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PrecalculusVersion | = 3byCarlStitz,Ph.D. JeffZeager,Ph.D.LakelandCommunityCollege LorainCountyCommunityCollegeJuly15,2011iiAcknowledgementsWhilethecoverofthistextbooklistsonlytwonames, thebookasitstandstodaywouldsimplynotexistifnotforthetirelessworkanddedicationofseveralpeople. Firstandforemost,wewishtothankourfamiliesfortheirpatienceandsupportduringthecreativeprocess. Wewouldalsoliketothankourstudents-thesoleinspirationforthework. Amongourcolleagues, wewishtothankRichBasich, Bill Previts, andIrinaLomonosov, whonotonlywereearlyadoptersof thetextbook, butalsocontributedmaterialstotheproject. Special thanksgotoKatieCimperman,Terry Dykstra, Frank LeMay, and Rich Hagen who provided valuable feedback from the classroom.ThanksalsotoDavidStumpf, IvanaGorgievska, JorgeGerszonowicz, KathrynArocho, HeatherBubnick, andFlorinMuscutariufortheirunwaiveringsupport(andsometimesdefense!) of theproject. From outside the classroom, we wish to thank Don Anthan and Ken White, who designedthe electric circuit applications used in the text, as well as Drs. Wendy Marley and Marcia BallingerfortheLorainCCCenrollmentdatausedinthetext. Theauthorsarealsoindebtedtothegoodfolksatourschools bookstores, GwenSevtis(LakelandCC)andChrisCallahan(LorainCCC),forworkingwithustogetprintedcopiestothestudentsasinexpensivelyaspossible. WewouldalsoliketothankLakelandfolksJeri Dickinson, MaryAnnBlakeley, JessicaNovak, andCorrieBergeronfortheirenthusiasmandpromotionoftheproject. Theadministrationatbothschoolshavealsobeenverysupportiveoftheproject,sofromLakeland,wewishtothankDr. MorrisW.Beverage, Jr., President, Dr. FredLaw, Provost, DeansDonAnthanandDr. SteveOluic, andtheBoardofTrustees. FromLorainCountyCommunityCollege, wewhichtothankDr. RoyA.Church, Dr. Karen Wells, and the Board of Trustees. From the Ohio Board of Regents, we wish tothank former Chancellor Eric Fingerhut, Darlene McCoy, Associate Vice Chancellor of AordabilityandEciency, andKellyBernard. FromOhioLINK,wewishtothankSteveAcker, JohnMagill,and Stacy Brannan. We also wish to thank the good folks at WebAssign,most notably Chris Hall,COO, andJoel Hollenbeck(formerVPofSales.) Last, butcertainlynotleast, wewishtothankall thefolkswhohavecontactedusovertheinterwebs, most notablyDimitri MoonenandJoelWordsworth, who gave us great feedback, and Antonio Olivares who helped debug the source code.TableofContentsPreface ix1 RelationsandFunctions 11.1 SetsofRealNumbersandTheCartesianCoordinatePlane . . . . . . . . . . . . . 11.1.1 SetsofNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 TheCartesianCoordinatePlane . . . . . . . . . . . . . . . . . . . . . . . . 61.1.3 DistanceinthePlane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.5 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2.1 GraphsofEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.2.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.3 IntroductiontoFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531.4 FunctionNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551.4.1 ModelingwithFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631.4.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691.5 FunctionArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841.5.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871.6 GraphsofFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931.6.1 GeneralFunctionBehavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071.6.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1141.7 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1201.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1401.7.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144iv TableofContents2 LinearandQuadraticFunctions 1512.1 LinearFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1512.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1632.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1692.2 AbsoluteValueFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1732.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1832.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1842.3 QuadraticFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1882.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2002.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2032.4 InequalitieswithAbsoluteValueandQuadraticFunctions . . . . . . . . . . . . . . 2082.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2202.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2222.5 Regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2252.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2302.5.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2333 PolynomialFunctions 2353.1 GraphsofPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2353.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2463.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2503.2 TheFactorTheoremandTheRemainderTheorem. . . . . . . . . . . . . . . . . . 2573.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2653.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2673.3 RealZerosofPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2693.3.1 ForThoseWishingtouseaGraphingCalculator. . . . . . . . . . . . . . . 2703.3.2 ForThoseWishingNOTtouseaGraphingCalculator . . . . . . . . . . . 2733.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2803.3.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2823.4 ComplexZerosandtheFundamentalTheoremofAlgebra . . . . . . . . . . . . . . 2863.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2943.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2964 RationalFunctions 3014.1 IntroductiontoRationalFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3014.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3144.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3164.2 GraphsofRationalFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3204.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3334.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3354.3 RationalInequalitiesandApplications . . . . . . . . . . . . . . . . . . . . . . . . . 3424.3.1 Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3504.3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353TableofContents v4.3.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3565 FurtherTopicsinFunctions 3595.1 FunctionComposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3595.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3695.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3725.2 InverseFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3785.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3945.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3965.3 OtherAlgebraicFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3975.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4075.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116 ExponentialandLogarithmicFunctions 4176.1 IntroductiontoExponentialandLogarithmicFunctions . . . . . . . . . . . . . . . 4176.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4296.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4336.2 PropertiesofLogarithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4376.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4456.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4476.3 ExponentialEquationsandInequalities . . . . . . . . . . . . . . . . . . . . . . . . . 4486.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4566.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4586.4 LogarithmicEquationsandInequalities . . . . . . . . . . . . . . . . . . . . . . . . 4596.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4666.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4686.5 ApplicationsofExponentialandLogarithmicFunctions . . . . . . . . . . . . . . . 4696.5.1 ApplicationsofExponentialFunctions. . . . . . . . . . . . . . . . . . . . . 4696.5.2 ApplicationsofLogarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 4776.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4826.5.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4907 HookedonConics 4957.1 IntroductiontoConics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4957.2 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4987.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5027.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5037.3 Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5057.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5127.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5137.4 Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5167.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5257.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527vi TableofContents7.5 Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5417.5.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5448 SystemsofEquationsandMatrices 5498.1 SystemsofLinearEquations: GaussianElimination . . . . . . . . . . . . . . . . . . 5498.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5628.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5648.2 SystemsofLinearEquations: AugmentedMatrices . . . . . . . . . . . . . . . . . . 5678.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5748.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5768.3 MatrixArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5788.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5958.4 SystemsofLinearEquations: MatrixInverses . . . . . . . . . . . . . . . . . . . . . 5988.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6098.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6128.5 DeterminantsandCramersRule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6148.5.1 DenitionandPropertiesoftheDeterminant . . . . . . . . . . . . . . . . . 6148.5.2 CramersRuleandMatrixAdjoints . . . . . . . . . . . . . . . . . . . . . . 6188.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6238.5.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6278.6 PartialFractionDecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6288.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6358.6.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6368.7 SystemsofNon-LinearEquationsandInequalities . . . . . . . . . . . . . . . . . . . 6378.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6468.7.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6489 SequencesandtheBinomialTheorem 6519.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6589.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6609.2 SummationNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6709.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6729.3 MathematicalInduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6739.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6789.3.2 SelectedAnswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6799.4 TheBinomialTheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6819.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6919.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692TableofContents vii10FoundationsofTrigonometry 69310.1 AnglesandtheirMeasure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69310.1.1 ApplicationsofRadianMeasure: CircularMotion . . . . . . . . . . . . . . 70610.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70910.1.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71210.2 TheUnitCircle: CosineandSine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71710.2.1 BeyondtheUnitCircle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73010.2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73610.2.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74010.3 TheSixCircularFunctionsandFundamentalIdentities . . . . . . . . . . . . . . . . 74410.3.1 BeyondtheUnitCircle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75210.3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75910.3.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76610.4 TrigonometricIdentities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77010.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78210.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78710.5 GraphsoftheTrigonometricFunctions. . . . . . . . . . . . . . . . . . . . . . . . . 79010.5.1 GraphsoftheCosineandSineFunctions . . . . . . . . . . . . . . . . . . . 79010.5.2 GraphsoftheSecantandCosecantFunctions . . . . . . . . . . . . . . . . 80010.5.3 GraphsoftheTangentandCotangentFunctions . . . . . . . . . . . . . . . 80410.5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80910.5.5 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81110.6 TheInverseTrigonometricFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 81910.6.1 InversesofSecantandCosecant: TrigonometryFriendlyApproach. . . . . 82710.6.2 InversesofSecantandCosecant: CalculusFriendlyApproach. . . . . . . . 83010.6.3 CalculatorsandtheInverseCircularFunctions. . . . . . . . . . . . . . . . . 83310.6.4 SolvingEquationsUsingtheInverseTrigonometricFunctions. . . . . . . . 83810.6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84110.6.6 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84910.7 TrigonometricEquationsandInequalities . . . . . . . . . . . . . . . . . . . . . . . 85710.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87110.7.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87411ApplicationsofTrigonometry 87911.1 ApplicationsofSinusoids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87911.1.1 HarmonicMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88311.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88911.1.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89211.2 TheLawofSines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89411.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90211.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90611.3 TheLawofCosines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908viii TableofContents11.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91411.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91611.4 PolarCoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91711.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92811.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93011.5 GraphsofPolarEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93611.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95611.5.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96111.6 HookedonConicsAgain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97111.6.1 RotationofAxes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97111.6.2 ThePolarFormofConics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97911.6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98411.6.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98511.7 PolarFormofComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98911.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100211.7.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100511.8 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101011.8.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102511.8.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102911.9 TheDotProductandProjection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103211.9.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104111.9.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104311.10 ParametricEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104611.10.1Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105711.10.2Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061Index 1067PrefaceThankyouforyourinterestinourbook, butmoreimportantly, thankyoufortakingthetimetoreadthePreface. IalwaysreadthePrefacesof thetextbookswhichIuseinmyclassesbecauseIbelieveitisinthePrefacewhereIbegintounderstandtheauthors-whotheyare, whattheirmotivationforwritingthebookwas, andwhattheyhopethereaderwill getoutof readingthetext. PedagogicalissuessuchascontentorganizationandhowprofessorsandstudentsshouldbestuseabookcanusuallybegleanedoutofitsTableofContents,butthereasonsbehindthechoicesauthorsmakeshouldbesharedinthePreface. Also, Ifeel thatthePrefaceofatextbookshoulddemonstratetheauthors loveof their disciplineandpassionfor teaching, sothat I comeawaybelieving that they really want to help students and not just make money. Thus, I thank my fellowPreface-readersagainforgivingmetheopportunitytosharewithyoutheneedandvisionwhichguidedthecreationofthisbookandpassionwhichbothCarlandIholdforMathematicsandtheteachingofit.Carl andI arenativesof NortheastOhio. Wemetingraduateschool atKentStateUniversityin1997. InishedmyPh.DinPureMathematicsinAugust1998andstartedteachingatLorainCounty Community College in Elyria, Ohio just two days after graduation. Carl earned his Ph.D inPure Mathematics in August 2000 and started teaching at Lakeland Community College in Kirtland,Ohiothatsamemonth. Ourschoolsarefairlysimilarinsizeandmissionandeachservesasimilarpopulationof students. Thestudents rangeinagefromabout 16(Ohiohas aPost-SecondaryEnrollment Option program which allows high school students to take college courses for free whilestillinhighschool.) toover65. Manyofthenon-traditionalstudentsarereturningtoschoolinordertochangecareers. Amajorityofthestudentsatbothschoolsreceivesomesortofnancialaid, beitscholarshipsfromtheschools foundations, state-fundedgrantsorfederal nancial aidlikestudent loans, andmanyof themhavelivesbusiedbyfamilyandjobdemands. SomewillbetakingtheirAssociatedegreesandentering(orre-entering)theworkforcewhileotherswill becontinuingontoafour-year collegeor university. Despitetheir manydierences, our studentsshareonecommonattribute: theydonotwanttospend $200onaCollegeAlgebrabook.The challenge of reducing the cost of textbooks is one that many states, including Ohio, are takingquiteseriously. Indeed, state-level leadershavestartedtoworkwithfacultyfromseveral of thecollegesanduniversitiesinOhioandwiththemajorpublishersaswell. Thatprocesswill takeconsiderable time soCarl andI came upwithaplanof our own. We decidedthat the bestwaytohelpourstudentsrightnowwastowriteourownCollegeAlgebrabookandgiveitawayelectronicallyforfree. WeweregrantedsabbaticalsfromourrespectiveinstitutionsfortheSpringx Prefacesemesterof2009andactuallybeganwritingthetextbookonDecember16,2008. Usinganopen-sourcetexteditorcalledTexNicCenterandanopen-sourcedistributionof LaTeXcalledMikTex2.7, Carl and I wrote and edited all of the text, exercises and answers and created all of the graphs(usingMetapostwithinLaTeX)forVersion0.9inabouteightmonths. (Wechoosetocreateatextinonlyblackandwhitetokeepprintingcoststoaminimumforthosestudentswhopreferaprintededition. ThissomewhatSpartanpagelayoutstandsinsharprelief totheexplosionofcolors foundinmost other CollegeAlgebratexts, but neither Carl nor I believethefour-colorprint adds anythingof value.) I usedthe bookinthree sections of College Algebraat LorainCountyCommunityCollegeintheFall of 2009andCarlscolleague, Dr. Bill Previts, taughtasectionof CollegeAlgebraat Lakelandwiththebookthat semester as well. Students hadtheoptionofdownloadingthebookasa.pdflefromourwebsitewww.stitz-zeager.comorbuyingalow-costprintedversionfromourcolleges respectivebookstores. (Bygivingthisbookawayforfreeelectronically,weendthecycleofneweditionsappearingevery18monthstocurtailtheusedbookmarket.) DuringThanksgivingbreakinNovember2009, manyadditional exerciseswrittenbyDr. Previtswereaddedandthetypographical errorsfoundbyourstudentsandotherswerecorrected. On December 10, 2009, Version2 was released. The book remains free for download atourwebsiteandbyusingLulu.comasanon-demandprintingservice,ourbookstoresarenowabletoprovideaprintededitionforjustunder $19. NeitherCarl norIhave, orwill ever, receiveanyroyaltiesfromtheprintededitions. Asacontributionbacktotheopen-sourcecommunity, all oftheLaTeXlesusedtocompilethebookareavailableforfreeunderaCreativeCommonsLicenseonourwebsiteaswell. Thatway,anyonewhowouldliketorearrangeoreditthecontentfortheirclassescandosoaslongasitremainsfree.Theonlydisadvantagetonotworkingforapublisheristhatwedonthaveapaideditorial sta.Whatwehaveinstead, beyondourselves, isfriends, colleaguesandunknownpeopleintheopen-source community who alert us to errors they nd as they read the textbook. What we gain in nothavingtoreporttoapublishersodramaticallyoutweighsthelackofthepaidstathatwehaveturneddowneveryoertopublishourbook. (AsofthewritingofthisPreface, wevehadthreeoers.) Bymaintainingthisbookbyourselves,CarlandIretainallcreativecontrolandkeepthebook our own. We control the organization, depth and rigor of the content which means we can resistthepressuretodiminishtherigorandhomogenizethecontentsoastoappeal toamassmarket.Acasual glancethroughtheTableof Contentsof mostof themajorpublishers CollegeAlgebrabooksrevealsnearlyisomorphiccontentinbothorderanddepth. OurTableofContentsshowsadierentapproach, onethatmightbelabeledFunctionsFirst. TotrulyuseTheRuleof Four,thatis,inordertodiscusseachnewconceptalgebraically,graphically,numericallyandverbally,itseemscompletelyobvioustousthatonewouldneedtointroducefunctionsrst. (Takeamomentandcompareour orderingtotheclassicequations rst, thentheCartesianPlaneandTHENfunctionsapproachseeninmostof themajorplayers.) Wethenintroduceaclassof functionsanddiscusstheequations,inequalities(withaheavyemphasisonsigndiagrams)andapplicationswhich involve functions in that class. The material is presented at a level that denitely prepares astudent for Calculus while giving them relevant Mathematics which can be used in other classes aswell. GraphingcalculatorsareusedsparinglyandonlyasatooltoenhancetheMathematics,nottoreplaceit. Theanswerstonearlyallofthecomputationalhomeworkexercisesaregiveninthexitextandwehavegonetogreatlengthstowritesomeverythoughtprovokingdiscussionquestionswhoseanswers arenot given. Onewill noticethat our exercisesets aremuchshorter thanthetraditional setsofnearly100drill andkillquestionswhichbuildskill devoidofunderstanding.Ourexperiencehasbeenthatstudentscandoabout15-20homeworkexercisesanightsoweverycarefullychosesmallersetsofquestionswhichcoverallofthenecessaryskillsandgetthestudentsthinkingmoredeeplyabouttheMathematicsinvolved.Criticsof theOpenEducational Resourcemovementmightquipthatopen-sourceiswherebadcontentgoestodie,towhichIsaythis: takeaseriouslookatwhatweoerourstudents. Lookthroughafewsectionstoseeifwhatwevewrittenisbadcontentinyouropinion. Iseethisopen-sourcebooknotassomethingwhichisfreeandwortheverypenny,butrather,asahighqualityalternative to the business as usual of the textbook industry and I hope that you agree. If you haveany comments, questions or concerns please feel free to contact me at [email protected] or Carlatcarl@stitz-zeager.com.JeZeagerLorainCountyCommunityCollegeJanuary25,2010xii PrefaceChapter1RelationsandFunctions1.1 Sets of Real Numbers and The Cartesian Coordinate Plane1.1.1 SetsofNumbersWhiletheauthorswouldlikenothingmorethantodelvequicklyanddeeplyintothesheerexcite-mentthatisPrecalculus,experience1hastaughtusthatabriefrefresheronsomebasicnotionsiswelcome, if notcompletelynecessary, atthisstage. Tothatend, wepresentabrief summaryofsettheoryandsomeoftheassociatedvocabularyandnotationsweuseinthetext. LikeallgoodMathbooks,webeginwithadenition.Denition1.1. Asetisawell-denedcollectionofobjectswhicharecalledtheelementsoftheset. Here, well-dened meansthatitispossibletodetermineifsomethingbelongstothecollectionornot,withoutprejudice.Forexample,thecollectionoflettersthatmakeupthewordsmolkoiswell-denedandisaset,but the collection of the worst math teachers in the world is not well-dened,and so is not a set.2Ingeneral,therearethreewaystodescribesets. TheyareWaystoDescribeSets1. TheVerbalMethod: Useasentencetodeneaset.2. TheRosterMethod: Beginwithaleftbrace,listeachelementofthesetonlyonceandthenendwitharightbrace.3. TheSet-Builder Method: Acombinationof theverbal androster methods usingadummyvariablesuchasx.For example, let S be the set described verballyas the set of letters that make up the word smolko.A roster description of Swould be s, m, o, l, k. Note that we listed o only once,even though it1. . . tobereadasgood,solidfeedbackfromcolleagues. . .2Foramorethought-provokingexample,considerthecollectionofallthingsthatdonotcontainthemselves-thisleadstothefamousRussellsParadox.2 RelationsandFunctionsappearstwiceinsmolko.Also,theorderoftheelementsdoesntmatter,so k, l, m, o, sisalsoarosterdescriptionofS. Aset-builderdescriptionofSis:x[ xisaletterinthewordsmolko.Thewaytoreadthisis: Thesetofelementsxsuchthatxisaletterinthewordsmolko. Ineachof theabovecases, wemayusethefamiliarequalssign= andwriteS= s, m, o, l, korS= x[ xisaletterinthewordsmolko.. Clearlymis inSandq is not inS. Weexpressthesesentimentsmathematicallybywritingm Sandq / S. Throughoutyourmathematicalupbringing,youhaveencounteredseveralfamoussetsofnumbers. Theyarelistedbelow.SetsofNumbers1. The EmptySet: = = x[ x ,= x. This is the set with no elements. Like the number0,itplaysavitalroleinmathematics.a2. TheNatural Numbers: N= 1, 2, 3, . . .Theperiodsof ellipsishereindicatethatthenaturalnumberscontain1,2,3,andsoforth.3. TheWholeNumbers: W= 0, 1, 2, . . .4. TheIntegers: Z = . . . , 1, 2, 1, 0, 1, 2, 3, . . .5. TheRational Numbers: Q =_ab [ a Zand b Z_. Rationalnumbersaretheratiosofintegers (provided the denominator is not zero!) It turns out that another way to describetherationalnumbersbis:Q = x[ xpossessesarepeatingorterminatingdecimalrepresentation.6. TheRealNumbers: R = x[ xpossessesadecimalrepresentation.7. TheIrrational Numbers: P= x[ xisanon-rationalrealnumber.Saidanotherway,anirrationalnumberisadecimalwhichneitherrepeatsnorterminates.c8. TheComplexNumbers: C = a +bi [ a,b Randi =1Despitetheirimportance,thecomplexnumbersplayonlyaminorroleinthetext.da. . . which,sadly,wewillnotexploreinthistext.bSeeSection9.2.cTheclassicexampleisthenumber(SeeSection10.1),butnumberslike2and0.101001000100001 . . .areothernerepresentatives.dTheyrstappearinSection3.4andreturninSection11.7.Itis importanttonotethateverynaturalnumberis awholenumber,which,inturn,isaninteger.Eachinteger is arational number (take b =1inthe above denitionforQ) andthe rationalnumbersareallrealnumbers,sincetheypossessdecimalrepresentations.3Ifwetakeb = 0inthe3Longdivision,anyone?1.1SetsofRealNumbersandTheCartesianCoordinatePlane 3abovedenitionof C, weseethateveryrealnumberisacomplexnumber. Inthissense, thesetsN, W, Z, Q, R,and CarenestedlikeMatryoshkadolls.Forthemostpart, thistextbookfocusesonsetswhoseelementscomefromthereal numbers R.Recall thatwemayvisualize Rasaline. Segmentsof thislinearecalledintervalsof numbers.Below is a summary of the so-called intervalnotation associated with given sets of numbers. Forintervalswithniteendpoints, welisttheleftendpoint, thentherightendpoint. Weusesquarebrackets, [ or], if theendpointisincludedintheinterval andusealled-inorclosed dottoindicate membership in the interval. Otherwise, we use parentheses, ( or ) and an open circle toindicatethattheendpointisnotpartoftheset. Iftheintervaldoesnothaveniteendpoints,weuse the symbols to indicate that the interval extends indenitely to the left and to indicatethattheinterval extendsindenitelytotheright. Sinceinnityisaconcept, andnotanumber,wealwaysuseparentheseswhenusingthesesymbolsinintervalnotation, anduseanappropriatearrowtoindicatethattheintervalextendsindenitelyinone(orboth)directions.IntervalNotationLetaandbberealnumberswitha < b.SetofRealNumbers IntervalNotation RegionontheRealNumberLinex[ a < x < b (a, b)a bx[ a x < b [a, b)a bx[ a < x b (a, b]a bx[ a x b [a, b]a bx[ x < b (, b)bx[ x b (, b]bx[ x > a (a, )ax[ x a [a, )aR(, )4 RelationsandFunctionsForanexample,considerthesetsofrealnumbersdescribedbelow.SetofRealNumbers IntervalNotation RegionontheRealNumberLinex[ 1 x < 3 [1, 3)1 3x[ 1 x 4 [1, 4]1 4x[ x 5 (, 5]5x[ x > 2 (2, )2Wewilloftenhaveoccasiontocombinesets. Therearetwobasicwaystocombinesets: intersec-tionandunion. Wedenebothoftheseconceptsbelow.Denition1.2. SupposeAandBaretwosets. TheintersectionofAandB: A B= x[ x Aandx B TheunionofAandB: A B= x[ x Aorx B(orboth)Saiddierently,theintersectionoftwosetsistheoverlapofthetwosetstheelementswhichthesetshaveincommon. Theunionoftwosetsconsistsofthetotalityoftheelementsineachofthesets, collectedtogether.4Forexample, if A= 1, 2, 3andB= 2, 4, 6, thenA B= 2andAB= 1, 2, 3, 4, 6. If A = [5, 3) and B= (1, ), then we can nd ABand ABgraphically.To nd AB,we shade the overlap of the two and obtain AB= (1, 3). To nd AB,we shadeeachofAandBanddescribetheresultingshadedregiontondA B= [5, ).5 1 3A = [5, 3),B= (1, )5 1 3A B= (1, 3)5 1 3A B= [5, )Whilebothintersectionandunionareimportant,wehavemoreoccasiontouseunioninthistextthanintersection, simplybecausemostof thesetsof real numberswewill beworkingwithareeitherintervalsorareunionsofintervals,asthefollowingexampleillustrates.4ThereaderisencouragedtoresearchVennDiagramsforanicegeometricinterpretationoftheseconcepts.1.1SetsofRealNumbersandTheCartesianCoordinatePlane 5Example1.1.1. Expressthefollowingsetsofnumbersusingintervalnotation.1. x[ x 2orx 2 2. x[ x ,= 33. x[ x ,= 3 4. x[ 1 < x 3orx = 5Solution.1. Thebestwaytoproceedhereistographthesetofnumbersonthenumberlineandgleantheanswerfromit. Theinequalityx 2correspondstotheinterval (, 2] andtheinequalityx 2correspondstotheinterval[2, ). Sincewearelookingtodescribetherealnumbersxinoneoftheseortheother,wehave x[ x 2orx 2 = (, 2] [2, ).2 2(, 2] [2, )2. Fortheset x[ x ,=3, weshadetheentirereal numberlineexceptx=3, whereweleaveanopencircle. This divides thereal number lineintotwointervals, (, 3) and(3, ).Sincethevalues of xcouldbeineither oneof theseintervals or theother, wehavethatx[ x ,= 3 = (, 3) (3, )3(, 3) (3, )3. Fortheset x[ x ,= 3,weproceedasbeforeandexcludebothx = 3andx = 3fromourset. Thisbreaksthenumberlineintothreeintervals, (, 3), (3, 3)and(3, ). Sincethesetdescribesreal numberswhichcomefromtherst, secondor thirdinterval, wehavex[ x ,= 3 = (, 3) (3, 3) (3, ).3 3(, 3) (3, 3) (3, )4. Graphingtheset x[ 1 < x 3orx = 5,wegetoneinterval,(1, 3]alongwithasinglenumber, orpoint, 5. Whilewecouldexpressthelatteras[5, 5] (Canyouseewhy?), wechoosetowriteouransweras x[ 1 < x 3orx = 5 = (1, 3] 5.1 3 5(1, 3] 56 RelationsandFunctions1.1.2 TheCartesianCoordinatePlaneInordertovisualizethepureexcitementthatisPrecalculus, weneedtouniteAlgebraandGe-ometry. Simplyput, wemustndawaytodrawalgebraicthings. Letsstartwithpossiblythegreatest mathematical achievement of all time: the CartesianCoordinatePlane.5Imagine tworealnumberlinescrossingatarightangleat0asdrawnbelow.xy4 3 2 1 1 2 3 443211234Thehorizontal numberlineisusuallycalledthex-axiswhilethevertical numberlineisusuallycalled the y-axis.6As with the usual number line, we imagine these axes extending o indenitelyinbothdirections.7Havingtwonumberlinesallowsustolocatethepositionsofpointsoofthenumberlinesaswellaspointsonthelinesthemselves.Forexample,considerthepointPonthenextpage. Tousethenumbersontheaxestolabelthispoint, we imagine dropping a vertical line from the x-axis to Pand extending a horizontal line fromthey-axistoP. Thisprocessissometimescalledprojecting thepointPtothex-(respectivelyy-)axis. WethendescribethepointPusingtheorderedpair(2, 4). Therstnumberintheorderedpair is calledtheabscissaor x-coordinateandthesecondis calledtheordinateory-coordinate.8Takentogether,theorderedpair(2, 4)comprisetheCartesiancoordinates9of thepoint P. Inpractice, thedistinctionbetweenapoint andits coordinates is blurred; forexample, weoftenspeakofthepoint(2, 4). Wecanthinkof(2, 4)asinstructionsonhowto5SonamedinhonorofReneDescartes.6Thelabelscanvarydependingonthecontextofapplication.7Usuallyextendingotowards innityis indicatedbyarrows, but here, the arrows are usedtoindicate thedirectionofincreasingvaluesofxandy.8Again, thenamesofthecoordinatescanvarydependingonthecontextoftheapplication. If, forexample, thehorizontal axisrepresentedtimewemightchoosetocall itthet-axis. Therstnumberintheorderedpairwouldthenbethet-coordinate.9AlsocalledtherectangularcoordinatesofPseeSection11.4formoredetails.1.1SetsofRealNumbersandTheCartesianCoordinatePlane 7reachPfromtheorigin(0, 0)bymoving2unitstotherightand4unitsdownwards. Noticethattheorderintheorderedpairisimportant ifwewishtoplotthepoint(4, 2), wewouldmovetotheleft4unitsfromtheoriginandthenmoveupwards2units,asbelowontheright.xyP4 3 2 1 1 2 3 443211234xyP(2, 4)(4, 2)4 3 2 1 1 2 3 443211234WhenwespeakoftheCartesianCoordinatePlane, wemeanthesetofall possibleorderedpairs(x, y)asxandytakevaluesfromtherealnumbers. BelowisasummaryofimportantfactsaboutCartesiancoordinates.ImportantFactsabouttheCartesianCoordinatePlane (a, b)and(c, d)representthesamepointintheplaneifandonlyifa = candb = d. (x, y)liesonthex-axisifandonlyify= 0. (x, y)liesonthey-axisifandonlyifx = 0. Theoriginisthepoint(0, 0). Itistheonlypointcommontobothaxes.Example1.1.2. Plotthefollowingpoints: A(5, 8), B_52, 3_, C(5.8, 3), D(4.5, 1), E(5, 0),F(0, 5),G(7, 0),H(0, 9),O(0, 0).10Solution. Toplotthesepoints,westartattheoriginandmovetotherightifthex-coordinateispositive;totheleftifitisnegative. Next,wemoveupifthey-coordinateispositiveordownifitisnegative. Ifthex-coordinateis0, westartattheoriginandmovealongthey-axisonly. Ifthey-coordinateis0wemovealongthex-axisonly.10TheletterOisalmostalwaysreservedfortheorigin.8 RelationsandFunctionsxyA(5, 8)B_52, 3_C(5.8, 3)D(4.5, 1)E(5, 0)F(0, 5)G(7, 0)H(0, 9)O(0, 0)987654321 1 2 3 4 5 6 7 8 9987654321123456789The axes divide the plane intofour regions calledquadrants. Theyare labeledwithRomannumeralsandproceedcounterclockwisearoundtheplane:xyQuadrantIx > 0,y> 0QuadrantIIx < 0,y> 0QuadrantIIIx < 0,y< 0QuadrantIVx > 0,y< 04 3 2 1 1 2 3 4432112341.1SetsofRealNumbersandTheCartesianCoordinatePlane 9For example, (1, 2) lies in Quadrant I, (1, 2) in Quadrant II, (1, 2) in Quadrant III and (1, 2)inQuadrantIV. Ifapointotherthantheoriginhappenstolieontheaxes, wetypicallyrefertothat point as lying on the positive or negative x-axis (if y= 0) or on the positive or negative y-axis(if x=0). Forexample, (0, 4)liesonthepositivey-axiswhereas(117, 0)liesonthenegativex-axis. Suchpointsdonotbelongtoanyofthefourquadrants.One of the most important concepts in all of Mathematics is symmetry.11There are many types ofsymmetryinMathematics,butthreeofthemcanbediscussedeasilyusingCartesianCoordinates.Denition1.3. Twopoints(a, b)and(c, d)intheplanearesaidtobe symmetricaboutthex-axisifa = candb = d symmetricaboutthey-axisifa = candb = d symmetricabouttheoriginifa = candb = dSchematically,0 xyP(x, y) Q(x, y)S(x, y) R(x, y)Inthe above gure, P andSare symmetric about the x-axis, as are QandR; P andQaresymmetricaboutthey-axis,asareRandS;andPandRaresymmetricabouttheorigin,asareQandS.Example1.1.3. Let Pbe the point (2, 3). Find the points which are symmetric to Pabout the:1. x-axis 2. y-axis 3. originCheckyouranswerbyplottingthepoints.Solution. ThegureafterDenition1.3givesusagoodwaytothinkaboutndingsymmetricpointsintermsoftakingtheoppositesofthex-and/ory-coordinatesofP(2, 3).11AccordingtoCarl. Jethinkssymmetryisoverrated.10 RelationsandFunctions1. Tondthepointsymmetricaboutthex-axis,wereplacethey-coordinatewithitsoppositetoget(2, 3).2. Tondthepointsymmetricaboutthey-axis,wereplacethex-coordinatewithitsoppositetoget(2, 3).3. Tondthepointsymmetricabouttheorigin,wereplacethex-andy-coordinateswiththeiroppositestoget(2, 3).xyP(2, 3)(2, 3)(2, 3)(2, 3)3 2 1 1 2 3321123Onewaytovisualizetheprocessesinthepreviousexampleiswiththeconceptofareection. Ifwe start with our point (2, 3) and pretend that the x-axis is a mirror, then the reection of (2, 3)acrossthex-axiswouldlieat(2, 3). If wepretendthatthey-axisisamirror, thereectionof(2, 3)acrossthataxiswouldbe(2, 3). Ifwereectacrossthex-axisandthenthey-axis, wewould go from (2, 3) to (2, 3) then to (2, 3),and so we would end up at the point symmetricto(2, 3)abouttheorigin. Wesummarizeandgeneralizethisprocessbelow.ReectionsToreectapoint(x, y)aboutthe: x-axis,replaceywith y. y-axis,replacexwith x. origin,replacexwith xandywith y.1.1.3 DistanceinthePlaneAnotherimportantconceptinGeometryisthenotionoflength. IfwearegoingtouniteAlgebraandGeometryusingtheCartesianPlane, thenweneedtodevelopanalgebraicunderstandingofwhatdistanceintheplanemeans. Supposewehavetwopoints, P (x0, y0)andQ(x1, y1) , intheplane. By the distance d between Pand Q, we mean the length of the line segment joining PwithQ. (Remember, givenanytwodistinctpointsintheplane, thereisauniquelinecontainingboth1.1SetsofRealNumbersandTheCartesianCoordinatePlane 11points.) Our goal now is to create an algebraic formula to compute the distance between these twopoints. Considerthegenericsituationbelowontheleft.P (x0, y0)Q(x1, y1)dP (x0, y0)Q(x1, y1)d(x1, y0)Withalittlemoreimagination,wecanenvisionarighttrianglewhosehypotenusehaslengthdasdrawn above on the right. From the latter gure,we see that the lengths of the legs of the triangleare [x1 x0[and [y1 y0[sothePythagoreanTheoremgivesus[x1 x0[2+[y1 y0[2= d2(x1 x0)2+ (y1 y0)2= d2(Do you remember why we can replace the absolute value notation with parentheses?)By extractingthesquareroot of bothsides of thesecondequationandusingthefact that distanceis nevernegative,wegetEquation1.1. TheDistanceFormula: ThedistancedbetweenthepointsP (x0, y0) andQ(x1, y1)is:d =_(x1 x0)2+ (y1 y0)2ItisnotalwaysthecasethatthepointsPandQlendthemselvestoconstructingsuchatriangle.IfthepointsPandQarearrangedverticallyorhorizontally,ordescribetheexactsamepoint,wecannotusetheabovegeometricargumenttoderivethedistanceformula. ItislefttothereaderinExercise35toverifyEquation1.1forthesecases.Example1.1.4. FindandsimplifythedistancebetweenP(2, 3)andQ(1, 3).Solution.d =_(x1 x0)2+ (y1 y0)2=_(1 (2))2+ (3 3)2=9 + 36= 35Sothedistanceis35.12 RelationsandFunctionsExample1.1.5. Find all of the points with x-coordinate 1 which are 4 units from the point (3, 2).Solution. Weshallsoonseethatthepointswewishtondareonthelinex = 1,butfornowwelljustviewthemaspointsoftheform(1, y). Visually,(1, y)(3, 2)xydistanceis4units2 3321123We require that the distance from (3, 2) to (1, y) be 4. The Distance Formula,Equation 1.1,yieldsd =_(x1 x0)2+ (y1 y0)24 =_(1 3)2+ (y 2)24 =_4 + (y 2)242=__4 + (y 2)2_2squaringbothsides16 = 4 + (y 2)212 = (y 2)2(y 2)2= 12y 2 = 12 extractingthesquarerooty 2 = 23y = 2 23Weobtaintwoanswers: (1, 2 + 23)and(1, 2 23).Thereaderisencouragedtothinkaboutwhytherearetwoanswers.Relatedtondingthedistancebetweentwopointsistheproblemofndingthemidpointofthelinesegmentconnectingtwopoints. Giventwopoints,P (x0, y0)andQ(x1, y1),themidpointMof Pand Q is dened to be the point on the line segment connecting Pand Q whose distance fromPisequaltoitsdistancefromQ.1.1SetsofRealNumbersandTheCartesianCoordinatePlane 13P (x0, y0)Q(x1, y1)MIfwethinkofreachingMbygoinghalfwayoverandhalfwayupwegetthefollowingformula.Equation1.2. TheMidpoint Formula: ThemidpointMof thelinesegmentconnectingP (x0, y0)andQ(x1, y1)is:M=_x0 +x12, y0 +y12_If we let d denote the distance between Pand Q, we leave it as Exercise 36 to show that the distancebetweenPandMisd/2whichisthesameasthedistancebetweenMandQ. ThissucestoshowthatEquation1.2givesthecoordinatesofthemidpoint.Example1.1.6. FindthemidpointofthelinesegmentconnectingP(2, 3)andQ(1, 3).Solution.M =_x0 +x12, y0 +y12_=_(2) + 12, 3 + (3)2_ =_12, 02_=_12, 0_Themidpointis_12, 0_.WeclosewithamoreabstractapplicationoftheMidpointFormula. WewillrevisitthefollowingexampleinExercise72inSection2.1.Example 1.1.7. If a ,= b, prove that the line y= x equally divides the line segment with endpoints(a, b)and(b, a).Solution. Toprovetheclaim,weuseEquation1.2tondthemidpointM =_a +b2, b +a2_=_a +b2, a +b2_Sincethexandycoordinatesofthispointarethesame,wendthatthemidpointliesontheliney= x,asrequired.14 RelationsandFunctions1.1.4 Exercises1. Fillinthechartbelow:SetofRealNumbers IntervalNotation RegionontheRealNumberLinex[ 1 x < 5[0, 3)2 7x[ 5 < x 0(3, 3)5 7x[ x 3(, 9)4x[ x 3InExercises2-7, ndtheindicatedintersectionorunionandsimplifyif possible. Expressyouranswersinintervalnotation.2. (1, 5] [0, 8) 3. (1, 1) [0, 6] 4. (, 4] (0, )5. (, 0) [1, 5] 6. (, 0) [1, 5] 7. (, 5] [5, 8)InExercises8-19,writethesetusingintervalnotation.8. x[ x ,= 5 9. x[ x ,= 1 10. x[ x ,= 3,41.1SetsofRealNumbersandTheCartesianCoordinatePlane 1511. x[ x ,= 0,2 12. x[ x ,= 2, 2 13. x[ x ,= 0, 414. x[ x 1 or x 1 15. x[ x < 3 or x 2 16. x[ x 3 or x > 017. x[ x 5 or x = 6 18. x[ x > 2 or x = 1 19. x[ 3 < x < 3 or x = 420. Plot and label the points A(3, 7), B(1.3, 2), C(,10), D(0, 8), E(5.5, 0), F(8, 4),G(9.2, 7.8)andH(7, 5)intheCartesianCoordinatePlanegivenbelow.xy9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 998765432112345678921. ForeachpointgiveninExercise20above Identifythequadrantoraxisin/onwhichthepointlies. Findthepointsymmetrictothegivenpointaboutthex-axis. Findthepointsymmetrictothegivenpointaboutthey-axis. Findthepointsymmetrictothegivenpointabouttheorigin.16 RelationsandFunctionsIn Exercises 22 - 29, nd the distance d between the points and the midpoint Mof the line segmentwhichconnectsthem.22. (1, 2),(3, 5) 23. (3, 10),(1, 2)24._12, 4_,_32, 1_25._23, 32_,_73, 2_26._245, 65_,_115, 195_. 27._2,3_,_8, 12_28._245,12_,_20,27_. 29. (0, 0),(x, y)30. Findallofthepointsoftheform(x, 1)whichare4unitsfromthepoint(3, 2).31. Findallofthepointsonthey-axiswhichare5unitsfromthepoint(5, 3).32. Findallofthepointsonthex-axiswhichare2unitsfromthepoint(1, 1).33. Findallofthepointsoftheform(x, x)whichare1unitfromtheorigin.34. Letsassumeforamomentthatwearestandingattheoriginandthepositivey-axispointsdueNorthwhilethepositivex-axispointsdueEast. OurSasquatch-o-metertellsusthatSasquatch is 3 miles West and 4 miles South of our current position. What are the coordinatesof his position?How far away is he from us?If he runs 7 miles due East what would his newpositionbe?35. VerifytheDistanceFormula1.1forthecaseswhen:(a) Thepointsarearrangedvertically. (Hint: UseP(a, y0)andQ(a, y1).)(b) Thepointsarearrangedhorizontally. (Hint: UseP(x0, b)andQ(x1, b).)(c) Thepointsareactuallythesamepoint. (Youshouldntneedahintforthisone.)36. Verifythe Midpoint Formulabyshowingthe distance betweenP(x1, y1) andMandthedistancebetweenMandQ(x2, y2)arebothhalfofthedistancebetweenPandQ.37. ShowthatthepointsA, BandCbelowaretheverticesofarighttriangle.(a) A(3, 2), B(6, 4),andC(1, 8) (b) A(3, 1), B(4, 0)andC(0, 3)38. Find a point D(x, y) such that the points A(3, 1), B(4, 0), C(0, 3) and Dare the cornersofasquare. Justifyyouranswer.39. Discusswithyourclassmateshowmanynumbersareintheinterval(0, 1).40. Theworldisnotat.12ThustheCartesianPlanecannotpossiblybetheendof thestory.DiscusswithyourclassmateshowyouwouldextendCartesianCoordinatestorepresentthethree dimensional world. What would the Distance and Midpoint formulas look like, assumingthoseconceptsmakesenseatall?12Therearethosewhodisagreewiththisstatement. LookthemupontheInternetsometimewhenyourebored.1.1SetsofRealNumbersandTheCartesianCoordinatePlane 171.1.5 Answers1.SetofRealNumbers IntervalNotation RegionontheRealNumberLinex[ 1 x < 5 [1, 5)1 5x[ 0 x < 3 [0, 3)0 3x[ 2 < x 7 (2, 7]2 7x[ 5 < x 0 (5, 0]5 0x[ 3 < x < 3 (3, 3)3 3x[ 5 x 7 [5, 7]5 7x[ x 3 (, 3]3x[ x < 9 (, 9)9x[ x > 4 (4, )4x[ x 3 [3, )32. (1, 5] [0, 8) = [0, 5] 3. (1, 1) [0, 6] = (1, 6]4. (, 4] (0, ) = (0, 4] 5. (, 0) [1, 5] = 6. (, 0) [1, 5] = (, 0) [1, 5] 7. (, 5] [5, 8) = 58. (, 5) (5, ) 9. (, 1) (1, )10. (, 3) (3, 4) (4, ) 11. (, 0) (0, 2) (2, )12. (, 2) (2, 2) (2, ) 13. (, 4) (4, 0) (0, 4) (4, )18 RelationsandFunctions14. (, 1] [1, ) 15. (, 3) [2, )16. (, 3] (0, ) 17. (, 5] 618. 1 1 (2, ) 19. (3, 3) 420. Therequiredpoints A(3, 7), B(1.3, 2), C(,10), D(0, 8), E(5.5, 0), F(8, 4),G(9.2, 7.8),andH(7, 5)areplottedintheCartesianCoordinatePlanebelow.xyA(3, 7)B(1.3, 2)C(,10)D(0, 8)E(5.5, 0)F(8, 4)G(9.2, 7.8)H(7, 5)9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 99876543211234567891.1SetsofRealNumbersandTheCartesianCoordinatePlane 1921. (a) ThepointA(3, 7)is inQuadrantIII symmetricaboutx-axiswith(3, 7) symmetricabouty-axiswith(3, 7) symmetricaboutoriginwith(3, 7)(b) ThepointB(1.3, 2)is inQuadrantIV symmetricaboutx-axiswith(1.3, 2) symmetric about y-axis with (1.3, 2) symmetricaboutoriginwith(1.3, 2)(c) ThepointC(,10)is inQuadrantI symmetric about x-axis with (, 10) symmetric about y-axis with (,10) symmetric about origin with(, 10)(d) ThepointD(0, 8)is onthepositivey-axis symmetricaboutx-axiswith(0, 8) symmetricabouty-axiswith(0, 8) symmetricaboutoriginwith(0, 8)(e) ThepointE(5.5, 0)is onthenegativex-axis symmetricaboutx-axiswith(5.5, 0) symmetricabouty-axiswith(5.5, 0) symmetricaboutoriginwith(5.5, 0)(f) ThepointF(8, 4)is inQuadrantII symmetricaboutx-axiswith(8, 4) symmetricabouty-axiswith(8, 4) symmetricaboutoriginwith(8, 4)(g) ThepointG(9.2, 7.8)is inQuadrantIV symmetricaboutx-axiswith(9.2, 7.8) symmetric about y-axis with(9.2, 7.8) symmetric about origin with (9.2, 7.8)(h) ThepointH(7, 5)is inQuadrantI symmetricaboutx-axiswith(7, 5) symmetricabouty-axiswith(7, 5) symmetricaboutoriginwith(7, 5)22. d = 5,M=_1,72_23. d = 410,M= (1, 4)24. d =26,M=_1,32_25. d =372,M=_56,74_26. d =74,M=_1310, 1310_27. d = 35,M=_22, 32_28. d =83,M=_45,532_29. d =_x2+y2,M=_x2,y2_30. (3 +7, 1),(3 7, 1) 31. (0, 3)32. (1 +3, 0),(1 3, 0) 33._22, 22_,_22,22_34. (3, 4),5miles,(4, 4)37. (a) ThedistancefromAtoBis [AB[ =13, thedistancefromAtoCis [AC[ =52,andthedistancefromBtoCis [BC[ =65. Since_13_2+_52_2=_65_2, weareguaranteedbytheconverseofthePythagoreanTheoremthatthetriangleisarighttriangle.(b) Showthat [AC[2+[BC[2= [AB[220 RelationsandFunctions1.2 RelationsFrom one point of view,1all of Precalculus can be thought of as studying sets of points in the plane.WiththeCartesianPlanenowfreshinourmemorywecandiscussthosesetsinmoredetail andasusual,webeginwithadenition.Denition1.4. Arelationisasetofpointsintheplane.Since relations are sets, we can describe them using the techniques presented in Section 1.1.1. Thatis, we can describe a relation verbally, using the roster method, or using set-builder notation. Sincethe elements in a relation are points in the plane, we often try to describe the relation graphically oralgebraically as well. Depending on the situation, one method may be easier or more convenient touse than another. As an example, consider the relation R = (2, 1), (4, 3), (0, 3). As written, Ris described using the roster method. Since R consists of points in the plane, we follow our instinctandplotthepoints. DoingsoproducesthegraphofR.(2, 1)(4, 3)(0, 3)xy4 3 2 1 1 2 3 443211234ThegraphofR.Inthefollowingexample,wegraphavarietyofrelations.Example1.2.1. Graphthefollowingrelations.1. A = (0, 0), (3, 1), (4, 2), (3, 2) 2. HLS1= (x, 3) [ 2 x 43. HLS2= (x, 3) [ 2 x < 4 4. V= (3, y) [ yisarealnumber5. H= (x, y) [ y= 2 6. R = (x, y) [ 1 < y 31Carls,ofcourse.1.2Relations 21Solution.1. TographA,wesimplyplotallofthepointswhichbelongtoA,asshownbelowontheleft.2. Dontletthenotationinthispartfoolyou. ThenameofthisrelationisHLS1,justlikethenameoftherelationinnumber1wasA. Thelettersandnumbersarejustpartofitsname,just like the numbers and letters of the phrase King George III were part of Georges name.Inwords, (x, 3) [ 2 x 4readsthesetofpoints(x, 3)suchthat 2 x 4. Allofthesepointshavethesamey-coordinate,3,butthex-coordinateisallowedtovarybetween2and4,inclusive. SomeofthepointswhichbelongtoHLS1includesomefriendlypointslike: (2, 3),(1, 3),(0, 3),(1, 3),(2, 3),(3, 3),and(4, 3). However,HLS1alsocontainsthepoints (0.829, 3),_56, 3_, (, 3), andsoon. It is impossible2tolist all of thesepoints,whichiswhythevariablexisused. Plottingseveral friendlyrepresentativepointsshouldconvinceyouthatHLS1describesthehorizontal linesegmentfromthepoint(2, 3)uptoandincludingthepoint(4, 3).xy4 3 2 1 1 2 3 41234ThegraphofAxy4 3 2 1 1 2 3 41234ThegraphofHLS13. HLS2ishauntinglysimilartoHLS1. Infact, theonlydierencebetweenthetwoisthatinsteadof2 x 4wehave2 x < 4. Thismeansthatwestillgetahorizontallinesegmentwhichincludes(2, 3)andextendsto(4, 3),butwedonot include(4, 3)becauseofthestrictinequalityx < 4. Howdowedenotethisonourgraph?Itisacommonmistaketomake the graph start at (2, 3) end at (3, 3) as pictured below on the left. The problem withthisgraphisthatweareforgettingaboutthepointslike(3.1, 3), (3.5, 3), (3.9, 3), (3.99, 3),andsoforth. Thereisnorealnumberthatcomesimmediatelybefore4,sotodescribethesetofpointswewant, wedrawthehorizontal linesegmentstartingat(2, 3)anddrawanopencircleat(4, 3)asdepictedbelowontheright.2Reallyimpossible. Theinterestedreaderisencouragedtoresearchcountableversusuncountablesets.22 RelationsandFunctionsxy4 3 2 1 1 2 3 41234ThisisNOTthecorrectgraphofHLS2xy4 3 2 1 1 2 3 41234ThegraphofHLS24. Next, wecometotherelationV , describedasthesetof points(3, y)suchthatyisarealnumber. All of thesepointshaveanx-coordinateof 3, but they-coordinateisfreetobewhateveritwantstobe, withoutrestriction.3Plottingafewfriendly pointsof V shouldconvince you that all the points of Vlie on the vertical line4x = 3. Since there is no restrictiononthey-coordinate,weputarrowsontheendoftheportionofthelinewedrawtoindicateitextendsindenitelyinbothdirections. ThegraphofV isbelowontheleft.5. Though written slightly dierently, the relation H= (x, y) [ y= 2 is similar to the relationVabove in that only one of the coordinates, in this case the y-coordinate, is specied, leavingxtobefree. Plottingsomerepresentativepointsgivesusthehorizontalliney= 2.xy1 2 3 443211234ThegraphofVxy4 3 2 1 1 2 3 44321ThegraphofH6. Forourlastexample, weturntoR= (x, y) [ 1 4 8. (x, 3) [ x 49. (1, y) [ y> 1 10. (2, y) [ y 511. (2, y) [ 3 < y 4 12. (3, y) [ 4 y< 313. (x, 2) [ 2 x < 3 14. (x, 3) [ 4 < x 415. (x, y) [ x > 2 16. (x, y) [ x 317. (x, y) [ y< 4 18. (x, y) [ x 3,y< 219. (x, y) [ x > 0,y< 4 20. (x, y) [ 2 x 23, < y 92InExercises21-30,describethegivenrelationusingeithertherosterorset-buildermethod.21.xy4 3 2 1 111234RelationA22.xy1 2 3 4 1 2 3 4123RelationB30 RelationsandFunctions23.xy1 2 332112345RelationC24.xy3 2 14321123RelationD25.xy4 3 2 1 1 2 3 4123RelationE26.xy3 2 1 1 2 31234RelationF27.xy3 2 1 1 2 3321123RelationG28.xy4 3 2 1 1 2 3321123RelationH1.2Relations 3129.xy1 1 2 3 4 5112345RelationI30.xy4 3 2 1 1 2 3 4 532112RelationJInExercises31-36,graphthegivenline.31. x = 2 32. x = 333. y= 3 34. y= 235. x = 0 36. y= 0Some relations are fairly easy to describe in words or with the roster method but are rather dicult,ifnotimpossible,tograph. Discusswithyourclassmateshowyoumightgraphtherelationsgivenin Exercises 37 - 40. Please note that in the notation below we are using the ellipsis, . . . , to denotethatthelistdoesnotend,butrather,continuestofollowtheestablishedpatternindenitely. FortherelationsinExercises37and38,givetwoexamplesofpointswhichbelongtotherelationandtwopointswhichdonotbelongtotherelation.37. (x, y) [ xisanoddinteger,andyisaneveninteger.38. (x, 1) [ xisanirrationalnumber 39. (1, 0), (2, 1), (4, 2), (8, 3), (16, 4), (32, 5), . . .40. . . . , (3, 9), (2, 4), (1, 1), (0, 0), (1, 1), (2, 4), (3, 9), . . .ForeachequationgiveninExercises41-52: Findthex-andy-intercept(s)ofthegraph,ifanyexist. FollowtheprocedureinExample 1.2.3tocreateatableofsamplepointsonthegraphoftheequation. Plotthesamplepointsandcreatearoughsketchofthegraphoftheequation. Testforsymmetry. Iftheequationappearstofailanyofthesymmetrytests,ndapointonthegraphoftheequationwhosereectionfailstobeonthegraphaswasdoneattheendofExample1.2.432 RelationsandFunctions41. y= x2+ 1 42. y= x22x 843. y= x3x 44. y=x343x45. y=x 2 46. y= 2x + 4 247. 3x y= 7 48. 3x 2y= 1049. (x + 2)2+y2= 16 50. x2y2= 151. 4y29x2= 36 52. x3y= 4The procedures which we have outlined in the Examples of this section and used in Exercises 41 - 52all rely on the fact that the equations were well-behaved. Not everything in Mathematics is quitesotame, asthefollowingequationswill showyou. DiscusswithyourclassmateshowyoumightapproachgraphingtheequationsgiveninExercises53- 56. Whatdicultiesarisewhentryingtoapplythevarioustestsandproceduresgiveninthissection? Formoreinformation, includingpicturesof thecurves, eachcurvenameisalinktoitspageatwww.wikipedia.org. Foramuchlongerlistoffascinatingcurves,clickhere.53. x3+y33xy= 0 FoliumofDescartes 54. x4= x2+y2KampyleofEudoxus55. y2= x3+ 3x2Tschirnhausencubic 56. (x2+y2)2= x3+y3Crookedegg57. Withthehelpofyourclassmates,ndexamplesofequationswhosegraphspossess symmetryaboutthex-axisonly symmetryaboutthey-axisonly symmetryabouttheoriginonly symmetryaboutthex-axis,y-axis,andoriginCan you nd an example of an equation whose graph possesses exactly twoof the symmetrieslistedabove?Whyorwhynot?1.2Relations 331.2.3 Answers1.xy3 2 1 1 2 31234567892.xy2 1 1 23211233.xy2 1 1 2123412344.xy654321 1 2 3 4 5 66543211234565.xy2 1 1 212346.xy1 2 312345678934 RelationsandFunctions7.xy4 3 2 1 1 2 3 4318.xy1 2 3 4 1 2 3 41239.xy1 1 21234567810.xy1 2 33211234511.xy3 2 1321123412.xy1 2 343211231.2Relations 3513.xy4 3 2 1 1 2 3 412314.xy4 3 2 1 1 2 3 412315.xy1 2 132112316.xy1 2 332112317.xy3 2 1 1 2 3123418.xy1 2 332112319.xy1 1 2 3123420.xy2 1 11234536 RelationsandFunctions21. A = (4, 1), (2, 1), (0, 3), (1, 4) 22. B= (x, 3) [ x 323. C= (2, y) [ y> 3 24. D = (2, y) [ 4 y< 325. E= (x, 2) [ 4 < x 3 26. F= (x, y) [ y 027. G = (x, y) [ x > 2 28. H= (x, y) [ 3 < x 229. I= (x, y) [ x 0,y 0 30. J= (x, y) [ 4 < x < 5, 3 < y< 231.xy3 2 1123123Thelinex = 232.xy1 2 3123123Thelinex = 333.xy3 2 1 1 2 3123Theliney= 334.xy3 2 1 1 2 3123Theliney= 235.xy3 2 1 1 2 3123123Thelinex = 0isthey-axis36.xy3 2 1 1 2 3123123Theliney= 0isthex-axis1.2Relations 3741. y= x2+ 1Thegraphhasnox-interceptsy-intercept: (0, 1)x y (x, y)2 5 (2, 5)1 2 (1, 2)0 1 (0, 1)1 2 (1, 2)2 5 (2, 5)xy21 1 212345Thegraphisnotsymmetricaboutthex-axis(e.g. (2, 5)isonthegraphbut(2, 5)isnot)Thegraphissymmetricaboutthey-axisThegraphisnotsymmetricabouttheorigin(e.g. (2, 5)isonthegraphbut(2, 5)isnot)42. y= x22x 8x-intercepts: (4, 0),(2, 0)y-intercept: (0, 8)x y (x, y)3 7 (3, 7)2 0 (2, 0)1 5 (1, 5)0 8 (0, 8)1 9 (1, 9)2 8 (2, 8)3 5 (3, 5)4 0 (4, 0)5 7 (5, 7)xy321 12345987654321234567Thegraphisnotsymmetricaboutthex-axis(e.g. (3, 7)isonthegraphbut(3, 7)isnot)Thegraphisnotsymmetricaboutthey-axis(e.g. (3, 7)isonthegraphbut(3, 7)isnot)Thegraphisnotsymmetricabouttheorigin(e.g. (3, 7)isonthegraphbut(3, 7)isnot)38 RelationsandFunctions43. y= x3xx-intercepts: (1, 0), (0, 0), (1, 0)y-intercept: (0, 0)x y (x, y)2 6 (2, 6)1 0 (1, 0)0 0 (0, 0)1 0 (1, 0)2 6 (2, 6)xy21 1 2654321123456Thegraphisnotsymmetricaboutthex-axis. (e.g. (2, 6)isonthegraphbut(2, 6)isnot)Thegraphisnotsymmetricaboutthey-axis. (e.g. (2, 6)isonthegraphbut(2, 6)isnot)Thegraphissymmetricabouttheorigin.44. y=x343xx-intercepts:_23, 0_, (0, 0)y-intercept: (0, 0)x y (x, y)4 4 (4, 4)394_3,94_2 4 (2, 4)1114_1,114_0 0 (0, 0)1 114_1, 114_2 4 (2, 4)3 94_3, 94_4 4 (4, 4)xy4321 1 2 3 412341234Thegraphisnotsymmetricaboutthex-axis(e.g. (4, 4)isonthegraphbut(4, 4)isnot)Thegraphisnotsymmetricaboutthey-axis(e.g. (4, 4)isonthegraphbut(4, 4)isnot)Thegraphissymmetricabouttheorigin1.2Relations 3945. y=x 2x-intercept: (2, 0)Thegraphhasnoy-interceptsx y (x, y)2 0 (2, 0)3 1 (3, 1)6 2 (6, 2)11 3 (11, 3)xy1 2 3 4 5 6 7 8 9 1011123Thegraphisnotsymmetricaboutthex-axis(e.g. (3, 1)isonthegraphbut(3, 1)isnot)Thegraphisnotsymmetricaboutthey-axis(e.g. (3, 1)isonthegraphbut(3, 1)isnot)Thegraphisnotsymmetricabouttheorigin(e.g. (3, 1)isonthegraphbut(3, 1)isnot)46. y= 2x + 4 2x-intercept: (3, 0)y-intercept: (0, 2)x y (x, y)4 2 (4, 2)3 0 (3, 0)2 22 2_2,2 2_1 23 2_2,3 2_0 2 (0, 2)1 25 2_2,5 2_xy4321 1 2321123Thegraphisnotsymmetricaboutthex-axis(e.g. (4, 2)isonthegraphbut(4, 2)isnot)Thegraphisnotsymmetricaboutthey-axis(e.g. (4, 2)isonthegraphbut(4, 2)isnot)Thegraphisnotsymmetricabouttheorigin(e.g. (4, 2)isonthegraphbut(4, 2)isnot)40 RelationsandFunctions47. 3x y= 7Re-writeas: y= 3x 7.x-intercept: (73, 0)y-intercept: (0, 7)x y (x, y)2 13 (2, 13)1 10 (1, 10)0 7 (0, 7)1 4 (1, 4)2 1 (2, 1)3 2 (3, 2)xy21 1 2 313121110987654321123Thegraphisnotsymmetricaboutthex-axis(e.g. (3, 2)isonthegraphbut(3, 2)isnot)Thegraphisnotsymmetricaboutthey-axis(e.g. (3, 2)isonthegraphbut(3, 2)isnot)Thegraphisnotsymmetricabouttheorigin(e.g. (3, 2)isonthegraphbut(3, 2)isnot)48. 3x 2y= 10Re-writeas: y=3x102.x-intercepts:_103 , 0_y-intercept: (0, 5)x y (x, y)2 8 (2, 8)1 132_1, 132_0 5 (0, 5)1 72_1, 72_2 2 (2, 2)xy321 1 2 3 498765432112Thegraphisnotsymmetricaboutthex-axis(e.g. (2, 2)isonthegraphbut(2, 2)isnot)Thegraphisnotsymmetricaboutthey-axis(e.g. (2, 2)isonthegraphbut(2, 2)isnot)Thegraphisnotsymmetricabouttheorigin(e.g. (2, 2)isonthegraphbut(2, 2)isnot)1.2Relations 4149. (x + 2)2+y2= 16Re-writeasy= _16 (x + 2)2.x-intercepts: (6, 0),(2, 0)y-intercepts:_0, 23_x y (x, y)6 0 (6, 0)4 23_4, 23_2 4 (2, 4)0 23_0, 23_2 0 (2, 0)xy7654321 1 2 35432112345Thegraphissymmetricaboutthex-axisThegraphisnotsymmetricaboutthey-axis(e.g. (6, 0)isonthegraphbut(6, 0)isnot)Thegraphisnotsymmetricabouttheorigin(e.g. (6, 0)isonthegraphbut(6, 0)isnot)50. x2y2= 1Re-writeas: y= x21.x-intercepts: (1, 0), (1, 0)Thegraphhasnoy-interceptsx y (x, y)3 8 (3, 8)2 3 (2, 3)1 0 (1, 0)1 0 (1, 0)2 3 (2, 3)3 8 (3, 8)xy321 1 2 3321123Thegraphissymmetricaboutthex-axisThegraphissymmetricaboutthey-axisThegraphissymmetricabouttheorigin42 RelationsandFunctions51. 4y29x2= 36Re-writeas: y= 9x2+362.Thegraphhasnox-interceptsy-intercepts: (0, 3)x y (x, y)4 35_4, 35_2 32_2, 32_0 3 (0, 3)2 32_2, 32_4 35_4, 35_xy4321 1 2 3 476543211234567Thegraphissymmetricaboutthex-axisThegraphissymmetricaboutthey-axisThegraphissymmetricabouttheorigin52. x3y= 4Re-writeas: y= 4x3.Thegraphhasnox-interceptsThegraphhasnoy-interceptsx y (x, y)212(2,12)1 4 (1, 4)1232 (12, 32)1232 (12, 32)1 4 (1, 4)2 12(2, 12)xy2 1 1 2324432Thegraphisnotsymmetricaboutthex-axis(e.g. (1, 4)isonthegraphbut(1, 4)isnot)Thegraphisnotsymmetricaboutthey-axis(e.g. (1, 4)isonthegraphbut(1, 4)isnot)Thegraphissymmetricabouttheorigin1.3IntroductiontoFunctions 431.3 IntroductiontoFunctionsOneofthecoreconceptsinCollegeAlgebraisthefunction. Therearemanywaystodescribeafunctionandwebeginbydeningafunctionasaspecialkindofrelation.Denition1.6. Arelationinwhicheachx-coordinateismatchedwithonlyoney-coordinateissaidtodescribeyasafunctionofx.Example1.3.1. Whichofthefollowingrelationsdescribeyasafunctionofx?1. R1= (2, 1), (1, 3), (1, 4), (3, 1) 2. R2= (2, 1), (1, 3), (2, 3), (3, 1)Solution. AquickscanofthepointsinR1revealsthatthex-coordinate1ismatchedwithtwodierent y-coordinates: namely3and4. HenceinR1, yis not afunctionof x. Ontheotherhand, everyx-coordinate inR2occurs onlyonce whichmeans eachx-coordinate has onlyonecorrespondingy-coordinate. So,R2doesrepresentyasafunctionofx.Notethatinthepreviousexample, therelationR2containedtwodierentpointswiththesamey-coordinates, namely(1, 3)and(2, 3). Remember, inordertosayyisafunctionof x, wejustneedtoensurethesamex-coordinateisntusedinmorethanonepoint.1Toseewhatthefunctionconceptmeansgeometrically,wegraphR1andR2intheplane.xy2 1 1 2 311234ThegraphofR1xy2 1 1 2 311234ThegraphofR2Thefactthatthex-coordinate1ismatchedwithtwodierenty-coordinatesinR1presentsitselfgraphicallyasthepoints(1, 3)and(1, 4)lyingonthesamevertical line, x=1. If weturnourattention to the graph of R2,we see that no two points of the relation lie on the same vertical line.WecangeneralizethisideaasfollowsTheorem 1.1. The Vertical Line Test: A set of points in the plane represents y as a functionofxifandonlyifnotwopointslieonthesameverticalline.1Wewillhaveoccasionlaterinthetexttoconcernourselveswiththeconceptofxbeingafunctionofy. Inthiscase,R1representsxasafunctionofy;R2doesnot.44 RelationsandFunctionsItisworthtakingsometimetomeditateontheVertical LineTest; itwill checktoseehowwellyouunderstandtheconceptoffunctionaswellastheconceptofgraph.Example1.3.2. Use the Vertical Line Test to determine which of the following relations describesyasafunctionofx.xy1 2 311234ThegraphofRxy1 111234ThegraphofSSolution. Looking at the graph of R, we can easily imagine a vertical line crossing the graph morethanonce. Hence, Rdoesnotrepresentyasafunctionof x. However, inthegraphof S, everyverticallinecrossesthegraphatmostonce,soSdoesrepresentyasafunctionofx.Intheprevioustest,wesaythatthegraphoftherelationRfailstheVerticalLineTest,whereasthe graph of Spasses the Vertical Line Test. Note that in the graph of R there are innitely manyvertical lines which cross the graph more than once. However, to fail the Vertical Line Test, all youneedisoneverticallinethattsthebill,asthenextexampleillustrates.Example1.3.3. Use the Vertical Line Test to determine which of the following relations describesyasafunctionofx.xy1 111234ThegraphofS1xy1 111234ThegraphofS21.3IntroductiontoFunctions 45Solution. Both S1and S2are slight modications to the relation Sin the previous example whosegraphwedeterminedpassedtheVertical LineTest. InbothS1andS2, itistheadditionof thepoint (1, 2) which threatens to cause trouble. In S1, there is a point on the curve with x-coordinate1 just below (1, 2),which means that both (1, 2) and this point on the curve lie on the vertical linex = 1. (Seethepicturebelowandtheleft.) Hence,thegraphofS1failstheVerticalLineTest,soyis not a function of x here. However, in S2notice that the point with x-coordinate 1 on the curvehas been omitted,leaving an open circle there. Hence,the vertical line x = 1 crosses the graph ofS2onlyatthepoint(1, 2). Indeed,anyverticallinewillcrossthegraphatmostonce,sowehavethatthegraphofS2passestheVerticalLineTest. Thusitdescribesyasafunctionofx.xy111234S1andthelinex = 1xy1 111234ThegraphofGforEx. 1.3.4SupposearelationFdescribesyasafunctionof x. Thesetsof x- andy-coordinatesaregivenspecialnameswhichwedenebelow.Denition1.7. SupposeFisarelationwhichdescribesyasafunctionofx. Thesetofthex-coordinatesofthepointsinFiscalledthedomainofF. Thesetofthey-coordinatesofthepointsinFiscalledtherangeofF.We demonstrate nding the domain and range of functions given to us either graphically or via therostermethodinthefollowingexample.Example1.3.4. FindthedomainandrangeofthefunctionF= (3, 2), (0, 1), (4, 2), (5, 2)andofthefunctionGwhosegraphisgivenaboveontheright.Solution. ThedomainofFisthesetofthex-coordinatesofthepointsinF,namely 3, 0, 4, 5andtherangeofFisthesetofthey-coordinates,namely 1, 2.Todeterminethedomainandrangeof G, weneedtodeterminewhichxandyvaluesoccurascoordinates of points on the given graph. To nd the domain, it may be helpful to imagine collapsingthecurvetothex-axisanddeterminingtheportionofthex-axisthatgetscovered. Thisiscalledprojectingthecurvetothex-axis. Beforewestartprojecting, weneedtopayattentiontotwo46 RelationsandFunctionssubtle notations on the graph: the arrowhead on the lower left corner of the graph indicates that thegraph continues to curve downwards to the left forever more;and the open circle at (1, 3) indicatesthatthepoint(1, 3)isntonthegraph,butallpointsonthecurveleadinguptothatpointare.projectdownprojectupxy1 111234ThegraphofGxy1 111234ThegraphofGWeseefromthegurethatifweprojectthegraphofGtothex-axis,wegetallrealnumberslessthan1. Usingintervalnotation,wewritethedomainofGas(, 1). TodeterminetherangeofG,weprojectthecurvetothey-axisasfollows:projectleftprojectrightxy1 111234ThegraphofGxy1 111234ThegraphofGNote that even though there is an open circle at (1, 3), we still include the yvalue of 3 in our range,sincethepoint(1, 3)isonthegraphof G. Weseethattherangeof Gisall real numberslessthanorequalto4,or,inintervalnotation,(, 4].1.3IntroductiontoFunctions 47Allfunctionsarerelations,butnotallrelationsarefunctions. Thustheequationswhichdescribedthe relations in Section1.2 may or may not describe y as a function of x. The algebraic representationof functions is possibly the most important way to view them so we need a process for determiningwhetherornotanequationofarelationrepresentsafunction. (WedelaythediscussionofndingthedomainofafunctiongivenalgebraicallyuntilSection1.4.)Example1.3.5. Determinewhichequationsrepresentyasafunctionofx.1. x3+y2= 1 2. x2+y3= 1 3. x2y= 1 3ySolution. Foreachoftheseequations,wesolveforyanddeterminewhethereachchoiceofxwilldetermineonlyonecorrespondingvalueofy.1.x3+y2= 1y2= 1 x3_y2=1 x3extractsquarerootsy = 1 x3If wesubstitutex=0intoourequationfory, wegety= 1 03= 1, sothat(0, 1)and(0, 1)areonthegraphofthisequation. Hence,thisequationdoesnotrepresentyasafunctionofx.2.x2+y3= 1y3= 1 x23_y3=31 x2y =31 x2Foreverychoiceof x, theequationy=31 x2returnsonlyonevalueof y. Hence, thisequationdescribesyasafunctionofx.3.x2y = 1 3yx2y + 3y = 1y_x2+ 3_= 1 factory =1x2+ 3For each choice of x, there is only one value for y, so this equation describes y as a function of x.Wecouldtrytouseourgraphingcalculatortoverifyourresponsestothepreviousexample, butwe immediately run into trouble. The calculators Y= menu requires that the equation be of theformy=someexpressionof x. If wewantedtoverifythattherstequationinExample1.3.548 RelationsandFunctionsdoesnotrepresentyasafunctionofx, wewouldneedtoentertwoseparateexpressionsintothecalculator: oneforthepositivesquarerootandoneforthenegativesquarerootwefoundwhensolvingtheequationfory. Aspredicted,theresultinggraphshownbelowclearlyfailstheVerticalLineTest,sotheequationdoesnotrepresentyasafunctionofx.Thus in order to use the calculator to show that x3+y2= 1 does not represent yas a function of xweneededtoknowanalyticallythatywasnotafunctionofxsothatwecouldusethecalculatorproperly. Therearemoreadvancedgraphingutilities out therewhichcandoimplicit functionplots, but you need to know even more Algebra to make them work properly. Do you get the pointweretryingtomakehere?Webelieveitisinyourbestinteresttolearntheanalyticwayofdoingthingssothatyouarealwayssmarterthanyourcalculator.1.3IntroductiontoFunctions 491.3.1 ExercisesIn Exercises 1 - 12,determine whether or not the relation represents yas a function of x. Find thedomainandrangeofthoserelationswhicharefunctions.1. (3, 9), (2, 4), (1, 1), (0, 0), (1, 1), (2, 4), (3, 9)2. (3, 0), (1, 6), (2, 3), (4, 2), (5, 6), (4, 9), (6, 2)3. (3, 0), (7, 6), (5, 5), (6, 4), (4, 9), (3, 0)4. (1, 2), (4, 4), (9, 6), (16, 8), (25, 10), (36, 12), . . .5. (x, y) [ xisanoddinteger,andyisaneveninteger6. (x, 1) [ xisanirrationalnumber7. (1, 0), (2, 1), (4, 2), (8, 3), (16, 4), (32, 5), . . . 8. . . . , (3, 9), (2, 4), (1, 1), (0, 0), (1, 1), (2, 4), (3, 9), . . . 9. (2, y) [ 3 < y< 4 10. (x, 3) [ 2 x < 411. _x, x2_ [ xisarealnumber 12. _x2, x_ [ xisarealnumberInExercises13-32, determinewhetherornottherelationrepresentsyasafunctionofx. Findthedomainandrangeofthoserelationswhicharefunctions.13.xy4 3 2 1 11123414.xy4 3 2 1 11123450 RelationsandFunctions15.xy2 1 1 21234516.xy3 2 1 1 2 332112317.xy1 2 3 4 5 6 7 8 912318.xy4 3 2 1 1 2 3 4123419.xy4 3 2 1 1 2 3 4 53211220.xy5 4 3 2 1 1 2 321123421.xy321 1 2 35432112345678922.xy54321 1 2 3 4 554321123451.3IntroductiontoFunctions 5123.xy54321 1 2 3 4 5543211234524.xy1 1 2 3 4 5 6543211234525.xy2 1 1 2123426.xy2 1 1 2123427.xy2 1 1 2123428.xy2 1 1 2123429.xy2 1 1 2122130.xy3 2 1 1 2 3122152 RelationsandFunctions31.xy2 1 1 2122132.xy2 1 1 21221InExercises33-47,determinewhetherornottheequationrepresentsyasafunctionofx.33. y= x3x 34. y=x 2 35. x3y= 436. x2y2= 137. y=xx2938. x = 639. x = y2+ 4 40. y= x2+ 4 41. x2+y2= 442. y=4 x243. x2y2= 4 44. x3+y3= 445. 2x + 3y= 4 46. 2xy= 4 47. x2= y248. ExplainwhythepopulationPof Sasquatchinagivenareaisafunctionof timet. Whatwouldbetherangeofthisfunction?49. Explainwhytherelationbetweenyourclassmatesandtheiremail addressesmaynotbeafunction. WhataboutphonenumbersandSocialSecurityNumbers?TheprocessgiveninExample 1.3.5fordeterminingwhetheranequationofarelationrepresentsyasafunctionofxbreaksdownifwecannotsolvetheequationforyintermsofx. However,thatdoes not prevent us from proving that an equation fails to represent yas a function of x. What wereally need is two points with the same x-coordinate and dierent y-coordinates which both satisfytheequationsothatthegraphoftherelationwouldfail theVertical LineTest1.1. DiscusswithyourclassmateshowyoumightndsuchpointsfortherelationsgiveninExercises50-53.50. x3+y33xy= 0 51. x4= x2+y252. y2= x3+ 3x253. (x2+y2)2= x3+y31.3IntroductiontoFunctions 531.3.2 Answers1. Functiondomain= 3, 2, 1,0,1,2,3range= 0,1,4,92. Notafunction3. Functiondomain= 7, 3, 3, 4, 5, 6range= 0, 4, 5, 6, 94. Functiondomain= 1, 4, 9, 16, 25, 36, . . .= x[ xisaperfectsquarerange= 2, 4, 6, 8, 10, 12, . . .= y [ yisapositiveeveninteger5. Notafunction 6. Functiondomain= x xisirrationalrange= 17. Functiondomain= x x=2nforsomewholenumbernrange= y yisanywholenumber8. Functiondomain= x xisanyintegerrange= y y=n2forsomeintegern9. Notafunction 10. Functiondomain=[2, 4),range= 311. Functiondomain=(, )range=[0, )12. Notafunction13. Functiondomain= 4, 3, 2, 1,0,1range= 1,0,1,2,3,414. Notafunction15. Functiondomain=(, )range=[1, )16. Notafunction17. Functiondomain=[2, )range=[0, )18. Functiondomain=(, )range=(0, 4]19. Notafunction 20. Functiondomain=[5, 3) (3, 3)range=(2, 1) [0, 4)54 RelationsandFunctions21. Functiondomain=[2, )range=[3, )22. Notafunction23. Functiondomain=[5, 4)range=[4, 4)24. Functiondomain=[0, 3) (3, 6]range=(4, 1] [0, 4]25. Functiondomain=(, )range=(, 4]26. Functiondomain=(, )range=(, 4]27. Functiondomain=[2, )range=(, 3]28. Functiondomain=(, )range=(, )29. Functiondomain=(, 0] (1, )range=(, 1] 230. Functiondomain=[3, 3]range=[2, 2]31. Notafunction 32. Functiondomain=(, )range= 233. Function 34. Function 35. Function36. Notafunction 37. Function 38. Notafunction39. Notafunction 40. Function 41. Notafunction42. Function 43. Notafunction 44. Function45. Function 46. Function 47. Notafunction1.4FunctionNotation 551.4 FunctionNotationInDenition1.6, wedescribedafunctionas aspecial kindof relation oneinwhicheachx-coordinateismatchedwithonlyoney-coordinate. Inthissection,wefocusmoreontheprocessbywhichthexismatchedwiththey. Ifwethinkofthedomainofafunctionasasetof inputsandtherangeasasetof outputs,wecanthinkofafunctionfasaprocessbywhicheachinputxismatchedwithonlyoneoutputy. Sincetheoutputiscompletelydeterminedbytheinputxandtheprocessf, wesymbolizetheoutputwithfunctionnotation: f(x), readf of x. Inotherwords, f(x)istheoutputwhichresultsbyapplyingtheprocessf totheinputx. Inthiscase, the parentheses here do not indicate multiplication, as they do elsewhere in Algebra. This cancause confusion if the context is not clear, so you must read carefully. This relationship is typicallyvisualizedusingadiagramsimilartotheonebelow.fxDomain(Inputs)y= f(x)Range(Outputs)Thevalueofyiscompletelydependentonthechoiceofx. Forthisreason, xisoftencalledtheindependentvariable, or argument of f, whereas yis often called the dependentvariable.Asweshall see, theprocessof afunctionf isusuallydescribedusinganalgebraicformula. Forexample, suppose a function ftakes a real number and performs the following two steps, in sequence1. multiplyby32. add4Ifwechoose5asourinput,instep1wemultiplyby3toget(5)(3) = 15. Instep2,weadd4toour result from step 1 which yields 15 +4 = 19. Using function notation, we would write f(5) = 19to indicate that the resultofapplying the process fto the input5 gives the output19. Ingeneral,ifweusexfortheinput, applyingstep1produces3x. Followingwithstep2produces3x + 4asournaloutput. Henceforaninputx,wegettheoutputf(x) = 3x + 4. Noticethattocheckourformulaforthecasex=5, wereplacetheoccurrenceof xintheformulaforf(x)with5togetf(5) = 3(5) + 4 = 15 + 4 = 19,asrequired.56 RelationsandFunctionsExample1.4.1. Supposeafunctiongisdescribedbyapplyingthefollowingsteps,insequence1. add42. multiplyby3Determineg(5)andndanexpressionforg(x).Solution. Startingwith5,step1gives5 + 4 = 9. Continuingwithstep2,weget(3)(9) = 27. Tond a formula for g(x), we start with our input x. Step 1 produces x+4. We now wish to multiplythisentirequantityby3,soweuseaparentheses: 3(x +4) = 3x +12. Hence,g(x) = 3x +12. Wecancheckourformulabyreplacingxwith5togetg(5) = 3(5) + 12 = 15 + 12 = 27 .Mostof thefunctionswewill encounterinCollegeAlgebrawill bedescribedusingformulasliketheoneswedevelopedforf(x)andg(x)above. EvaluatingformulasusingthisfunctionnotationisakeyskillforsuccessinthisandmanyotherMathcourses.Example1.4.2. Letf(x) = x2+ 3x + 41. Findandsimplifythefollowing.(a) f(1),f(0),f(2)(b) f(2x),2f(x)(c) f(x + 2),f(x) + 2,f(x) +f(2)2. Solvef(x) = 4.Solution.1. (a) Tondf(1),wereplaceeveryoccurrenceofxintheexpressionf(x)with 1f(1) = (1)2+ 3(1) + 4= (1) + (3) + 4= 0Similarly,f(0) = (0)2+ 3(0) + 4 = 4,andf(2) = (2)2+ 3(2) + 4 = 4 + 6 + 4 = 6.(b) Tondf(2x),wereplaceeveryoccurrenceofxwiththequantity2xf(2x) = (2x)2+ 3(2x) + 4= (4x2) + (6x) + 4= 4x2+ 6x + 4Theexpression2f(x)meanswemultiplytheexpressionf(x)by22f(x) = 2_x2+ 3x + 4_= 2x2+ 6x + 81.4FunctionNotation 57(c) Tondf(x + 2),wereplaceeveryoccurrenceofxwiththequantityx + 2f(x + 2) = (x + 2)2+ 3(x + 2) + 4= _x2+ 4x + 4_+ (3x + 6) + 4= x24x 4 + 3x + 6 + 4= x2x + 6Tondf(x) + 2,weadd2totheexpressionforf(x)f(x) + 2 =_x2+ 3x + 4_+ 2= x2+ 3x + 6Fromourworkabove,weseef(2) = 6sothatf(x) +f(2) =_x2+ 3x + 4_+ 6= x2+ 3x + 102. Sincef(x) = x2+3x +4,theequationf(x) = 4isequivalentto x2+3x +4 = 4. Solvingweget x2+ 3x=0, orx(x + 3)=0. Wegetx=0orx=3, andwecanverifytheseanswersbycheckingthatf(0) = 4andf(3) = 4.AfewnotesaboutExample1.4.2areinorder. Firstnotethedierencebetweentheanswersforf(2x)and2f(x). Forf(2x),wearemultiplyingtheinput by2;for2f(x),wearemultiplyingtheoutput by2. Aswesee, wegetentirelydierentresults. Alongtheselines, notethatf(x + 2),f(x)+2 and f(x)+f(2) are three dierentexpressions as well. Even though function notation usesparentheses, as does multiplication, there is nogeneral distributive property of function notation.Finally, note the practice of usingparentheses whensubstitutingone algebraic expressionintoanother;wehighlyrecommendthispracticeasitwillreducecarelesserrors.Supposenowwewishtondr(3)forr(x) =2xx29. Substitutiongivesr(3) =2(3)(3)29=60,whichisundened. (Whyisthis,again?) Thenumber3isnotanallowableinputtothefunctionr; inotherwords, 3isnotinthedomainof r. Whichotherreal numbersareforbiddeninthisformula?We think back to arithmetic. The reason r(3) is undened is because substitution resultsinadivisionby0. Todeterminewhichothernumbersresultinsuchatransgression, wesetthedenominatorequalto0andsolvex29 = 0x2= 9x2=9 extractsquarerootsx = 358 RelationsandFunctionsAs long as we substitute numbers other than 3 and 3, the expression r(x) is a real number. Hence,wewriteourdomainininterval notation1as(, 3) (3, 3) (3, ). Whenaformulaforafunctionisgiven,weassumethatthefunctionisvalidforallrealnumberswhichmakearithmeticsense when substituted into the formula. This set of numbers is often called the implieddomain2ofthefunction. Atthisstage,thereareonlytwomathematicalsinsweneedtoavoid: divisionby0 and extracting even roots of negative numbers. The following example illustrates these concepts.Example1.4.3. Findthedomain3ofthefollowingfunctions.1. g(x) =4 3x 2. h(x) =54 3x3. f(x) =21 4xx 34. F(x) =42x + 1x215. r(t) =46 t + 36. I(x) =3x2xSolution.1. Thepotentialdisasterforgisiftheradicand4isnegative. Toavoidthis,weset4 3x 0.Fromthis,weget3x 4orx 43. Whatthisshowsisthataslongasx 43,theexpression4 3x 0,andtheformulag(x)returnsarealnumber. Ourdomainis_,43.2. Theformulaforh(x)ishauntinglyclosetothatof g(x)withonekeydierence whereastheexpressionforg(x)includesanevenindexedroot(namelyasquareroot), theformulaforh(x)involvesanoddindexedroot(thefthroot). Sinceoddrootsofrealnumbers(evennegativereal numbers)arereal numbers, thereisnorestrictionontheinputstoh. Hence,thedomainis(, ).3. In the expression for f, there are two denominators. We need to make sure neither of them is0. Tothatend,weseteachdenominatorequalto0andsolve. Forthesmalldenominator,wegetx 3 = 0orx = 3. Forthelargedenominator1SeetheExercisesforSection1.1.2or,implicitdomain3Thewordimpliedis,well,implied.4Theradicandistheexpressioninsidetheradical.1.4FunctionNotation 591 4xx 3= 01 =4xx 3(1)(x 3) =_4x$$$x 3_$$$$(x 3) cleardenominatorsx 3 = 4x3 = 3x1 = xSowegettworeal numberswhichmakedenominators0, namelyx= 1andx=3. Ourdomainisallrealnumbersexcept 1and3: (, 1) (1, 3) (3, ).4. Inndingthedomainof F, wenoticethatwehavetwopotentiallyhazardousissues: notonlydowehaveadenominator, wehaveafourth(even-indexed)root. Ourstrategyistodeterminetherestrictionsimposedbyeachpartandselectthereal numberswhichsatisfyboth conditions. To satisfy the fourth root, we require 2x+1 0. From this we get 2x 1orx 12. Next,weroundupthevaluesofxwhichcouldcausetroubleinthedenominatorbysettingthedenominatorequalto0. Wegetx21 = 0,orx = 1. Hence,inorderforarealnumberxtobeinthedomainofF, x 12butx ,= 1. Inintervalnotation, thissetis_12, 1_ (1, ).5. Dontbeputobythet here. Itisanindependentvariablerepresentingareal number,justlikexdoes,andissubjecttothesamerestrictions. Asinthepreviousproblem,wehavedoubledangerhere: wehaveasquarerootandadenominator. Tosatisfythesquareroot,weneedanon-negativeradicandsowesett + 3 0togett 3. Settingthedenominatorequal tozerogives6 t + 3=0, ort + 3=6. Squaringbothsidesgivest + 3=36, ort = 33. Sincewesquaredbothsidesinthecourseofsolvingthisequation,weneedtocheckouranswer.5Sureenough, whent=33, 6 t + 3=6 36=0, sot=33will causeproblemsinthedenominator. Atlastwecanndthedomainof r: weneedt 3, butt ,= 33. Ournalansweris[3, 33) (33, ).6. ItstemptingtosimplifyI(x) =3x2x= 3x,and,sincetherearenolongeranydenominators,claim that there are no longer any restrictions. However, in simplifying I(x), we are assumingx ,=0, since00isundened.6Proceedingasbefore, wendthedomainof I tobeall realnumbersexcept0: (, 0) (0, ).Itisworthreiteratingtheimportanceof ndingthedomainof afunctionbefore simplifying, asevidencedbythefunctionIinthepreviousexample. EventhoughtheformulaI(x)simpliesto5Doyourememberwhy?Considersquaringbothsidestosolvet + 1 = 2.6Moreprecisely,thefraction00isanindeterminantform. Calculusisrequiredtamesuchbeasts.60 RelationsandFunctions3x,itwouldbeinaccuratetowriteI(x) = 3xwithoutaddingthestipulationthatx ,= 0. Itwouldbeanalogoustonotreportingtaxableincomeorsomeothersinofomission.1.4.1 ModelingwithFunctionsThe importance of Mathematics to our society lies in its value to approximate, or model real-worldphenomenon. Whetheritbeusedtopredictthehightemperatureonagivenday, determinethehours of daylight on a given day, or predict population trends of various and sundry real and myth-icalbeasts,7Mathematicsissecondonlytoliteracyintheimportancehumanitysdevelopment.8ItisimportanttokeepinmindthatanytimeMathematicsisusedtoapproximatereality, therearealwayslimitationstothemodel. Forexample, supposegrapesareonsaleatthelocalmarketfor$1.50perpound. Thenonepoundofgrapescosts$1.50,twopoundsofgrapescost$3.00,andsoforth. Supposewewanttodevelopaformulawhichrelatesthecostof buyinggrapestotheamountofgrapesbeingpurchased. Sincethesetwoquantitiesvaryfromsituationtosituation,weassignthemvariables. Letcdenotethecostofthegrapesandletgdenotetheamountofgrapespurchased. Tondthecostcofthegrapes,wemultiplytheamountofgrapesgbytheprice$1.50dollarsperpoundtogetc = 1.5gIn order for the units to be correct in the formula, g must be measured in poundsof grapes in whichcasethecomputedvalueof cismeasuredindollars. Sincewereinterestedinndingthecostcgivenanamountg, wethinkof gastheindependentvariableandcasthedependentvariable.Usingthelanguageoffunctionnotation,wewritec(g) = 1.5gwhere g is the amount of grapes purchased (in pounds) and c(g) is the cost (in dollars). For example,c(5) represents the cost, in dollars, to purchase 5 pounds of grapes. In this case, c(5) = 1.5(5) = 7.5,so it would cost $7.50. If, on the other hand, we wanted to nd the amountof grapes we can purchasefor$5,wewouldneedtosetc(g) = 5andsolveforg. Inthiscase,c(g) = 1.5g,sosolvingc(g) = 5is equivalent to solving 1.5g= 5 Doing so gives g=51.5= 3.3. This means we can purchase exactly3.3poundsof grapesfor$5. Of course, youwouldbehard-pressedtobuyexactly3.3poundsofgrapes,9andthisleadsustoournexttopicofdiscussion,theapplieddomain10ofafunction.Eventhough, mathematically, c(g)=1.5ghasnodomainrestrictions(therearenodenominatorsandnoeven-indexedradicals), therearecertainvaluesof gthatdontmakeanyphysical sense.Forexample, g= 1correspondstopurchasing 1poundsof grapes.11Also, unlessthelocalmarket mentioned is the State of California (or some other exporter of grapes), it also doesnt makemuchsenseforg=500,000,000, either. Sotherealityofthesituationlimitswhatgcanbe, and7SeeSections2.5,11.1,and6.5,respectively.8InCarlshumbleopinion,ofcourse. . .9Youcouldgetclose... withinacertainspeciedmarginoferror,perhaps.10or,explicitdomain11Maybethismeansreturningapoundofgrapes?1.4FunctionNotation 61theselimitsdeterminetheapplieddomainofg. Typically, anapplieddomainisstatedexplicitly.Inthiscase, itwouldbecommontoseesomethinglikec(g)=1.5g, 0 g 100, meaningthenumber of pounds of grapes purchased is limited from 0 up to 100. The upper bound here, 100 mayrepresenttheinventoryofthemarket,orsomeotherlimitassetbylocalpolicyorlaw. Evenwiththisrestriction, ourmodel hasitslimitations. Aswesawabove, itisvirtuallyimpossibletobuyexactly3.3poundsofgrapessothatourcostisexactly$5. Inthiscase, beingsensibleshoppers,wewouldmostlikelyrounddownandpurchase3poundsofgrapesorhoweverclosethemarketscalecanreadto3.3withoutbeingover. Itistimeforamoresophisticatedexample.Example1.4.4. Theheighthinfeetofamodelrocketabovethegroundtsecondsafterlift-oisgivenbyh(t) =_ 5t2+ 100t, if 0 t 200, if t > 201. Findandinterpreth(10)andh(60).2. Solveh(t) = 375andinterpretyouranswers.Solution.1. We rst note that the independent variable here is t, chosenbecause it represents time.Secondly,thefunctionisbrokenupintotworules: oneformulaforvaluesoftbetween0and20inclusive,andanotherforvaluesoftgreaterthan20. Sincet = 10satisestheinequality0 t 20, we use the rst formulalisted, h(t) = 5t2+ 100t, tondh(10). We geth(10) = 5(10)2+ 100(10) = 500. Sincetrepresentsthenumberofsecondssincelift-oandh(t)istheheightabovethegroundinfeet,theequationh(10) = 500meansthat10secondsafter lift-o, the model rocket is 500 feet above the ground. To nd h(60), we note that t = 60satisest>20, soweusetheruleh(t)=0. Thisfunctionreturnsavalueof 0regardlessofwhatvalueissubstitutedinfort,soh(60) = 0. Thismeansthat60secondsafterlift-o,therocketis0feetabovetheground; inotherwords, aminuteafterlift-o, therockethasalreadyreturnedtoEarth.2. Since the function h is dened in pieces, we need to solve h(t) = 375 in pieces. For 0 t 20,h(t) = 5t2+ 100t,soforthesevaluesoft,wesolve 5t2+ 100t = 375. Rearrangingterms,weget5t2100t + 375 = 0,andfactoringgives5(t 5)(t 15) = 0. Ouranswersaret = 5andt = 15,andsincebothofthesevaluesoftliebetween0and20,wekeepbothsolutions.Fort>20, h(t)=0, andinthiscase, therearenosolutionsto0=375. Intermsof themodelrocket,solvingh(t) = 375correspondstondingwhen,ifever,therocketreaches375feetabovetheground. Ourtwoanswers,t = 5andt = 15correspondtotherocketreachingthisaltitudetwiceonce5secondsafterlaunch,andagain15secondsafterlaunch.1212Whatgoesup. . .62 RelationsandFunctionsThe type of function in the previous example is called a piecewise-dened function, or piecewisefunctionforshort. Manyreal-worldphenomena(e.g. postal rates,13incometaxformulas14)aremodeledbysuchfunctions.By the way,if we wanted to avoid using a piecewise function in Example 1.4.4,we could have usedh(t) = 5t2+100t on the explicit domain 0 t 20 because after 20 seconds, the rocket is on thegroundandstopsmoving. Inmanycases,though,piecewisefunctionsareyouronlychoice,soitsbesttounderstandthemwell.Mathematical modeling is not a one-section topic. Its not even a one-coursetopic as is evidenced byundergraduateandgraduatecoursesinmathematicalmodelingbeingoeredatmanyuniversities.Thus our goal in this section cannot possibly be to tell you the whole story. What we can do is getyoustarted. Aswestudynewclassesoffunctions,wewillseewhatphenomenatheycanbeusedtomodel. Inthatrespect, mathematical modelingcannotbeatopicinabook, butrather, mustbeathemeof thebook. Fornow, wehaveyouexploresomeverybasicmodelsintheExercisesbecauseyouneedtocrawltowalktorun. Aswelearnmoreaboutfunctions,wellhelpyoubuildyourownmodelsandgetyouonyourwaytoapplyingMathematicstoyourworld.13SeetheUnitedS