1000 0 0143 0 0000 1 0000 0 3 7 7 5 3.Solvethefollowingsystemsoflinearequationsusingthetechniquesdiscussedinthissection. Compareandcontrastthesetechniqueswiththoseyouusedtosolvethesystemsinthe ExercisesinSection8.1. a )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x + y =17 x + y =5 b 8 < : x + y + z =3 2 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x +5 y +7 z =7 c 8 < : 4 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =5 2 y +6 z =30 x + z =5 d 8 > > < > > : x 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(x 4 =0 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 + x 3 =0 )]TJ/F53 10.9091 Tf 8.485 0 Td [(x 3 + x 4 =1

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452SystemsofEquationsandMatrices 4.ThepriceforadmissionintotheStitz-ZeagerSasquatchMuseumandResearchStationis $ 15 foradultsand $ 8forkids13yearsoldandyounger.WhentheZahlenreichfamilyvisitsthe museumtheirbillis $ 38andwhentheNullsatzfamilyvisitstheirbillis $ 39.Onedayboth familieswenttogetherandtookanadultbabysitteralongtowatchthekidsandthetotal admissionchargewas $ 92.Laterthatsummer,theadultsfrombothfamilieswentwithout thekidsandthebillwas $ 45.Isthatenoughinformationtodeterminehowmanyadults andchildrenareineachfamily?Ifnot,statewhethertheresultingsystemisinconsistentor consistentdependent.Inthelattercase,giveatleasttwoplausiblesolutions. 5.UsethetechniqueinExample8.2.3tondthelinebetweenthepoints )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 4and ; 1. Howdoesyouranswercomparetotheslope-interceptformofthelineinEquation2.3? 6.Withthehelpofyourclassmates,ndatleasttwodierentrowechelonformsforthematrix 12 3 412 8

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8.2SystemsofLinearEquations:AugmentedMatrices453 8.2.2Answers 1.aReducedrowechelonform bNeither cRowechelonformonly dReducedrowechelonform eReducedrowechelonform fRowechelonformonly 2.a )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 7 b )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 20 ; 19 c )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 t +4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 t )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 ; 2 ;t forallrealnumbers t dInconsistent e s )]TJ/F53 10.9091 Tf 10.909 0 Td [(t +7 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 s +3 t +2 ;s;t forallrealnumbers s and t f )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 t )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ; 4 t +20 ;t forallrealnumbers t gInconsistent 3.a )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 7 b ; 2 ; 0 c )]TJ/F53 10.9091 Tf 8.485 0 Td [(t +5 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 t +15 ;t forallrealnumbers t d ; 2 ; 3 ; 4 4.Let x 1 and x 2 bethenumbersofadultsandchildren,respectively,intheZahlenreichfamily andlet x 3 and x 4 bethenumbersofadultsandchildren,respectively,intheNullsatzfamily. Thesystemofequationsdeterminedbythegiveninformationis 8 > > < > > : 15 x 1 +8 x 2 =38 15 x 3 +8 x 4 =39 15 x 1 +8 x 2 +15 x 3 +8 x 4 =77 15 x 1 +15 x 3 =45 WesubtractedthecostofthebabysitterinE3sotheconstantis77,not92.Thissystemis consistentdependentanditssolutionis )]TJ/F34 7.9701 Tf 8.312 -4.541 Td [(8 15 t + 2 5 ; )]TJ/F53 10.9091 Tf 8.484 0 Td [(t +4 ; )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(8 15 t + 13 5 ;t .Ourvariablesrepresentnumbersofadultsandchildrensotheymustbewholenumbers.Runningthroughthe values t =0 ; 1 ; 2 ; 3 ; 4yieldsonlyonesolutionwhereallfourvariablesarewholenumbers; t =3givesus ; 1 ; 1 ; 3.Thusthereare2adultsand1childintheZahlenreichsand1adult and3kidsintheNullsatzs.

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454SystemsofEquationsandMatrices 8.3MatrixArithmetic InSection8.2,weusedaspecialclassofmatrices,theaugmentedmatrices,toassistusinsolving systemsoflinearequations.Inthissection,westudymatricesasmathematicalobjectsoftheir ownaccord,temporarilydivorcedfromsystemsoflinearequations.Todosoconvenientlyrequires somemorenotation.Whenwewrite A =[ a ij ] m n ,wemean A isan m by n matrix 1 and a ij isthe entryfoundinthe i throwand j thcolumn.Schematically,wehave j countscolumns fromlefttoright )462()222()223()222()222()222()222()223()222()222()222()223()222()222()462(! A = 2 6 6 6 4 a 11 a 12 a 12 a 21 a 22 a 2 n . . . . a m 1 a m 2 a mn 3 7 7 7 5 ? ? ? ? ? ? ? ? y i countsrows fromtoptobottom Withthisnewnotationwecandenewhatitmeansfortwomatricestobeequal. Definition 8.6 MatrixEquality: Twomatricesaresaidtobe equal iftheyarethesamesize andtheircorrespondingentriesareequal.Morespecically,if A =[ a ij ] m n and B =[ b ij ] p r ,we write A = B provided 1. m = p and n = r 2. a ij = b ij forall1 i m andall1 j n Essentially,twomatricesareequaliftheyarethesamesizeandtheyhavethesamenumbersin thesamespots. 2 Forexample,thetwo2 3matricesbeloware,despiteappearances,equal. 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(29 25117 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 = ln 3 p )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 e 2ln 125 2 = 3 3 2 13log : 001 Nowthatwehaveanagreeduponunderstandingofwhatitmeansfortwomatricestoequaleach other,wemaybegindeningarithmeticoperationsonmatrices.Ourrstoperationisaddition. Definition 8.7 MatrixAddition: Giventwomatricesofthesamesize,thematrixobtained byaddingthecorrespondingentriesofthetwomatricesiscalledthe sum ofthetwomatrices. Morespecically,if A =[ a ij ] m n and B =[ b ij ] m n ,wedene A + B =[ a ij ] m n +[ b ij ] m n =[ a ij + b ij ] m n Asanexample,considerthesumbelow. 1 Recallthatmeans A has m rowsand n columns. 2 Criticsmaywellask:Whynotleaveitatthat?WhytheneedforallthenotationinDenition8.6?Itisthe authors'attempttoexposeyoutothewonderfulworldofmathematicalprecision.

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8.3MatrixArithmetic455 2 4 23 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 81 3 5 = 2 4 2+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(13+4 4+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0+8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7+1 3 5 = 2 4 17 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 5 Itisworththereader'stimetothinkwhatwouldhavehappenedhadwereversedtheorderofthe summandsabove.Aswewouldexpect,wearriveatthesameanswer.Ingeneral, A + B = B + A formatrices A and B ,providedtheyarethesamesizesothatthesumisdenedintherstplace. Thisisthe commutativelaw ofmatrixaddition.Toseewhythisistrueingeneral,weappealto thedenitionofmatrixaddition.Given A =[ a ij ] m n and B =[ b ij ] m n A + B =[ a ij ] m n +[ b ij ] m n =[ a ij + b ij ] m n =[ b ij + a ij ] m n =[ b ij ] m n +[ a ij ] m n = B + A wherethesecondequalityisthedenitionof A + B ,thethirdequalityholdsbythecommutative lawofrealnumberaddition,andthefourthequalityisthedenitionof B + A .Inotherwords, matrixadditioniscommutativebecauserealnumberadditionis.Asimilarargumentshowsthe associativelaw ofmatrixadditionalsoholds,inheritedinturnfromtheassociativelawofreal numberaddition.Specically,formatrices A B ,and C ofthesamesize, A + B + C = A + B + C Inotherwords,whenaddingmorethantwomatrices,itdoesn'tmatterhowtheyaregrouped.This meansthatwecanwrite A + B + C withoutparenthesesandthereisnoambiguityastowhatthis means. 3 Thesepropertiesandmorearesummarizedinthefollowingtheorem. Theorem 8.3 PropertiesofMatrixAddition CommutativeProperty: Forall m n matrices, A + B = B + A AssociativeProperty: Forall m n matrices, A + B + C = A + B + C IdentityProperty: If0 m n isthe m n matrixwhoseentriesareall0,then0 m n is calledthe m n additiveidentity andforall m n matrices A A +0 m n =0 m n + A = A InverseProperty: Foreverygiven m n matrix A ,thereisauniquematrixdenoted )]TJ/F53 10.9091 Tf 8.485 0 Td [(A calledthe additiveinverseof A suchthat A + )]TJ/F53 10.9091 Tf 8.485 0 Td [(A = )]TJ/F53 10.9091 Tf 8.485 0 Td [(A + A =0 m n Theidentitypropertyiseasilyveriedbyresortingtothedenitionofmatrixaddition;justasthe number0istheadditiveidentityforrealnumbers,thematrixcomprisedofall0'sdoesthesamejob formatrices.Theinversepropertyisabitcurious-justwhatismeantby )]TJ/F53 10.9091 Tf 8.485 0 Td [(A ?Algebrasuggests tousthat,justlike )]TJ/F53 10.9091 Tf 8.485 0 Td [(x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x )]TJ/F53 10.9091 Tf 8.485 0 Td [(A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 A ,butnowweareleftwiththequestion,justwhatis meantby )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 A ?Howdowemultiplyamatrixbyarealnumber?Let'sthinkbacktoelementary schoolwhenyourstlearnedmultiplication.Youmayrecallthenotionthatmultiplication,atleast byanaturalnumber,isaformof`rapidaddition'inthat2+2+2=3 2.Weknowfromalgebra 4 3 Atechnicaldetailwhichissadlylostonmostreaders. 4 TheDistributiveProperty,inparticular.

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456SystemsofEquationsandMatrices that3 x = x + x + x ,soitseemsnaturalthatgivenamatrix A ,wewoulddene3 A = A + A + A If A =[ a ij ] m n ,wehave 3 A = A + A + A =[ a ij ] m n +[ a ij ] m n +[ a ij ] m n =[ a ij + a ij + a ij ] m n =[3 a ij ] m n Inotherwords,multiplyingthe matrix inthisfashionby3isthesameasmultiplying eachentry by3.Thisleadsustothefollowingdenition. Definition 8.8 Scalar a Multiplication: Wedenetheproductofarealnumberandamatrix tobethematrixobtainedbymultiplyingeachofitsentriesbysaidrealnumber.Morespecically, if k isarealnumberand A =[ a ij ] m n ,wedene kA = k [ a ij ] m n =[ ka ij ] m n a Theword`scalar'herereferstorealnumbers.`Scalarmultiplication'inthiscontextmeanswearemultiplying amatrixbyarealnumberascalar. Returningtoourdiscussionconcerningadditiveinverses,wegetfromDenition8.8that )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1[ a ij ] m n =[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 a ij ] m n =[ )]TJ/F53 10.9091 Tf 8.485 0 Td [(a ij ] m n Inotherwords,multiplying A by )]TJ/F15 10.9091 Tf 8.485 0 Td [(1givesthematrixwhoseentriesaretheadditiveinversesofthe entriesin A .Thusitmakessensethat )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 A = )]TJ/F53 10.9091 Tf 8.485 0 Td [(A istheadditiveinverseofthe matrix A ,andto check,wendtheirsumisthezeromatrix,asrequired: A + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 A =[ a ij ] m n +[ )]TJ/F53 10.9091 Tf 8.485 0 Td [(a ij ] m n =[ a ij + )]TJ/F53 10.9091 Tf 8.485 0 Td [(a ij ] m n =[0] m n =0 m n X Withtheadditiveinverseofamatrixwellinhand,wecannowdenematrixsubtraction.Youmay rememberfromarithmeticthat a )]TJ/F53 10.9091 Tf 11.018 0 Td [(b = a + )]TJ/F53 10.9091 Tf 8.484 0 Td [(b ;thatis,subtractioncanberewrittenas`adding theopposite.'Weextendthisconcepttomatrices.Fortwomatrices A and B ofthesamesize,we dene A )]TJ/F53 10.9091 Tf 10.909 0 Td [(B = A + )]TJ/F53 10.9091 Tf 8.485 0 Td [(B .Atthelevelofentries,thisamountsto A )]TJ/F53 10.9091 Tf 10.909 0 Td [(B = A + )]TJ/F53 10.9091 Tf 8.485 0 Td [(B =[ a ij ] m n +[ )]TJ/F53 10.9091 Tf 8.485 0 Td [(b ij ] m n =[ a ij + )]TJ/F53 10.9091 Tf 8.484 0 Td [(b ij ] m n =[ a ij )]TJ/F53 10.9091 Tf 10.909 0 Td [(b ij ] m n Thustosubtracttwomatricesofequalsize,wesubtracttheircorrespondingentries.Surprised? Onemaywellwonderwhytheword`scalar'isusedfor`realnumber.'Ithaseverythingtodowith `scaling'factors. 5 Apoint P x;y intheplanecanberepresentedbyitspositionmatrix, P as follows x;y $ P = x y Supposewetakethepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1andmultiplyitspositionmatrixby3.Wehave 3 P =3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 = 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 5 SeeSection1.8.

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8.3MatrixArithmetic457 whichcorrespondstothepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 ; 3.Wecanimaginetaking )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 1to )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 ; 3inthisfashionas adilationbyafactorof3inboththehorizontalandverticaldirections.Doingthistoallpoints x;y intheplane,therefore,hastheeectofmagnifyingscalingtheplanebyafactorof3. Asdidmatrixaddition,scalarmultiplicationinheritsmanypropertiesfromrealnumberarithmetic. Belowwesummarizetheseproperties. Theorem 8.4 PropertiesofScalarMultiplication AssociativeProperty: Forevery m n matrix A andscalars k and r kr A = k rA IdentityProperty: Forall m n matrices A ,1 A = A AdditiveInverseProperty: Forall m n matrices A )]TJ/F53 10.9091 Tf 8.485 0 Td [(A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 A DistributivePropertyofScalarMultiplicationoverScalarAddition: Forevery m n matrix A andscalars k and r k + r A = kA + rA DistributivePropertyofScalarMultiplicationoverMatrixAddition: Forall m n matrices A and B scalars k k A + B = kA + kB ZeroProductProperty: If A isan m n matrixand k isascalar,then kA =0 m n ifandonlyif k =0or A =0 m n Aswiththeotherresultsinthissection,Theorem8.4canbeprovedusingthedenitionsofscalar multiplicationandmatrixaddition.Forexample,toprovethat k A + B = kA + kB forascalar k and m n matrices A and B ,westartbyadding A and B ,thenmultiplyingby k andseeinghow thatcompareswiththesumof kA and kB k A + B = k )]TJ/F15 10.9091 Tf 5 -8.837 Td [([ a ij ] m n +[ b ij ] m n = k [ a ij + b ij ] m n =[ k a ij + b ij ] m n =[ ka ij + kb ij ] m n Asfor kA + kB ,wehave kA + kB = k [ a ij ] m n + k [ b ij ] m n =[ ka ij ] m n +[ kb ij ] m n =[ ka ij + kb ij ] m n X whichestablishestheproperty.Theremainingpropertiesarelefttothereader.Thepropertiesin Theorems8.3and8.4establishanalgebraicsystemthatletsustreatmatricesandscalarsmoreor lessaswewouldrealnumbersandvariables,asournextexampleillustrates. Example 8.3.1 Solveforthematrix A :3 A )]TJ/F55 10.9091 Tf 10.367 15.382 Td [( 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 35 +5 A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(42 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 + 1 3 912 )]TJ/F15 10.9091 Tf 8.485 0 Td [(339

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458SystemsofEquationsandMatrices Solution. 3 A )]TJ/F55 10.9091 Tf 10.909 15.382 Td [( 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 35 +5 A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(42 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 + 1 3 912 )]TJ/F15 10.9091 Tf 8.485 0 Td [(339 3 A + )]TJ/F55 10.9091 Tf 10.303 15.382 Td [( 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 35 +5 A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(42 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 + )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 3 )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 3 )]TJ/F34 7.9701 Tf 6.196 -4.541 Td [(1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F34 7.9701 Tf 6.195 -4.541 Td [(1 3 # 3 A + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 35 +5 A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(42 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 + 34 )]TJ/F15 10.9091 Tf 8.485 0 Td [(113 3 A + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 35 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 A = )]TJ/F15 10.9091 Tf 8.484 0 Td [(16 511 3 A + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 A = )]TJ/F15 10.9091 Tf 8.484 0 Td [(16 511 3 A + )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 A = )]TJ/F15 10.9091 Tf 8.484 0 Td [(16 511 3 A + )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 A + )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(16 511 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 A + )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 + )]TJ/F55 10.9091 Tf 10.303 15.382 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(16 511 + )]TJ/F55 10.9091 Tf 10.303 15.382 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 A +0 2 2 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(16 511 )]TJ/F55 10.9091 Tf 10.909 15.382 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(311 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 A = 35 817 )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F34 7.9701 Tf 9.681 4.296 Td [(1 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 A = )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 2 35 817 \000 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 A = )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 2 )]TJ/F54 10.9091 Tf 5 -8.837 Td [()]TJ/F34 7.9701 Tf 9.68 4.296 Td [(1 2 )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 )]TJ/F54 10.9091 Tf 5 -8.836 Td [()]TJ/F34 7.9701 Tf 9.68 4.295 Td [(1 2 # 1 A = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(5 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(17 2 # A = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(3 2 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(5 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(17 2 # Thereaderisencouragedtocheckouranswerintheoriginalequation. Whilethesolutiontothepreviousexampleiswritteninexcruciatingdetail,inpracticemanyof thestepsaboveareomitted.Wehavespelledouteachstepinthisexampletoencouragethereader

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8.3MatrixArithmetic459 tojustifyeachstepusingthedenitionsandpropertieswehaveestablishedthusfarformatrix arithmetic.ThereaderisencouragedtosolvetheequationinExample8.3.1astheywouldany otherlinearequation. Wenowturnourattentionto matrixmultiplication -thatis,multiplyingamatrixbyanother matrix.Basedonthe`nosurprises'trendsofarinthesection,youmayexpectthatinorderto multiplytomatrices,theymustbeofthesamesizeandyoundtheproductmymultiplyingthe correspondingentries.Whilethiskindofproductisusedinotherareaofmathematics, 6 wedene matrixmultiplicationtoserveusinsolvingsystemsoflinearequations.Tothatend,webeginby deningtheproductofarowandacolumn.Wemotivatethegeneraldenitionwithanexample. Considerthetwomatrices A and B below. A = 20 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1035 B = 2 4 312 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 48 )]TJ/F15 10.9091 Tf 8.485 0 Td [(59 50 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 3 5 Let R 1denotetherstrowof A and C 1denotetherstcolumnof B .Tondthe`product'of R 1 with C 1,denoted R 1 C 1,werstndtheproductoftherstentryin R 1andtherstentryin C 1. Next,weaddtothattheproductofthesecondentryin R 1andthesecondentryin C 1.Finally, wetakethatsumandweaddtothattheproductofthelastentryin R 1andthelastentryin C 1. Usingentrynotation, R 1 C 1= a 11 b 11 + a 12 b 21 + a 13 b 31 =++ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=6+0+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(5=1. Wecanvisualizethisschematicallyasfollows 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1035 2 4 312 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 48 )]TJ/F15 10.9091 Tf 8.485 0 Td [(59 50 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 3 5 )445()223()222()222()222()223()222()222()445(! 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 5 ? ? ? ? ? y | {z } )445()223()222()222()222()223()222()222()445(! 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 5 ? ? ? ? ? y | {z } )445()223()222()222()222()223()222()222()445(! 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 5 ? ? ? ? ? y | {z } a 11 b 11 + a 12 b 21 + a 13 b 31 ++ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 Tond R 2 C 3where R 2denotesthesecondrowof A and C 3denotesthethirdcolumnof B ,we proceedsimilarly.Westartwithndingtheproductoftherstentryof R 2withtherstentryin C 3thenaddtoittheproductofthesecondentryin R 2withthesecondentryin C 3,andsoforth. Usingentrynotation,wehave R 2 C 3= a 21 b 13 + a 22 b 23 + a 23 b 33 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(10+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(5+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= )]TJ/F15 10.9091 Tf 8.485 0 Td [(45. Schematically, 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 3 5 2 4 31 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 48 )]TJ/F15 10.9091 Tf 8.485 0 Td [(59 50 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 3 5 6 SeethisarticleontheHadamardProduct .

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460SystemsofEquationsandMatrices )445()223()222()222()222()223()222()222()445(! )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 35 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ? ? ? ? ? y | {z } )445()223()222()222()222()223()222()222()445(! )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 3 5 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ? ? ? ? ? y | {z } )445()223()222()222()222()223()222()222()445(! )]TJ/F15 10.9091 Tf 8.485 0 Td [(103 5 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ? ? ? ? ? y | {z } a 21 b 13 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(10= )]TJ/F15 10.9091 Tf 8.485 0 Td [(20+ a 22 b 23 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5= )]TJ/F15 10.9091 Tf 8.485 0 Td [(15+ a 23 b 33 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 Generalizingthisprocess,wehavethefollowingdenition. Definition 8.9 ProductofaRowandaColumn: Suppose A =[ a ij ] m n and B =[ b ij ] n r Let Ri denotethe i throwof A andlet Cj denotethe j thcolumnof B .The productof R i and C j ,denoted R i C j istherealnumberdenedby Ri Cj = a i 1 b 1 j + a i 2 b 2 j + :::a in b nj Notethatinordertomultiplyarowbyacolumn,thenumberofentriesintherowmustmatch thenumberofentriesinthecolumn.Wearenowinthepositiontodenematrixmultiplication. Definition 8.10 MatrixMultiplication: Suppose A =[ a ij ] m n and B =[ b ij ] n r .Let Ri denotethe i throwof A andlet Cj denotethe j thcolumnof B .The productof A and B denoted AB ,isthematrixdenedby AB =[ Ri Cj ] m r thatis AB = 2 6 6 6 4 R 1 C 1 R 1 C 2 :::R 1 Cr R 2 C 1 R 2 C 2 :::R 2 Cr . . . . Rm C 1 Rm C 2 :::Rm Cr 3 7 7 7 5 ThereareanumberofsubtletiesinDenition8.10whichwarrantcloserinspection.Firstand foremost,Denition8.10tellsusthatthe ij -entryofamatrixproduct AB isthe i throwof A timesthe j thcolumnof B .Inorderforthistobedened,thenumberofentriesintherowsof A mustmatchthenumberofentriesinthecolumnsof B .Thismeansthatthenumberofcolumns of A mustmatchthenumberofrowsof B 7 Inotherwords,tomultiply A times B ,thesecond dimensionof A mustmatchtherstdimensionof B ,whichiswhyinDenition8.10, A m n isbeing multipliedbyamatrix B n r .Furthermore,theproductmatrix AB hasasmanyrowsas A andas manycolumnsof B .Asaresult,whenmultiplyingamatrix A m n byamatrix B n r ,theresultis thematrix AB m r .Returningtoourexamplematricesbelow,weseethat A isa2 3 matrixand B isa3 4matrix.Thismeansthattheproductmatrix AB isdenedandwillbea2 4matrix. A = 20 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1035 B = 2 4 312 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 48 )]TJ/F15 10.9091 Tf 8.485 0 Td [(59 50 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 3 5 7 Thereaderisencouragedtothinkthisthroughcarefully.

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8.3MatrixArithmetic461 Using Ri todenotethe i throwof A and Cj todenotethe j thcolumnof B ,weform AB according toDenition8.10. AB = R 1 C 1 R 1 C 2 R 1 C 3 R 1 C 4 R 2 C 1 R 2 C 2 R 2 C 3 R 2 C 4 = 126 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 714 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4547 Notethattheproduct BA isnotdened,since B isa3 4 matrixwhile A isa2 3matrix; B has morecolumnsthan A hasrows,andsoitisnotpossibletomultiplyarowof B byacolumnof A Evenwhenthedimensionsof A and B arecompatiblesuchthat AB and BA arebothdened,the product AB and BA aren'tnecessarilyequal. 8 Inotherwords, AB maynotequal BA .Although thereisnocommutativepropertyofmatrixmultiplicationingeneral,severalotherrealnumber propertiesareinheritedbymatrixmultiplication,asillustratedinournexttheorem. Theorem 8.5 PropertiesofMatrixMultiplication Let A B and C bematricessuchthat allofthematrixproductsbelowaredenedandlet k bearealnumber. AssociativePropertyofMatrixMultiplication: AB C = A BC AssociativePropertywithScalarMultiplication: k AB = kA B = A kB IdentityProperty: Foranaturalnumber k ,the k k identitymatrix ,denoted I k ,is denedby I k =[ d ij ] k k where d ij = 1 ; if i = j 0 ; otherwise Forall m n matrices, I m A = AI n = A DistributivePropertyofMatrixMultiplicationoverMatrixAddition: A B C = AB AC and A B C = AC BC TheonepropertyinTheorem8.5whichbegsfurtherinvestigationis,withoutdoubt,themultiplicativeidentity.Theentriesinamatrixwhere i = j comprisewhatiscalledthe maindiagonal ofthematrix.Theidentitymatrixhas1'salongitsmaindiagonaland0'severywhereelse.Afew examplesofthematrix I k mentionedinTheorem8.5aregivenbelow.Thereaderisencouragedto seehowtheymatchthedenitionoftheidentitymatrixpresentedthere. [1] 10 01 2 4 100 010 001 3 5 2 6 6 4 1000 0100 0010 0001 3 7 7 5 I 1 I 2 I 3 I 4 8 Andmaynotevenhavethesamedimensions.Forexample,if A isa2 3matrixand B isa3 2matrix,then AB isdenedandisa2 2matrixwhile BA isalsodened...butisa3 3matrix!

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462SystemsofEquationsandMatrices Theidentitymatrixisanexampleofwhatiscalleda squarematrix asithasthesamenumber ofrowsascolumns.Notethattoinordertoverifythattheidentitymatrixactsasamultiplicative identity,somecaremustbetakendependingontheorderofthemultiplication.Forexample,take thematrix2 3matrix A fromearlier A = 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1035 Inorderfortheproduct I k A tobedened, k =2;similarly,for AI k tobedened, k =3.Weleave ittothereadertoshow I 2 A = A and AI 3 = A .Inotherwords, 10 01 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1035 = 20 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1035 and 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1035 2 4 100 010 001 3 5 = 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1035 WhiletheproofsofthepropertiesinTheorem8.5arecomputationalinnature,thenotationbecomes quiteinvolvedveryquickly,sotheyarelefttoacourseinLinearAlgebra.Thefollowingexample providessomepracticewithmatrixmultiplicationanditsproperties.Asusual,somevaluable lessonsaretobelearned. Example 8.3.2 1.Find AB for A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(117 462 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 and B = 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(32 15 )]TJ/F15 10.9091 Tf 8.484 0 Td [(43 3 5 2.Find C 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 C +10 I 2 for C = 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 34 3.Suppose M isa4 4matrix.UseTheorem8.5toexpand M )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 4 M +3 I 4 Solution. 1.Wehave AB = )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(117 462 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(32 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(43 3 5 = 00 00 2.Justas x 2 means x timesitself, C 2 denotesthematrix C timesitself.Weget

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8.3MatrixArithmetic463 C 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 C +10 I 2 = 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 34 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 34 +10 10 01 = 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 34 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 34 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(510 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 + 100 010 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 1510 + 510 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 = 00 00 3.Weexpand M )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 4 M +3 I 4 withthesamepedanticzealweshowedinExample8.3.1. Thereaderisencouragedtodeterminewhichpropertyofmatrixarithmeticisusedaswe proceedfromonesteptothenext. M )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 4 M +3 I 4 = M )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 I 4 M + M )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 4 I 4 = MM )]TJ/F15 10.9091 Tf 10.909 0 Td [( I 4 M + M I 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [( I 4 I 4 = M 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 4 M +3 MI 4 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 I 4 I 4 = M 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 M +3 M )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 4 I 4 = M 2 + M )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 I 4 Example8.3.2illustratessomeinterestingfeaturesofmatrixmultiplication.Firstnotethatin part1,neither A nor B isthezeromatrix,yettheproduct AB isthezeromatrix.Hence,the thezeroproductpropertyenjoyedbyrealnumbersandscalarmultiplicationdoesnotholdfor matrixmultiplication.Parts2and3introduceustopolynomialsinvolvingmatrices.Thereaderis encouragedtostepbackandcompareourexpansionofthematrixproduct M )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 4 M +3 I 4 in part3withtheproduct x )]TJ/F15 10.9091 Tf 11.037 0 Td [(2 x +3fromrealnumberalgebra.Theexercisesexplorethiskind ofparallelfurther. Aswementionedearlier,apoint P x;y inthe xy -planecanberepresentedasa2 1position matrix.Wenowshowthatmatrixmultiplicationcanbeusedtorotatethesepoints,andhence graphsofequations. Example 8.3.3 Let R = p 2 2 )]TJ/F40 7.9701 Tf 9.681 10.993 Td [(p 2 2 p 2 2 p 2 2 # 1.Plot P ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(2, Q ; 0, S ; 3,and T )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3intheplaneaswellasthepoints RP RQ RS ,and RT .Plotthelines y = x and y = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x asguides.Whatdoes R appeartobedoing tothesepoints? 2.Ifapoint P isonthehyperbola x 2 )]TJ/F53 10.9091 Tf 10.894 0 Td [(y 2 =4,showthatthepoint RP isonthecurve y = 2 x .

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464SystemsofEquationsandMatrices Solution. For P ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,thepositionmatrixis P = 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ,and RP = p 2 2 )]TJ/F40 7.9701 Tf 9.681 10.993 Td [(p 2 2 p 2 2 p 2 2 #" 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 # = 2 p 2 0 Wehavethat R takes ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(2to p 2 ; 0.Similarly,wend ; 0ismovedto p 2 ; 2 p 2, ; 3 ismovedto )]TJ/F34 7.9701 Tf 9.681 4.395 Td [(3 p 2 2 ; 3 p 2 2 ,and )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3ismovedto ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 p 2.Plottingtheseinthecoordinate planealongwiththelines y = x and y = )]TJ/F53 10.9091 Tf 8.485 0 Td [(x ,weseethatthematrix R isrotatingthesepoints counterclockwiseby45 P RP Q RQ S RS T RT x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1123 )]TJ/F34 7.9701 Tf 6.587 0 Td [(4 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 4 Foragenericpoint P x;y onthehyperbola x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 2 =4,wehave RP = p 2 2 )]TJ/F40 7.9701 Tf 9.68 10.993 Td [(p 2 2 p 2 2 p 2 2 #" x y # = p 2 2 x )]TJ/F40 7.9701 Tf 12.105 10.993 Td [(p 2 2 y p 2 2 x + p 2 2 y # whichmeans R takes x;y to p 2 2 x )]TJ/F40 7.9701 Tf 12.105 10.993 Td [(p 2 2 y; p 2 2 x + p 2 2 y .Toshowthatthispointisonthecurve y = 2 x ,wereplace x with p 2 2 x )]TJ/F40 7.9701 Tf 12.105 10.993 Td [(p 2 2 y and y with p 2 2 x + p 2 2 y andsimplify.

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8.3MatrixArithmetic465 y = 2 x p 2 2 x + p 2 2 y ? = 2 p 2 2 x )]TJ/F41 5.9776 Tf 7.782 8.201 Td [(p 2 2 y p 2 2 x )]TJ/F40 7.9701 Tf 12.105 10.993 Td [(p 2 2 y p 2 2 x + p 2 2 y ? = 2 p 2 2 x )]TJ/F40 7.9701 Tf 12.105 10.993 Td [(p 2 2 y p 2 2 x )]TJ/F40 7.9701 Tf 12.105 10.993 Td [(p 2 2 y p 2 2 x 2 )]TJ/F55 10.9091 Tf 10.909 12.109 Td [( p 2 2 y 2 ? =2 x 2 2 )]TJ/F37 7.9701 Tf 12.105 4.931 Td [(y 2 2 ? =2 x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 2 X =4 Since x;y isonthehyperbola x 2 )]TJ/F53 10.9091 Tf 11.129 0 Td [(y 2 =4,weknowthatthislastequationistrue.Sinceallof ourstepsarereversible,thislastequationisequivalenttoouroriginalequation,whichestablishes thepointis,indeed,onthegraphof y = 2 x .Thismeansthatthegraphof y = 2 x isahyperbola )]TJ/F15 10.9091 Tf 12.542 0 Td [(itisthehyperbola x 2 )]TJ/F53 10.9091 Tf 11.189 0 Td [(y 2 =4rotatedcounterclockwiseby45 9 Belowwehavethegraphof x 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y 2 =4solidlineand y = 2 x dashedlineforcomparison. x y )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1134 )]TJ/F34 7.9701 Tf 6.587 0 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 4 Whenwestartedthissection,wementionedthatwewouldtemporarilyconsidermatricesastheir ownentities,butthatthealgebradevelopedherewouldultimatelyallowustosolvesystemsof linearequations.Tothatend,considerthesystem 8 < : 3 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z =8 x +2 y )]TJ/F53 10.9091 Tf 10.909 0 Td [(z =4 2 x +3 y )]TJ/F15 10.9091 Tf 10.91 0 Td [(4 z =10 InSection8.2,weencodedthissystemintotheaugmentedmatrix 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 8 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 10 3 5 9 SeeSection7.5formoredetails.

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466SystemsofEquationsandMatrices Recallthattheentriestotheleftoftheverticallinecomefromthecoecientsofthevariablesin thesystem,whilethoseontherightcomprisetheassociatedconstants.Forthatreason,wemay formthe coecientmatrix A ,the unknownsmatrix X andthe constantmatrix B asbelow A = 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 5 X = 2 4 x y z 3 5 B = 2 4 8 4 10 3 5 Wenowconsiderthematrixequation AX = B AX = B 2 4 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(11 12 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 23 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 3 5 2 4 x y z 3 5 = 2 4 8 4 10 3 5 2 4 3 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y + z x +2 y )]TJ/F53 10.9091 Tf 10.909 0 Td [(z 2 x +3 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 z 3 5 = 2 4 8 4 10 3 5 Weseethatndingasolution x;y;z totheoriginalsystemcorrespondstondingasolution X forthematrixequation AX = B .Ifwethinkaboutsolvingtherealnumberequation ax = b ,we wouldsimply`divide'bothsidesby a .Isitpossibleto`divide'bothsidesofthematrixequation AX = B bythematrix A ?ThisisthecentraltopicofSection8.4.

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8.3MatrixArithmetic467 8.3.1Exercises 1.Usingthematrices A = 12 34 B = 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(52 C = 10 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(11 2 0 3 5 59 D = 2 4 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(4 3 0 68 3 5 E = 2 4 123 04 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 5 computethefollowingmatricesorstatethattheyareundened. a7 B )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 A b AB c BA d E + D e ED f CD +2 A g EDC h CDE i ABCED 2.Let A = abc def E 1 = 01 10 E 2 = 50 01 E 3 = 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 01 Compute E 1 A E 2 A and E 3 A .Whateectdideachofthe E i matriceshaveontherowsof A ?Create E 4 sothatitseecton A istomultiplythebottomrowby )]TJ/F15 10.9091 Tf 8.485 0 Td [(6.Howwouldyou extendthisideatomatriceswithmorethantworows? 3.InthesmallvillageofPedimaxusinthecountryofSasquatchia,all150residentsgetoneofthe twolocalnewspapers.Marketresearchhasshownthatinanygivenweek,90%ofthosewho subscribetothePedimaxusTribunewanttokeepgettingit,but10%wanttoswitchtothe SasquatchiaPicayune.OfthosewhoreceivethePicayune,80%wanttocontinuewithitand 20%wantswitchtotheTribune.Wecanexpressthissituationusingmatrices.Specically, let X bethe`statematrix'givenby X = T P where T isthenumberofpeoplewhogettheTribuneand P isthenumberofpeoplewhoget thePicayuneinagivenweek.Let Q bethe`transitionmatrix'givenby Q = 0 : 900 : 20 0 : 100 : 80 suchthat QX willbethestatematrixforthenextweek. aLet'sassumethatwhenPedimaxuswasfounded,all150residentsgottheTribune.Let's callthisWeek0.Thiswouldmean X = 150 0

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468SystemsofEquationsandMatrices Since10%ofthat150wanttoswitchtothePicayune,weshouldhavethatforWeek 1,135peoplegettheTribuneand15peoplegetthePicayune.Showthat QX inthis situationisindeed QX = 135 15 bAssumingthatthepercentagesstaythesame,wecangettothesubscriptionnumbers forWeek2bycomputing Q 2 X .HowmanypeoplegeteachpaperinWeek2? cExplainwhythetransitionmatrixdoeswhatwewantittodo. dIftheconditionsdonotchangefromweektoweek,then Q remainsthesameandwe havewhat'sknownasa StochasticProcess 10 becauseWeek n 'snumbersarefoundby computing Q n X .Chooseafewvaluesof n and,withthehelpofyourclassmatesand calculator,ndouthowmanypeoplegeteachpaperforthatweek.Youshouldstartto seeapatternas n !1 eIfyoudidn'tseethepattern,we'llhelpyouout.Let X s = 100 50 : Showthat QX s = X s Thisiscalledthe steadystate becausethenumberofpeople whogeteachpaperdidn'tchangeforthenextweek.Showthat Q n X X s as n !1 fNowlet S = 2 3 2 3 1 3 1 3 # Showthat Q n S as n !1 gShowthat SY = X s foranymatrix Y oftheform Y = y 150 )]TJ/F53 10.9091 Tf 10.909 0 Td [(y ThismeansthatnomatterhowthedistributionstartsinPedimaxus,if Q isapplied oftenenough,wealwaysendupwith100peoplegettingtheTribuneand50people gettingthePicayune. 4.Let z = a + bi and w = c + di bearbitrarycomplexnumbers.Associate z and w withthe matrices Z = ab )]TJ/F53 10.9091 Tf 8.485 0 Td [(ba and W = cd )]TJ/F53 10.9091 Tf 8.485 0 Td [(dc Showthatcomplexnumberaddition,subtractionandmultiplicationaremirroredbythe associated matrix arithmetic.Thatis,showthat Z + W Z )]TJ/F53 10.9091 Tf 10.916 0 Td [(W and ZW producematrices whichcanbeassociatedwiththecomplexnumbers z + w z )]TJ/F53 10.9091 Tf 10.909 0 Td [(w and zw ,respectively. 10 Morespecically,wehaveaMarkovChain,whichisaspecialtypeofstochasticprocess.

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8.3MatrixArithmetic469 5.Asquarematrixissaidtobean uppertriangularmatrix ifallofitsentriesbelowthe maindiagonalarezeroanditissaidtobea lowertriangularmatrix ifallofitsentries abovethemaindiagonalarezero.Forexample, E = 2 4 123 04 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 5 fromExercise1aboveisanuppertriangularmatrixwhereas F = 10 30 isalowertriangularmatrix.Zerosareallowedonthemaindiagonal.Discussthefollowing questionswithyourclassmates. aGiveanexampleofamatrixwhichisneitheruppertriangularnorlowertriangular. bIstheproductoftwo n n uppertriangularmatricesalwaysuppertriangular? cIstheproductoftwo n n lowertriangularmatricesalwayslowertriangular? dGiventhematrix A = 12 34 write A as LU where L isalowertriangularmatrixand U isanuppertriangularmatrix? eArethereanymatriceswhicharesimultaneouslyupperandlowertriangular? 6.Let A = 12 34 and B = 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(52 Compare A + B 2 to A 2 +2 AB + B 2 .Discusswithyourclassmateswhatconstraintsmust beplacedontwoarbitrarymatrices A and B sothatboth A + B 2 and A 2 +2 AB + B 2 exist. Whenwill A + B 2 = A 2 +2 AB + B 2 ?Ingeneral,whatisthecorrectformulafor A + B 2 ?

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470SystemsofEquationsandMatrices 8.3.2Answers 1.a7 B )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(29 )]TJ/F15 10.9091 Tf 8.485 0 Td [(47 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 b AB = )]TJ/F15 10.9091 Tf 8.485 0 Td [(101 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 c BA = )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 d E + D isundened e ED = 2 6 4 67 3 11 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(178 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(72 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 3 7 5 f CD +2 A = 238 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(126 863 15 361 5 # g EDC = 2 6 4 3449 15 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(407 6 99 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(9548 15 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(101 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(648 )]TJ/F15 10.9091 Tf 8.485 0 Td [(324 )]TJ/F15 10.9091 Tf 8.485 0 Td [(35 )]TJ/F15 10.9091 Tf 8.485 0 Td [(360 3 7 5 h CDE isundened i ABCED = )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(90749 15 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(28867 5 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(156601 15 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(47033 5 # 2. E 1 A = def abc E 1 interchanged R 1and R 2of A E 2 A = 5 a 5 b 5 c def E 2 multiplied R 1of A by5. E 3 A = a )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 db )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ec )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 f def E 3 replaced R 1in A with R 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 R 2. E 4 = 10 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6

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8.4SystemsofLinearEquations:MatrixInverses471 8.4SystemsofLinearEquations:MatrixInverses WeconcludedSection8.3byshowinghowwecanrewriteasystemoflinearequationsasthematrix equation AX = B where A and B areknownmatricesandthesolutionmatrix X oftheequation correspondstothesolutionofthesystem.Inthissection,wedevelopthemethodforsolvingsuch anequation.Tothatend,considerthesystem 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 y =16 3 x +4 y =7 Towritethisasamatrixequation,wefollowtheprocedureoutlinedonpage466.Wendthe coecientmatrix A ,theunknownsmatrix X andconstantmatrix B tobe A = 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 34 X = x y B = 16 7 Inordertomotivatehowwesolveamatrixequationlike AX = B ,werevisitsolvingasimilar equationinvolvingrealnumbers.Considertheequation3 x =5.Tosolve,wesimplydivideboth sidesby3andobtain x = 5 3 .Howcanwegoaboutdeningananalogousprocessformatrices? Toanswerthisquestion,wesolve3 x =5again,butthistime,wepayattentiontotheproperties ofrealnumbersbeingusedateachstep.Recallthatdividingby3isthesameasmultiplyingby 1 3 =3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ,theso-called multiplicativeinverse of3. 1 3 x =5 3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 x =3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 Multiplybythemultiplicativeinverseof3 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 3 x =3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 Associativepropertyofmultiplication 1 x =3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 Inverseproperty x =3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 MultiplicativeIdentity Ifwewishtocheckouranswer,wesubstitute x =3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 intotheoriginalequation 3 x ? =5 3 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(3 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ? =5 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(3 3 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ? =5Associativepropertyofmultiplication 1 5 ? =5Inverseproperty 5 X =5MultiplicativeIdentity ThinkingbacktoTheorem8.5,weknowthatmatrixmultiplicationenjoysbothanassociative propertyandamultiplicativeidentity.What'smissingfromthemixisamultiplicativeinversefor thecoecientmatrix A .Assumingwecanndsuchabeast,wecanmimicoursolutionandcheck to3 x =5asfollows 1 Everynonzerorealnumber a hasamultiplicativeinverse,denoted a )]TJ/F35 5.9776 Tf 5.757 0 Td [(1 ,suchthat a )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 a = a a )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 =1.

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472SystemsofEquationsandMatrices Solving AX = B Checkingouranswer AX = B A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 AX = A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 B )]TJ/F53 10.9091 Tf 5 -8.836 Td [(A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 A X = A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 B I 2 X = A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 B X = A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 B AX ? = B A )]TJ/F53 10.9091 Tf 5 -8.837 Td [(A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 B ? = B )]TJ/F53 10.9091 Tf 5 -8.836 Td [(AA )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 B ? = B I 2 B ? = B B X = B Thematrix A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 isread` A -inverse'andwewilldeneitformallylaterinthesection.Atthisstage, wehavenoideaifsuchamatrix A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 exists,butthatwon'tdeterusfromtryingtondit. 2 We want A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 tosatisfytwoequations, A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 A = I 2 and AA )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = I 2 ,making A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 necessarilya2 2 matrix. 3 Hence,weassume A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 hastheform A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = x 1 x 2 x 3 x 4 forrealnumbers x 1 x 2 x 3 and x 4 .Forreasonswhichwillbecomeclearlater,wefocusourattention ontheequation AA )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = I 2 .Wehave AA )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = I 2 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 34 x 1 x 2 x 3 x 4 = 10 01 2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 3 x 1 +4 x 3 3 x 2 +4 x 4 = 10 01 Thisgivesriseto two moresystemsofequations 2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 =1 3 x 1 +4 x 3 =0 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 =0 3 x 2 +4 x 4 =1 Atthispoint,itmayseemabsurdtocontinuewiththisventure.Afterall,theintentwastosolve one systemofequations,andindoingso,wehaveproduced two moretosolve.Remember,the objectiveofthisdiscussionistodevelopageneral method which,whenusedinthecorrectscenarios, allowsustodofarmorethanjustsolveasystemofequations.Ifwesetabouttosolvethesesystems usingaugmentedmatricesusingthetechniquesinSection8.2,weseethatnotonlydobothsystems havethesamecoecientmatrix,thiscoecientmatrixisnoneotherthanthematrix A itself.We willcomebacktothisobservationinamoment. 2 MuchlikeCarl'squesttondSasquatch. 3 Sincematrixmultiplicationisn'tnecessarilycommutative,atthisstage,thesearetwodierentequations.

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8.4SystemsofLinearEquations:MatrixInverses473 2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 =1 3 x 1 +4 x 3 =0 Encodeintoamatrix )414()222()223()222()222()222()223()222()222()222()223()222()414(! 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 34 0 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 =0 3 x 2 +4 x 4 =1 Encodeintoamatrix )414()222()223()222()222()222()223()222()222()222()223()222()414(! 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 34 1 Tosolvethesetwosystems,weuseGauss-JordanEliminationtoputtheaugmentedmatricesinto reducedrowechelonform.Weleavethedetailstothereader.Fortherstsystem,weget 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 1 34 0 GaussJordanElimination )427()222()222()222()223()222()222()222()223()222()222()222()223()222()222()427(! 10 4 17 01 )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(3 17 # whichgives x 1 = 4 17 and x 3 = )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(3 17 .Tosolvethesecondsystem,weusetheexactsamerow operations,inthesameorder,toputitsaugmentedmatrixintoreducedrowechelonformThink aboutwhythatworks.andweobtain 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 34 0 GaussJordanElimination )427()222()222()223()222()222()222()222()223()222()222()222()223()222()222()427(! 10 3 17 01 2 17 # whichmeans x 2 = 3 17 and x 4 = 2 17 .Hence, A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = x 1 x 2 x 3 x 4 = 4 17 3 17 )]TJ/F34 7.9701 Tf 11.797 4.296 Td [(3 17 2 17 # Wecanchecktoseethat A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 behavesasitshouldbycomputing AA )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 AA )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 = 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 34 4 17 3 17 )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(3 17 2 17 # = 10 01 = I 2 X Asanaddedbonus, A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 A = 4 17 3 17 )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(3 17 2 17 # 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 34 = 10 01 = I 2 X Wecannowreturntotheproblemathand.Fromourdiscussionatthebeginningofthesection onpage472,weknow X = A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 B = 4 17 3 17 )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(3 17 2 17 # 16 7 = 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 sothatournalsolutiontothesystemis x;y = ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2. Aswementioned,thepointofthisexercisewasnotjusttosolvethesystemoflinearequations,but todevelopageneralmethodfornding A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 .Wenowtakeastepbackandanalyzetheforegoing discussioninamoregeneralcontext.Insolvingfor A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ,weusedtwoaugmentedmatrices,bothof whichcontainedthesameentriesas A

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474SystemsofEquationsandMatrices 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 34 0 = A 1 0 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 34 1 = A 0 1 Wealsonotethatthereducedrowechelonformsoftheseaugmentedmatricescanbewrittenas 10 4 17 01 )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(3 17 # = I 2 x 1 x 3 # 10 3 17 01 2 17 # = I 2 x 2 x 4 # wherewehaveidentiedtheentriestotheleftoftheverticalbarastheidentity I 2 andtheentries totherightoftheverticalbarasthesolutionstooursystems.Thelongandshortofthesolution processcanbesummarizedas A 1 0 GaussJordanElimination )427()222()222()223()222()222()222()222()223()222()222()222()223()222()222()427(! I 2 x 1 x 3 A 0 1 GaussJordanElimination )427()222()222()223()222()222()222()222()223()222()222()222()223()222()222()427(! I 2 x 2 x 4 Sincetherowoperationsforbothprocessesarethesame,allofthearithmeticonthelefthandside oftheverticalbarisidenticalinbothproblems.Theonlydierencebetweenthetwoprocessesis whathappenstotheconstantstotherightoftheverticalbar.Aslongaswekeeptheseseparated intocolumns,wecancombineoureortsintoone`super-sized'augmentedmatrixanddescribethe aboveprocessas A 10 01 GaussJordanElimination )427()222()222()222()223()222()222()222()223()222()222()222()223()222()222()427(! I 2 x 1 x 2 x 3 x 4 Wehavetheidentitymatrix I 2 appearingastherighthandsideoftherstsuper-sizedaugmented matrixandthelefthandsideofthesecondsuper-sizedaugmentedmatrix.Tooursurpriseand delight,theelementsontherighthandsideofthesecondsuper-sizedaugmentedmatrixarenone otherthanthosewhichcomprise A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 .Hence,wehave A I 2 GaussJordanElimination )427()222()222()222()223()222()222()222()223()222()222()222()223()222()222()427(! I 2 A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 Inotherwords,theprocessofnding A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 foramatrix A canbeviewedasperformingaseriesof rowoperationswhichtransform A intotheidentitymatrixofthesamedimension.Wecanview thisprocessasfollows.Intryingtond A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ,wearetryingto`undo'multiplicationbythematrix A .Theidentitymatrixinthesuper-sizedaugmentedmatrix[ A j I ]keepsarunningmemoryofall ofthemovesrequiredto`undo' A .Thisresultsinexactlywhatwewant, A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 .Wearenowready

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8.4SystemsofLinearEquations:MatrixInverses475 toformalizeandgeneralizetheforegoingdiscussion.Webeginwiththeformaldenitionofan invertiblematrix. Definition 8.11 An n n matrix A issaidtobe invertible ifthereexistsamatrix A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 ,read ` A inverse',suchthat A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 A = AA )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = I n Notethat,accordingtoourdenition,invertiblematricesaresquare,andassuch,theconditions inDenition8.11forcethematrix A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 tobesamedimensionsas A ,thatis, n n .Sincenot allmatricesaresquare,notallmatricesareinvertible.However,justbecauseamatrixissquare doesn'tguaranteeitisinvertible.Seetheexercises.Ourrstresultsummarizessomeofthe importantcharacteristicsofinvertiblematricesandtheirinverses. Theorem 8.6 Suppose A isan n n matrix. 1.If A isinvertiblethen A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 isunique. 2. A isinvertibleifandonlyif AX = B hasauniquesolutionforevery n r matrix B TheproofsofthepropertiesinTheorem8.6relyonahealthymixofdenitionandmatrixarithmetic.Toestablishtherstproperty,weassumethat A isinvertibleandsupposethematrices B and C actasinversesfor A .Thatis, BA = AB = I n and CA = AC = I n .Weneedtoshowthat B and C are,infact,thesamematrix.Toseethis,wenotethat B = I n B = CA B = C AB = CI n = C Hence,anytwomatricesthatactlike A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 are,infact,thesamematrix. 4 Toprovethesecond propertyofTheorem8.6,wenotethatif A isinvertiblethenthediscussiononpage472shows thesolutionto AX = B tobe X = A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 B ,andsince A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 isunique,sois A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 B .Conversely,if AX = B hasauniquesolutionforevery n r matrix B ,then,inparticular,thereisaunique solution X 0 totheequation AX = I n .Thesolutionmatrix X 0 isourcandidatefor A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 .We have AX 0 = I n bydenition,butweneedtoalsoshow X 0 A = I n .Tothatend,wenotethat A X 0 A = AX 0 A = I n A = A .Inotherwords,thematrix X 0 A isasolutiontotheequation AX = A .Clearly, X = I n isalsoasolutiontotheequation AX = A ,andsinceweareassumingeverysuchequationasa unique solution,wemusthave X 0 A = I n .Hence,wehave X 0 A = AX 0 = I n sothat X 0 = A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 and A isinvertible.Theforegoingdiscussionjustiesourquesttond A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 using oursuper-sizedaugmentedmatrixapproach A I n GaussJordanElimination )427()222()222()222()223()222()222()222()223()222()222()222()223()222()222()427(! I n A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 Weare,inessence,tryingtondtheuniquesolutiontotheequation AX = I n usingrowoperations. Whatdoesallofthismeanforasystemoflinearequations?Theorem8.6tellsusthatifwewrite thesystemintheform AX = B ,thenifthecoecientmatrix A isinvertible,thereisonlyone solutiontothesystem )]TJ/F15 10.9091 Tf 12.163 0 Td [(thatis,if A isinvertible,thesystemisconsistentandindependent. 5 We 4 Ifthisproofsoundsfamiliar,itshould.SeethediscussionfollowingTheorem5.2onpage281. 5 Itcanbeshownthatamatrixisinvertibleifandonlyifwhenitservesasacoecientmatrixforasystemof equations,thesystemisalwaysconsistentindependent.ItamountstothesecondpropertyinTheorem8.6where thematrices B arerestrictedtobeing n 1matrices.Wenotefortheinterestedreaderthat,owingtohowmatrix multiplicationisdened,beingabletonduniquesolutionsto AX = B for n 1matrices B givesyouthesame statementaboutsolvingsuchequationsfor n r matrices )]TJ/F63 8.9664 Tf 10.035 0 Td [(sincewecanndauniquesolutiontothemonecolumn atatime.

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476SystemsofEquationsandMatrices alsoknowthattheprocessbywhichwend A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 isdeterminedcompletelyby A ,andnotbythe constantsin B .Thisanswersthequestionastowhywewouldbotherdoingrowoperationson asuper-sizedaugmentedmatrixtond A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 insteadofanordinaryaugmentedmatrixtosolvea system;bynding A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 wehavedonealloftherowoperationsweeverneedtodo,onceandforall, sincewecanquicklysolve any equation AX = B using one multiplication, A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 B .Weillustrate thisinournextexample. Example 8.4.1 Let A = 2 4 312 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 214 3 5 1.Userowoperationstond A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 .Checkyouranswerbynding A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 A and AA )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 2.Use A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 tosolvethefollowingsystemsofequations a 8 < : 3 x + y +2 z =26 )]TJ/F53 10.9091 Tf 8.485 0 Td [(y +5 z =39 2 x + y +4 z =117 b 8 < : 3 x + y +2 z =4 )]TJ/F53 10.9091 Tf 8.485 0 Td [(y +5 z =2 2 x + y +4 z =5 c 8 < : 3 x + y +2 z =1 )]TJ/F53 10.9091 Tf 8.485 0 Td [(y +5 z =0 2 x + y +4 z =0 Solution. 1.Webeginwithasuper-sizedaugmentedmatrixandproceedwithGauss-Jordanelimination. 2 4 312 100 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 010 214 001 3 5 Replace R 1 )426()223()222()222()222()222()427(! with 1 3 R 1 2 4 1 1 3 2 3 1 3 00 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 010 214 001 3 5 2 4 1 1 3 2 3 1 3 00 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 010 214 001 3 5 Replace R 3with )376()222()222()222()222()223()222()222()222()376(! )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 R 1+ R 3 2 4 1 1 3 2 3 1 3 00 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 010 0 1 3 8 3 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(2 3 01 3 5 2 4 1 1 3 2 3 1 3 00 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 010 0 1 3 8 3 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(2 3 01 3 5 Replace R 2 )472()222()223()222()222()222()222()473(! with )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 R 2 2 4 1 1 3 2 3 1 3 00 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 1 3 8 3 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(2 3 01 3 5 2 4 1 1 3 2 3 1 3 00 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 1 3 8 3 )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(2 3 01 3 5 Replace R 3with )375()223()222()222()222()223()222()222()222()376(! )]TJ/F35 5.9776 Tf 7.782 3.259 Td [(1 3 R 2+ R 3 2 4 1 1 3 2 3 1 3 00 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 00 13 3 )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(2 3 1 3 1 3 5 2 4 1 1 3 2 3 1 3 00 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 00 13 3 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(2 3 1 3 1 3 5 Replace R 3 )426()223()222()222()222()222()427(! with 3 13 R 3 2 4 1 1 3 2 3 1 3 00 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 001 )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(2 13 1 13 3 13 3 5 2 4 1 1 3 2 3 1 3 00 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 001 )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(2 13 1 13 3 13 3 5 Replace R 1with )]TJ/F35 5.9776 Tf 7.782 3.258 Td [(2 3 R 3+ R 1 )383()222()223()222()222()222()223()222()222()222()223()383(! Replace R 2with 5 R 3+ R 2 2 6 4 1 1 3 0 17 39 )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(2 39 )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(2 13 010 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(10 13 )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(8 13 15 13 001 )]TJ/F34 7.9701 Tf 11.798 4.296 Td [(2 13 1 13 3 13 3 7 5

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8.4SystemsofLinearEquations:MatrixInverses477 2 6 4 1 1 3 0 17 39 )]TJ/F34 7.9701 Tf 11.797 4.296 Td [(2 39 )]TJ/F34 7.9701 Tf 11.797 4.296 Td [(2 13 010 )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(10 13 )]TJ/F34 7.9701 Tf 11.797 4.296 Td [(8 13 15 13 001 )]TJ/F34 7.9701 Tf 11.797 4.296 Td [(2 13 1 13 3 13 3 7 5 Replace R 1with )376()222()222()222()222()223()222()222()222()376(! )]TJ/F35 5.9776 Tf 7.782 3.259 Td [(1 3 R 2+ R 1 2 6 4 100 9 13 2 13 )]TJ/F34 7.9701 Tf 11.797 4.296 Td [(7 13 010 )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(10 13 )]TJ/F34 7.9701 Tf 11.798 4.296 Td [(8 13 15 13 001 )]TJ/F34 7.9701 Tf 11.798 4.296 Td [(2 13 1 13 3 13 3 7 5 Wend A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = 2 6 4 9 13 2 13 )]TJ/F34 7.9701 Tf 11.797 4.296 Td [(7 13 )]TJ/F34 7.9701 Tf 9.68 4.296 Td [(10 13 )]TJ/F34 7.9701 Tf 11.797 4.296 Td [(8 13 15 13 )]TJ/F34 7.9701 Tf 11.797 4.296 Td [(2 13 1 13 3 13 3 7 5 .Tocheckouranswer,wecompute A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 A = 2 6 4 9 13 2 13 )]TJ/F34 7.9701 Tf 11.798 4.296 Td [(7 13 )]TJ/F34 7.9701 Tf 9.681 4.296 Td [(10 13 )]TJ/F34 7.9701 Tf 11.798 4.296 Td [(8 13 15 13 )]TJ/F34 7.9701 Tf 11.798 4.296 Td [(2 13 1 13 3 13 3 7 5 2 6 4 312 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 214 3 7 5 = 2 6 4 100 010 001 3 7 5 = I 3 X and AA )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 = 2 6 4 312 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 214 3 7 5 2 6 4 9 13 2 13 )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(7 13 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(10 13 )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(8 13 15 13 )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(2 13 1 13 3 13 3 7 5 = 2 6 4 100 010 001 3 7 5 = I 3 X 2.Eachofthesystemsinthisparthas A asitscoecientmatrix.Theonlydierencebetween thesystemsistheconstantswhichisthematrix B intheassociatedmatrixequation AX = B Wesolveeachofthemusingtheformula X = A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 B a X = A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 B = 2 6 4 9 13 2 13 )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(7 13 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(10 13 )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(8 13 15 13 )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(2 13 1 13 3 13 3 7 5 2 6 4 26 39 117 3 7 5 = 2 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(39 91 26 3 7 5 .Oursolutionis )]TJ/F15 10.9091 Tf 8.485 0 Td [(39 ; 91 ; 26. b X = A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 B = 2 6 4 9 13 2 13 )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(7 13 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(10 13 )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(8 13 15 13 )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(2 13 1 13 3 13 3 7 5 2 6 4 4 2 5 3 7 5 = 2 6 4 5 13 19 13 9 13 3 7 5 .Weget )]TJ/F34 7.9701 Tf 8.313 -4.541 Td [(5 13 ; 19 13 ; 9 13 c X = A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 B = 2 6 4 9 13 2 13 )]TJ/F34 7.9701 Tf 11.797 4.296 Td [(7 13 )]TJ/F34 7.9701 Tf 9.681 4.295 Td [(10 13 )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(8 13 15 13 )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(2 13 1 13 3 13 3 7 5 2 6 4 1 0 0 3 7 5 = 2 6 4 9 13 )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(10 13 )]TJ/F34 7.9701 Tf 11.797 4.295 Td [(2 13 3 7 5 .Wend )]TJ/F34 7.9701 Tf 8.312 -4.541 Td [(9 13 ; )]TJ/F34 7.9701 Tf 9.68 4.295 Td [(10 13 ; )]TJ/F34 7.9701 Tf 11.798 4.295 Td [(2 13 6 InExample8.4.1,weseethatndingoneinversematrixcanenableustosolveanentirefamily ofsystemsoflinearequations.Therearemanyexamplesofwherethiscomesinhandy`inthe wild',andwechoseourexampleforthissectionfromtheeldofelectronics.Wealsotakethis opportunitytointroducethestudenttohowwecancomputeinversematricesusingthecalculator. 6 Notethatthesolutionistherstcolumnofthe A )]TJ/F35 5.9776 Tf 5.756 0 Td [(1 .Thereaderisencouragedtomeditateonthis`coincidence'.

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478SystemsofEquationsandMatrices Example 8.4.2 Considerthecircuitdiagrambelow. 7 Wehavetwobatterieswithsourcevoltages VB 1 and VB 2 ,measuredinvolts V ,alongwithsixresistorswithresistances R 1 through R 6 ,measured inkiloohms, k .UsingOhm'sLaw andKirchho'sVoltageLaw ,wecanrelatethevoltagesupplied tothecircuitbythetwobatteriestothevoltagedropsacrossthesixresistorsinordertondthe four`mesh'currents: i 1 i 2 i 3 and i 4 ,measuredinmilliamps, mA .Ifwethinkofelectronsowing throughthecircuit,wecanthinkofthevoltagesourcesasprovidingthe`push'whichmakesthe electronsmove,theresistorsasobstaclesfortheelectronstoovercome,andthemeshcurrentasa netrateofowofelectronsaroundtheindicatedloops. Thesystemoflinearequationsassociatedwiththiscircuitis 8 > > < > > : R 1 + R 3 i 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(R 3 i 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(R 1 i 4 = VB 1 )]TJ/F53 10.9091 Tf 8.485 0 Td [(R 3 i 1 + R 2 + R 3 + R 4 i 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(R 4 i 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(R 2 i 4 =0 )]TJ/F53 10.9091 Tf 8.485 0 Td [(R 4 i 2 + R 4 + R 6 i 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(R 6 i 4 = )]TJ/F53 10.9091 Tf 8.485 0 Td [(VB 2 )]TJ/F53 10.9091 Tf 8.485 0 Td [(R 1 i 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(R 2 i 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(R 6 i 3 + R 1 + R 2 + R 5 + R 6 i 4 =0 1.Assumingtheresistancesareall1 k ,ndthemeshcurrentsifthebatteryvoltagesare a VB 1 =10 V and VB 2 =5 V b VB 1 =10 V and VB 2 =0 V c VB 1 =0 V and VB 2 =10 V d VB 1 =10 V and VB 2 =10 V 2.Assuming VB 1 =10 V and VB 2 =5 V ,ndthepossiblecombinationsofresistanceswhich wouldyieldthemeshcurrentsyoufoundin1a. 7 TheauthorswishtothankDonAnthanofLakelandCommunityCollegeforthedesignofthisexample.

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8.4SystemsofLinearEquations:MatrixInverses479 Solution. 1.Substitutingtheresistancevaluesintooursystemofequations,weget 8 > > < > > : 2 i 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(i 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(i 4 = VB 1 )]TJ/F53 10.9091 Tf 8.484 0 Td [(i 1 +3 i 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(i 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(i 4 =0 )]TJ/F53 10.9091 Tf 8.485 0 Td [(i 2 +2 i 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(i 4 = )]TJ/F53 10.9091 Tf 8.485 0 Td [(VB 2 )]TJ/F53 10.9091 Tf 8.484 0 Td [(i 1 )]TJ/F53 10.9091 Tf 10.909 0 Td [(i 2 )]TJ/F53 10.9091 Tf 10.909 0 Td [(i 3 +4 i 4 =0 Thiscorrespondstothematrixequation AX = B where A = 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 3 7 7 5 X = 2 6 6 4 i 1 i 2 i 3 i 4 3 7 7 5 B = 2 6 6 4 VB 1 0 )]TJ/F53 10.9091 Tf 8.485 0 Td [(VB 2 0 3 7 7 5 Whenweinputthematrix A intothecalculator,wend fromwhichwehave A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = 2 6 6 4 1 : 6251 : 251 : 1251 1 : 251 : 51 : 251 1 : 1251 : 251 : 6251 1111 3 7 7 5 Tosolvethefoursystemsgiventous,wend X = A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 B wherethevalueof B isdetermined bythegivenvaluesof VB 1 and VB 2 1a B = 2 6 6 4 10 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 0 3 7 7 5 ; 1b B = 2 6 6 4 10 0 0 0 3 7 7 5 ; 1c B = 2 6 6 4 0 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 3 7 7 5 ; 1d B = 2 6 6 4 10 0 10 0 3 7 7 5 aFor VB 1 =10 V and VB 2 =5 V ,thecalculatorgives i 1 =10 : 625 mA i 2 =6 : 25 mA i 3 =3 : 125 mA ,and i 4 =5 mA .

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480SystemsofEquationsandMatrices bBykeeping VB 1 =10 V andsetting VB 2 =0 V ,weareessentiallyremovingtheeect ofthesecondbattery.Weget i 1 =16 : 25 mA i 2 =12 : 5 mA i 3 =11 : 25 mA ,and i 4 =10 mA Solutionto1aSolutionto1b cPartcisasymmetricsituationtopartbinsomuchaswearezeroingout VB 1 and making VB 2 =10.Wend i 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 : 25 mA i 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 : 5 mA i 3 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(16 : 25 mA ,and i 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 mA ,wherethenegativesindicatethatthecurrentisowingintheopposite directionasisindicatedonthediagram.Thereaderisencouragedtostudythesymmetry here,andifneedbe,holdupamirrortothediagramtoliterally`see'whatishappening. dFor VB 1 =10 V and VB 2 =10 V ,weget i 1 =5 mA i 2 =0 mA i 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 mA ,and i 4 =0 mA .Themeshcurrents i 2 and i 4 beingzeroisaconsequenceofbothbatteries `pushing'inequalbutoppositedirections,causingthenetowofelectronsinthesetwo regionstocancelout. Solutionto1cSolutionto1d 2.Wenowturnthetablesandaregiven VB 1 =10 V VB 2 =5 V i 1 =10 : 625 mA i 2 =6 : 25 mA i 3 =3 : 125 mA and i 4 =5 mA andourunknownsaretheresistancevalues.Rewritingour systemofequations,weget 8 > > < > > : 5 : 625 R 1 +4 : 375 R 3 =10 1 : 25 R 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 : 375 R 3 +3 : 125 R 4 =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 : 125 R 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 : 875 R 6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 : 625 R 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 : 25 R 2 +5 R 5 +1 : 875 R 6 =0 Thecoecientmatrixforthissystemis4 6equationswith6unknownsandistherefore notinvertible.Wedoknow,however,thissystemisconsistent,sincesettingalltheresistancevaluesequalto1correspondstooursituationinproblem1a.Thismeanswehavean

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8.4SystemsofLinearEquations:MatrixInverses481 underdeterminedconsistentsystemwhichisnecessarilydependent.Tosolvethissystem,we encodeitintoanaugmentedmatrix 2 6 6 4 5 : 2504 : 375000 10 01 : 25 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 : 3753 : 12500 0 000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 : 1250 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 : 875 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 : 625 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 250051 : 875 0 3 7 7 5 andusethecalculatortowriteinreducedrowechelonform 2 6 6 4 100 : 7000 1 : 7 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 : 500 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 000100 : 6 1 : 6 000010 1 3 7 7 5 Decodingthissystemfromthematrix,weget 8 > > < > > : R 1 +0 : 7 R 3 =1 : 7 R 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 : 5 R 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 : 5 R 6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 R 4 +0 : 6 R 6 =1 : 6 R 5 =1 Wehavecansolvefor R 1 R 2 R 4 and R 5 leaving R 3 and R 6 asfreevariables.Labeling R 3 = s and R 6 = t ,wehave R 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 7 s +1 : 7, R 2 =3 : 5 s +1 : 5 t )]TJ/F15 10.9091 Tf 11.653 0 Td [(4, R 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 6 t +1 : 6 and R 5 =1.Sinceresistancevaluesarealwayspositive,weneedtorestrictourvaluesof s and t .Weknow R 3 = s> 0andwhenwecombinethatwith R 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 7 s +1 : 7 > 0, weget0 0andwith R 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 6 t +1 : 6 > 0,wend 0 0,wegraphthe line3 : 5 s +1 : 5 t )]TJ/F15 10.9091 Tf 11.656 0 Td [(4=0onthe st -planeandshadeaccordingly. 8 Imposingtheadditional conditions0 0 .Thereaderisencouragedtocheckthat thesolutionpresentedin1a,namelyallresistancevaluesequalto1,correspondstoapair s;t intheregion. 8 SeeSection2.4forareviewofthisprocedure.

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482SystemsofEquationsandMatrices t s )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1124 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 Theregionwhere3 : 5 s +1 : 5 t )]TJ/F34 7.9701 Tf 8.468 0 Td [(4 > 0 t s t = 8 3 s = 16 7 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 )]TJ/F34 7.9701 Tf 6.586 0 Td [(1124 )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 2 3 Theregionforourparameters s and t

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8.4SystemsofLinearEquations:MatrixInverses483 8.4.1Exercises 1.Findtheinverseofthematrixorstatethatthematrixisnotinvertible. a A = 12 34 b B = 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(53 c C = 615 1435 d D = 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 16 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 e E = 2 4 304 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(32 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 5 f F = 2 4 46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 34 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 126 3 5 g G = 2 4 123 2311 3419 3 5 h H = 2 6 6 4 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(287 )]TJ/F15 10.9091 Tf 8.485 0 Td [(50160 1041 3 7 7 5 2.Useamatrixinversetosolvethefollowingsystemsoflinearequations. a 3 x +7 y =26 5 x +12 y =39 b 3 x +7 y =0 5 x +12 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 c 3 x +7 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 5 x +12 y =5 3.Usetheinverseof E fromExercise1abovetosolvethefollowingsystemsoflinearequations. a 8 < : 3 x +4 z =1 2 x )]TJ/F53 10.9091 Tf 10.909 0 Td [(y +3 z =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x +2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 z =0 b 8 < : 3 x