GRAPHS OF PARENT FUNCTIONS

Linear Function Absolute Value Function Square Root Function

Domain: Domain: Domain:Range: Range: Range:x-intercept: Intercept: Intercept:y-intercept: Decreasing on Increasing on Increasing when Increasing on Decreasing when Even function

y-axis symmetry

Greatest Integer Function Quadratic (Squaring) Function Cubic Function

Domain: Domain: Domain:Range: the set of integers Range : Range:x-intercepts: in the interval Range : Intercept:y-intercept: Intercept: Increasing on Constant between each pair of Decreasing on for Odd function

consecutive integers Increasing on for Origin symmetryJumps vertically one unit at Increasing on for

each integer value Decreasing on for Even functiony-axis symmetryRelative minimum

relative maximum or vertex: �0, 0�

�a < 0�,�a > 0�,

a < 0�0, ��a < 0���, 0�

a > 0�0, ��a > 0���, 0�

���, ���0, 0��0, 0��0, 0����, 0��a < 0��0, 1�

���, ���0, ���a > 0����, �����, �����, ��

x

y

(0, 0)

f(x) = x3

−2−3 1 2 3

−2

−1

−3

2

3

x

y

−1−2 1 2 3 4

1

−1

−2

−3

2

3

f(x) = ax , a > 02

f(x) = ax , a < 02

x

y

1−1−2−3 2 3

−3

1

2

3

f(x) = x[[ ]]

f�x� � x3f�x� � ax2f�x� � �x�

m < 0�0, ��m > 0

�0, �����, 0��0, b��0, 0��0, 0���b�m, 0�

�0, ���0, �����, ���0, �����, �����, ��

x

y

−1 2 3 4−1

1

2

3

4

(0, 0)

f(x) = x

x

y

−1−2 2

−1

−2

1

2

(0, 0)

f(x) = ⏐x⏐x

y

(0, b)

bm( (− , 0 b

m( (− , 0

f(x) = mx + b,m > 0

f(x) = mx + b,m < 0

f�x� � xf �x� � x � �x,�x,

x � 0x < 0

f �x� � mx � b

Rational (Reciprocal) Function Exponential Function Logarithmic Function

Domain: Domain: Domain:Range: Range: Range:No intercepts Intercept: Intercept:Decreasing on and Increasing on Increasing on Odd function for Vertical asymptote: y-axisOrigin symmetry Decreasing on ContinuousVertical asymptote: y-axis for Reflection of graph of Horizontal asymptote: x-axis Horizontal asymptote: x-axis in the line

Continuous

Sine Function Cosine Function Tangent Function

Domain: all

Range:Period:x-intercepts:y-intercept:Vertical asymptotes:

Odd functionOrigin symmetry

x ��

2� n�

�0, 0��n�, 0�

����, ��

x ��

2� n�

2

1

3

π π2

π2

−

f(x) = tan x

x

y

3π2

−2

−3

2

3

π π π2

π π2

− −

f(x) = cos x

x2

y

1

−2

−3

2

3

π π π2

π−

f(x) = sin x

x2

y

f�x� � tan xf�x� � cos xf�x� � sin x

y � xf�x� � axf �x� � a�x

���, ��f �x� � ax

�0, �����, ���0, �����, 0��1, 0��0, 1�

���, ���0, �����, 0� � �0, �)�0, �����, �����, 0� � �0, �)

x

y

f(x) = loga x

1 2

−1

1

(1, 0)

x

y

(0, 1)f(x) = a−xf(x) = ax

x

y

f(x) =1x

−1 1 2 3

1

2

3

f�x� � loga x, a > 0, a � 1f�x� � ax, a > 0, a � 1f�x� �1x

Domain:Range:Period:x-intercepts:y-intercept:Odd functionOrigin symmetry

�0, 0��n�, 0�

2���1, 1�

���, �� Domain:Range:Period:

x-intercepts:

y-intercept:Even functiony-axis symmetry

�0, 1���

2� n�, 0

2���1, 1�

���, ��

Cosecant Function Secant Function Cotangent Function

Domain: all Range:Period:No interceptsVertical asymptotes:Odd functionOrigin symmetry

Inverse Sine Function Inverse Cosine Function Inverse Tangent Function

Domain:

Range:

Intercept:Odd functionOrigin symmetry

�0, 0���

�

2,

�

2���1, 1�

x

y

−1−2 1 2

f(x) = arctan x

2π−

2π

x

y

−1 1

f(x) = arccos x

π

x

y

−1 1

2

2

π

π−

f(x) = arcsin x

f�x� � arctan xf�x� � arccos xf�x� � arcsin x

x � n�

2����, �1� � �1, ��

x � n�

2

1

3

π ππ2

π2

− −

f(x) = cot x =

x

y

2π

1tan x

2

−2

−3

3

π ππ2

π2

−−x

y

3π 2π2

f(x) = sec x = 1

cos x

2

1

3

ππ π2

−

f(x) = csc x =

x

y

2π

1sin x

f�x� � cot xf�x� � sec xf�x� � csc x

Domain: all

Range:Period:y-intercept:Vertical asymptotes:

Even functiony-axis symmetry

x ��

2� n�

�0, 1�2����, �1� � �1, ��

x ��

2� n� Domain: all

Range:Period:

x-intercepts:

Vertical asymptotes:Odd functionOrigin symmetry

x � n�

��

2� n�, 0

����, ��

x � n�

Domain:Range:

y-intercept: �0,�

2 �0, ��

��1, 1� Domain:

Range:

Intercept:Horizontal asymptotes:

Odd functionOrigin symmetry

y � ±�

2

�0, 0���

�

2,

�

2 ���, ��

Precalculus with LimitsSecond Edition

Ron LarsonThe Pennsylvania State UniversityThe Behrend College

With the assistance of

David C. FalvoThe Pennsylvania State UniversityThe Behrend College

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

Precalculus with Limits, Second EditionRon Larson

Publisher: Charlie VanWagner

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© 2011, 2007 Brooks/Cole, Cengage Learning

ALL RIGHTS RESERVED. No part of this work covered by the copyrightherein may be reproduced, transmitted, stored, or used in any form or byany means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution,information networks, or information storage and retrieval systems,except as permitted under Section 107 or 108 of the 1976 United StatesCopyright Act, without the prior written permission of the publisher.

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Printed in the United States of America1 2 3 4 5 6 7 13 12 11 10 09

Library of Congress Control Number: 2009930251

Student Edition:

ISBN-13: 978-1-4390-4909-9

ISBN-10: 1-4390-4909-2

iii

A Word from the Author (Preface) vii

Functions and Their Graphs 1

1.1 Rectangular Coordinates 21.2 Graphs of Equations 131.3 Linear Equations in Two Variables 241.4 Functions 391.5 Analyzing Graphs of Functions 541.6 A Library of Parent Functions 661.7 Transformations of Functions 731.8 Combinations of Functions: Composite Functions 831.9 Inverse Functions 921.10 Mathematical Modeling and Variation 102Chapter Summary 114 Review Exercises 116Chapter Test 121 Proofs in Mathematics 122Problem Solving 123

Polynomial and Rational Functions 125

2.1 Quadratic Functions and Models 1262.2 Polynomial Functions of Higher Degree 1362.3 Polynomial and Synthetic Division 1502.4 Complex Numbers 1592.5 Zeros of Polynomial Functions 1662.6 Rational Functions 1812.7 Nonlinear Inequalities 194Chapter Summary 204 Review Exercises 206Chapter Test 210 Proofs in Mathematics 211Problem Solving 213

Exponential and Logarithmic Functions 215

3.1 Exponential Functions and Their Graphs 2163.2 Logarithmic Functions and Their Graphs 2273.3 Properties of Logarithms 2373.4 Exponential and Logarithmic Equations 2443.5 Exponential and Logarithmic Models 255Chapter Summary 268 Review Exercises 270Chapter Test 273 Cumulative Test for Chapters 1–3 274Proofs in Mathematics 276 Problem Solving 277

Contents

chapter 1

chapter 2

chapter 3

iv Contents

Trigonometry 279

4.1 Radian and Degree Measure 2804.2 Trigonometric Functions: The Unit Circle 2924.3 Right Triangle Trigonometry 2994.4 Trigonometric Functions of Any Angle 3104.5 Graphs of Sine and Cosine Functions 3194.6 Graphs of Other Trigonometric Functions 3304.7 Inverse Trigonometric Functions 3414.8 Applications and Models 351Chapter Summary 362 Review Exercises 364Chapter Test 367 Proofs in Mathematics 368Problem Solving 369

Analytic Trigonometry 371

5.1 Using Fundamental Identities 3725.2 Verifying Trigonometric Identities 3805.3 Solving Trigonometric Equations 3875.4 Sum and Difference Formulas 3985.5 Multiple-Angle and Product-to-Sum Formulas 405Chapter Summary 416 Review Exercises 418Chapter Test 421 Proofs in Mathematics 422Problem Solving 425

Additional Topics in Trigonometry 427

6.1 Law of Sines 4286.2 Law of Cosines 4376.3 Vectors in the Plane 4456.4 Vectors and Dot Products 4586.5 Trigonometric Form of a Complex Number 468Chapter Summary 478 Review Exercises 480Chapter Test 484 Cumulative Test for Chapters 4–6 485Proofs in Mathematics 487 Problem Solving 491

Systems of Equations and Inequalities 493

7.1 Linear and Nonlinear Systems of Equations 4947.2 Two-Variable Linear Systems 5057.3 Multivariable Linear Systems 5177.4 Partial Fractions 5307.5 Systems of Inequalities 538

chapter 4

chapter 5

chapter 6

chapter 7

Contents v

7.6 Linear Programming 549Chapter Summary 558 Review Exercises 560Chapter Test 565 Proofs in Mathematics 566Problem Solving 567

Matrices and Determinants 569

8.1 Matrices and Systems of Equations 5708.2 Operations with Matrices 5848.3 The Inverse of a Square Matrix 5998.4 The Determinant of a Square Matrix 6088.5 Applications of Matrices and Determinants 616Chapter Summary 628 Review Exercises 630Chapter Test 635 Proofs in Mathematics 636Problem Solving 637

Sequences, Series, and Probability 639

9.1 Sequences and Series 6409.2 Arithmetic Sequences and Partial Sums 6519.3 Geometric Sequences and Series 6619.4 Mathematical Induction 6719.5 The Binomial Theorem 6819.6 Counting Principles 6899.7 Probability 699Chapter Summary 712 Review Exercises 714Chapter Test 717 Cumulative Test for Chapters 7–9 718Proofs in Mathematics 720 Problem Solving 723

Topics in Analytic Geometry 725

10.1 Lines 72610.2 Introduction to Conics: Parabolas 73310.3 Ellipses 74210.4 Hyperbolas 75110.5 Rotation of Conics 76110.6 Parametric Equations 76910.7 Polar Coordinates 77710.8 Graphs of Polar Equations 78310.9 Polar Equations of Conics 791Chapter Summary 798 Review Exercises 800Chapter Test 803 Proofs in Mathematics 804Problem Solving 807

chapter 8

chapter 9

chapter 10

vi Contents

chapter 11

chapter 12

Analytic Geometry in Three Dimensions 809

11.1 The Three-Dimensional Coordinate System 81011.2 Vectors in Space 81711.3 The Cross Product of Two Vectors 82411.4 Lines and Planes in Space 831Chapter Summary 840 Review Exercises 842Chapter Test 844 Proofs in Mathematics 845Problem Solving 847

Limits and an Introduction to Calculus 849

12.1 Introduction to Limits 85012.2 Techniques for Evaluating Limits 86112.3 The Tangent Line Problem 87112.4 Limits at Infinity and Limits of Sequences 88112.5 The Area Problem 890Chapter Summary 898 Review Exercises 900Chapter Test 903 Cumulative Test for Chapters 10–12 904Proofs in Mathematics 906 Problem Solving 907

Appendix A Review of Fundamental Concepts of Algebra A1

A.1 Real Numbers and Their Properties A1A.2 Exponents and Radicals A14A.3 Polynomials and Factoring A27A.4 Rational Expressions A39A.5 Solving Equations A49A.6 Linear Inequalities in One Variable A63A.7 Errors and the Algebra of Calculus A73

Answers to Odd-Numbered Exercises and Tests A81

Index A211

Index of Applications (web)

Appendix B Concepts in Statistics (web)

B.1 Representing DataB.2 Measures of Central Tendency and DispersionB.3 Least Squares Regression

vii

In this edition, we continue to offer instructors and students a text that is pedagogicallysound, mathematically precise, and still comprehensible. There are many changes in themathematics, art, and design; the more significant changes are noted here.

• New Chapter Openers Each Chapter Opener has three parts, In Mathematics, InReal Life, and In Careers. In Mathematics describes an important mathematicaltopic taught in the chapter. In Real Life tells students where they will encounter thistopic in real-life situations. In Careers relates application exercises to a variety ofcareers.

• New Study Tips and Warning/Cautions Insightful information is given to students in two new features. The Study Tip provides students with useful information or suggestions for learning the topic. The Warning/Caution points outcommon mathematical errors made by students.

• New Algebra Helps Algebra Help directs students to sections of the textbookwhere they can review algebra skills needed to master the current topic.

• New Side-by-Side Examples Throughout the text, we present solutions to manyexamples from multiple perspectives—algebraically, graphically, and numerically.The side-by-side format of this pedagogical feature helps students to see that a problemcan be solved in more than one way and to see that different methods yield the sameresult. The side-by-side format also addresses many different learning styles.

A Word fromthe Author

Welcome to the Second Edition of Precalculus with Limits! We are proud to offer youa new and revised version of our textbook. With the Second Edition, we have listenedto you, our users, and have incorporated many of your suggestions for improvement.

1st Edition2nd Edition

• New Capstone Exercises Capstones are conceptual problems that synthesize keytopics and provide students with a better understanding of each section’s concepts. Capstone exercises are excellent for classroom discussion or test prep, andteachers may find value in integrating these problems into their reviews of the section.

• New Chapter Summaries The Chapter Summary now includes an explanationand/or example of each objective taught in the chapter.

• Revised Exercise Sets The exercise sets have been carefully and extensivelyexamined to ensure they are rigorous and cover all topics suggested by our users.Many new skill-building and challenging exercises have been added.

For the past several years, we’ve maintained an independent website—CalcChat.com—that provides free solutions to all odd-numbered exercises in the text.Thousands of students using our textbooks have visited the site for practice and helpwith their homework. For the Second Edition, we were able to use information fromCalcChat.com, including which solutions students accessed most often, to help guidethe revision of the exercises.

I hope you enjoy the Second Edition of Precalculus with Limits. As always, I welcome comments and suggestions for continued improvements.

viii A Word from the Author

ix

I would like to thank the many people who have helped me prepare the text and the supplements package. Their encouragement, criticisms, and suggestions have beeninvaluable.

Thank you to all of the instructors who took the time to review the changes in thisedition and to provide suggestions for improving it. Without your help, this book wouldnot be possible.

Reviewers

Chad Pierson, University of Minnesota-Duluth; Sally Shao, Cleveland State University;Ed Stumpf, Central Carolina Community College; Fuzhen Zhang, Nova SoutheasternUniversity; Dennis Shepherd, University of Colorado, Denver; Rhonda Kilgo,Jacksonville State University; C. Altay Özgener, Manatee Community CollegeBradenton; William Forrest, Baton Rouge Community College; Tracy Cook, Universityof Tennessee Knoxville; Charles Hale, California State Poly University Pomona; SamuelEvers, University of Alabama; Seongchun Kwon, University of Toledo; Dr. Arun K.Agarwal, Grambling State University; Hyounkyun Oh, Savannah State University;Michael J. McConnell, Clarion University; Martha Chalhoub, Collin CountyCommunity College; Angela Lee Everett, Chattanooga State Tech Community College;Heather Van Dyke, Walla Walla Community College; Gregory Buthusiem, BurlingtonCounty Community College; Ward Shaffer, College of Coastal Georgia; CarmenThomas, Chatham University; Emily J. Keaton

My thanks to David Falvo, The Behrend College, The Pennsylvania StateUniversity, for his contributions to this project. My thanks also to Robert Hostetler, TheBehrend College, The Pennsylvania State University, and Bruce Edwards, University ofFlorida, for their significant contributions to previous editions of this text.

I would also like to thank the staff at Larson Texts, Inc. who assisted with proof-reading the manuscript, preparing and proofreading the art package, and checking andtypesetting the supplements.

On a personal level, I am grateful to my spouse, Deanna Gilbert Larson, for herlove, patience, and support. Also, a special thanks goes to R. Scott O’Neil. If you have suggestions for improving this text, please feel free to write to me. Over the past twodecades I have received many useful comments from both instructors and students, andI value these comments very highly.

Ron Larson

Acknowledgments

x

Supplements

Supplements for the Instructor

Annotated Instructor’s Edition This AIE is the complete student text plus point-of-use annotations for the instructor, including extra projects, classroom activities, teachingstrategies, and additional examples. Answers to even-numbered text exercises,Vocabulary Checks, and Explorations are also provided.

Complete Solutions Manual This manual contains solutions to all exercises from thetext, including Chapter Review Exercises and Chapter Tests.

Instructor’s Companion Website This free companion website contains an abundanceof instructor resources.

PowerLecture™ with ExamView® The CD-ROM provides the instructor with dynamicmedia tools for teaching college algebra. PowerPoint® lecture slides and art slides ofthe figures from the text, together with electronic files for the test bank and a link to theSolution Builder, are available. The algorithmic ExamView allows you to create, deliver,and customize tests (both print and online) in minutes with this easy-to-use assessmentsystem. Enhance how your students interact with you, your lecture, and each other.

Solutions Builder This is an electronic version of the complete solutions manualavailable via the PowerLecture and Instructor’s Companion Website. It provides instructors with an efficient method for creating solution sets to homework or examsthat can then be printed or posted.

Online AIE to the Note Taking Guide This AIE includes the answers to all problemsin the innovative Note Taking Guide.

Supplements xi

Supplements for the Student

Student Companion Website This free companion website contains an abundance ofstudent resources.

Instructional DVDs Keyed to the text by section, these DVDs provide comprehensivecoverage of the course—along with additional explanations of concepts, sample problems, and applications—to help students review essential topics.

Student Study and Solutions Manual This guide offers step-by-step solutions for allodd-numbered text exercises, Chapter and Cumulative Tests, and Practice Tests withsolutions.

Premium eBook The Premium eBook offers an interactive version of the textbookwith search features, highlighting and note-making tools, and direct links to videos ortutorials that elaborate on the text discussions.

Enhanced WebAssign Enhanced WebAssign is designed for you to do your homework online. This proven and reliable system uses pedagogy and content found inLarson’s text, and then enhances it to help you learn Precalculus more effectively.Automatically graded homework allows you to focus on your learning and get interactive study assistance outside of class.

Note Taking Guide This is an innovative study aid, in the form of a notebook organizer, that helps students develop a section-by-section summary of key concepts.

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