College Algebra and Trigonometry Paper

782 Pages·2011·12.3 MB·English

by  Sheldon Axler

This page intentionally left blank
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Algebra &
Trigonometry
with
Student Solutions Manual
Sheldon Axler
San Francisco State University
JOHN WILEY & SONS, INC.
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About the Author
Sheldon AxlerisDeanof
theCollegeofScience&En-
gineeringatSanFrancisco
State University, where he
joinedthefacultyasChairoftheMathematicsDepartmentin1997.
AxlerwasvaledictorianofhishighschoolinMiami,Florida. Hereceived
hisABfromPrincetonUniversitywithhighesthonors,followedbyaPhDin
MathematicsfromtheUniversityofCaliforniaatBerkeley.
AsapostdoctoralMooreInstructoratMIT,Axlerreceivedauniversity-wide
teachingaward. Axlerwasthenanassistantprofessor,associateprofessor,
and professor at Michigan State University, where he received the first J.
SutherlandFrameTeachingAwardandtheDistinguishedFacultyAward.
AxlerreceivedtheLesterR.FordAwardforexpositorywritingfromthe
Mathematical Association of America in 1996. In addition to publishing
numerousresearchpapers,AxleristheauthorofLinearAlgebraDoneRight
(whichhasbeenadoptedasatextbookatover240universitiesandcolleges),
CollegeAlgebra,andPrecalculus: APreludetoCalculus;heisalsoco-author
ofHarmonicFunctionTheory (agraduate/research-levelbook).
AxlerhasservedasEditor-in-ChiefoftheMathematicalIntelligencer andas
AssociateEditoroftheAmericanMathematicalMonthly. Hehasbeenamem-
beroftheCounciloftheAmericanMathematicalSocietyandamemberof
theBoardofTrusteesoftheMathematicalSciencesResearchInstitute. Axler
currently serves on the editorial board of Springer’s series Undergraduate
TextsinMathematics,GraduateTextsinMathematics,andUniversitext.
v
Contents
About the Author v
Preface to the Instructor xvi
WileyPLUS xxii
Acknowledgments xxiii
Preface to the Student xxvi
1 The Real Numbers 1
1.1 The Real Line 2
ConstructionoftheRealLine 2
IsEveryRealNumberRational? 3
Problems 6
1.2 Algebra of the Real Numbers 7
CommutativityandAssociativity 7
TheOrderofAlgebraicOperations 8
TheDistributiveProperty 10
AdditiveInversesandSubtraction 11
MultiplicativeInversesandtheAlgebraofFractions 13
SymbolicCalculators 16
Exercises,Problems,andWorked-outSolutions 19
1.3 Inequalities, Intervals, and Absolute Value 24
PositiveandNegativeNumbers 24
LesserandGreater 25
Intervals 27
AbsoluteValue 30
Exercises,Problems,andWorked-outSolutions 33
Chapter Summary and Chapter Review Questions 40
vi
Contents vii
2 Combining Algebra and Geometry 41
2.1 The Coordinate Plane 42
Coordinates 42
GraphsofEquations 44
DistanceBetweenTwoPoints 46
Length,Perimeter,andCircumference 48
Exercises,Problems,andWorked-outSolutions 50
2.2 Lines 57
Slope 57
TheEquationofaLine 58
ParallelLines 61
PerpendicularLines 62
Midpoints 64
Exercises,Problems,andWorked-outSolutions 66
2.3 Quadratic Expressions and Conic Sections 75
CompletingtheSquare 75
TheQuadraticFormula 77
Circles 79
Ellipses 81
Parabolas 83
Hyperbolas 85
Exercises,Problems,andWorked-outSolutions 88
2.4 Area 98
Squares,Rectangles,andParallelograms 98
TrianglesandTrapezoids 99
Stretching 101
CirclesandEllipses 102
Exercises,Problems,andWorked-outSolutions 105
Chapter Summary and Chapter Review Questions 115
3 Functions and Their Graphs 117
3.1 Functions 118
DefinitionandExamples 118
TheGraphofaFunction 121
TheDomainofaFunction 124
TheRangeofaFunction 126
viii Contents
FunctionsviaTables 128
Exercises,Problems,andWorked-outSolutions 129
3.2 Function Transformations and Graphs 142
VerticalTransformations: Shifting,Stretching,andFlipping 142
HorizontalTransformations: Shifting,Stretching,Flipping 145
CombinationsofVerticalFunctionTransformations 149
EvenFunctions 152
OddFunctions 153
Exercises,Problems,andWorked-outSolutions 154
3.3 Composition of Functions 165
CombiningTwoFunctions 165
DefinitionofComposition 166
OrderMattersinComposition 169
DecomposingFunctions 170
ComposingMorethanTwoFunctions 171
FunctionTransformationsasCompositions 172
Exercises,Problems,andWorked-outSolutions 174
3.4 Inverse Functions 180
TheInverseProblem 180
One-to-oneFunctions 181
TheDefinitionofanInverseFunction 182
TheDomainandRangeofanInverseFunction 184
TheCompositionofaFunctionandItsInverse 185
CommentsAboutNotation 187
Exercises,Problems,andWorked-outSolutions 189
3.5 A Graphical Approach to Inverse Functions 197
TheGraphofanInverseFunction 197
GraphicalInterpretationofOne-to-One 199
IncreasingandDecreasingFunctions 200
InverseFunctionsviaTables 203
Exercises,Problems,andWorked-outSolutions 204
Chapter Summary and Chapter Review Questions 209
4 Polynomial and Rational Functions 213
4.1 Integer Exponents 214
PositiveIntegerExponents 214