Essentials of college algebra

Preface xiii vii Sets of Numbers ■ Order of Operations ■ Scientific Notation ■ Problem Solving One-Variable Data ■ Two-Variable Data ■ The Distance Formula ■ TheMidpoint Formula ■ Circle

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How many songs will my iPod hold?

The number of songs that will fit on your iPod depends on

the size of its memory We can use the concept of slope to

analyze memory requirements for storing music on iPods.

(See Example 2 from Section 1.4 on page 49 to learn more.)

Determining sunset times

Whether we are traveling cross-country, driving a boat on a lake, or designing a solar power plant, the time of sunset can be important If we know the sunset time on two different days, can we make predictions about sunset times on other days? Using a linear function, we can often make accurate estimates (See Example 7 from Section 2.4 on page 139.)

Going green

Many activities, such as driving a car, watching television, riding a jet ski, or flying in an airplane, emit carbon dioxide into the air A commercial airliner, for example, emits 150 pounds of carbon dioxide for each passenger who flies 240 miles Understanding our environmental impact on the planet is becoming increasingly important Functions can be used to model and predict carbon emissions (See the Chapter 2 opener on page 76 and Example 4 from Section 2.1 on page 81.)

Gary Rockswold teaches algebra in context, answering the question,

“Why am I learning this?”

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Mathematics can be used to improve an athlete’s performance Learn how parabolas, the angle of release, and the velocity of the basketball all play a role

in determining whether the ball goes through the hoop (See the Chapter 3 opener on page 169 to learn more.)

Modeling movement of weather

How do meteorologists know where a cold front will

be tomorrow? How do they know that one city will be

hit by a blizzard, and a city 100 miles away will get

only flurries? Scientists model weather systems and

make predictions by translating and transforming

graphs on a weather map (Learn more in Section 3.5

on pages 222 and 238.)

Waiting in line

Have you ever noticed how a slight increase in the number of cars exiting a parking garage or trying to get through a construction zone can make your wait much longer? These long lines of waiting cars are subject to a nonlinear effect To make predictions about traffic congestion, highway engineers often use rational functions (See the application and Example 2

in Section 4.6 on page 309 to learn more.)

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Boston Columbus Indianapolis New York San Francisco Upper Saddle River

Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto

Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo

Gary K Rockswold

Minnesota State University, Mankato

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Library of Congress Cataloging-in-Publication Data

Rockswold, Gary K.

Essentials of college algebra with modeling and visualization / Gary K Rockswold 4th ed.

p cm.

ISBN-13: 978-0-321-71528-9 (student edition)

ISBN-10: 0-321-71528-4 (student edition)

1 Algebra Textbooks 2 Algebraic functions Textbooks 3.

Mathematical models Textbooks I Title

QA154.3.R635 2012

512.9 dc22

2010040485 Copyright © 2012, 2008, 2006, 2002 Pearson Education, Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher Printed in the United States of America For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request to 617-671-3447, or e-mail at http://www.pearsoned.com/legal/permissions.htm.

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ISBN-10: 0-321-71528-4 ISBN-13: 978-0-321-71528-9

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Larry Pearson, who is deeply missed by many

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Preface xiii

vii

Sets of Numbers ■ Order of Operations ■ Scientific Notation ■

Problem Solving

One-Variable Data ■ Two-Variable Data ■ The Distance Formula ■ TheMidpoint Formula ■ Circles ■ Graphing with a Calculator (Optional)

Checking Basic Concepts for Sections 1.1 and 1.2 28

Basic Concepts ■ Representations of Functions ■ Formal Definition of aFunction ■ Graphing Calculators and Functions (Optional) ■ IdentifyingFunctions ■ Functions Represented by Diagrams and Equations

Constant Functions ■ Linear Functions ■ Slope as a Rate of Change ■

Nonlinear Functions

Increasing and Decreasing Functions ■ Interval Notation ■ Average Rate ofChange ■ The Difference Quotient

Checking Basic Concepts for Section 1.5 67

Chapter 1 Review Exercises 71 Chapter 1 Extended and Discovery Exercises 74

Functions as Models ■ Representations of Linear Functions ■ Modelingwith Linear Functions ■ Piecewise-Defined Functions ■ The GreatestInteger Function ■ Linear Regression (Optional)

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2.2 Equations of Lines 96

Forms for Equations of Lines ■ Finding Intercepts ■ Horizontal, Vertical,Parallel, and Perpendicular Lines ■ Modeling Data (Optional) ■ DirectVariation

Equations ■ Symbolic Solutions ■ Graphical and Numerical Solutions ■

Solving for a Variable ■ Problem-Solving Strategies

Inequalities ■ Compound Inequalities

The Absolute Value Function ■ Absolute Value Equations ■ Absolute Value Inequalities

Chapter 2 Review Exercises 161 Chapter 2 Extended and Discovery Exercises 164 Chapters 1–2 Cumulative Review Exercises 165

Basic Concepts ■ Completing the Square and the VertexFormula ■ Applications and Models ■ Quadratic Regression (Optional)

Quadratic Equations ■ Factoring ■ The Square Root Property ■

Completing the Square ■ The Quadratic Formula ■ The Discriminant ■

Problem Solving

Basic Concepts ■ Arithmetic Operations on Complex Numbers ■

Quadratic Equations with Complex Solutions

Basic Concepts ■ Graphical and Numerical Solutions ■ Symbolic Solutions

Vertical and Horizontal Shifts ■ Stretching and Shrinking ■ Reflection ofGraphs ■ Combining Transformations ■ Modeling with Transformations(Optional)

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Chapter 3 Review Exercises 244 Chapter 3 Extended and Discovery Exercises 246

Polynomial Functions ■ Identifying Extrema ■ Symmetry

Graphs of Polynomial Functions ■ Piecewise-Defined Polynomial Functions ■ Polynomial Regression (Optional)

Division by Monomials ■ Division by Polynomials ■ Synthetic Division

Factoring Polynomials ■ Graphs and Multiple Zeros ■ Rational Zeros (Optional) ■ Descartes’ Rule of Signs (Optional) ■ PolynomialEquations ■ Intermediate Value Property (Optional)

Fundamental Theorem of Algebra ■ Polynomial Equations with ComplexSolutions

Rational Functions ■ Vertical Asymptotes ■ Horizontal Asymptotes ■

Identifying Asymptotes ■ Graphs and Transformations of Rational Functions

■ Graphing Rational Functions by Hand (Optional)

Rational Equations ■ Variation ■ Polynomial Inequalities ■ RationalInequalities

Rational Exponents and Radical Notation ■ Equations Involving Radicals ■

Power Functions and Models ■ Equations Involving Rational Exponents ■

Power Regression (Optional)

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6 Systems of Equations and Inequalities 478

Functions of Two Variables ■ Systems of Equations in Two Variables ■ Types

of Linear Systems in Two Variables ■ The Method of Substitution ■ TheElimination Method ■ Graphical and Numerical Methods ■ Joint Variation

Systems of Linear and Nonlinear Inequalities ■ Linear Programming

Arithmetic Operations on Functions ■ Review of Function Notation ■

Composition of Functions

Inverse Operations and Inverse Functions ■ One-to-One Functions ■

Symbolic Representations of Inverse Functions ■ Other Representations ofInverse Functions

Linear and Exponential Growth ■ Exponential Growth and Decay ■

Compound Interest ■ The Natural Exponential Function ■ Exponential Models

The Common Logarithmic Function ■ Logarithms with Other Bases ■ BasicEquations ■ General Equations

Basic Properties of Logarithms ■ Expanding and Combining LogarithmicExpressions ■ Change of Base Formula

Exponential Equations ■ Logarithmic Equations

Exponential Model ■ Logarithmic Model ■ Logistic Model ■

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6.3 Systems of Linear Equations in Three Variables 511

Basic Concepts ■ Solving with Elimination and Substitution ■ Systems with

No Solutions ■ Systems with Infinitely Many Solutions

Representing Systems of Linear Equations with Matrices ■ Row-EchelonForm ■ Gaussian Elimination ■ Solving Systems of Linear Equations withTechnology (Optional)

Matrix Notation ■ Sums, Differences, and Scalar Multiples of Matrices ■

Matrix Products ■ Technology and Matrices (Optional)

Understanding Matrix Inverses ■ The Identity Matrix ■ Matrix Inverses ■

Finding Inverses Symbolically ■ Representing Linear Systems with MatrixEquations ■ Solving Linear Systems with Inverses

Definition and Calculation of Determinants ■ Cramer’s Rule ■ Area of Regions

Geometric Shapes in a Plane ■ The Pythagorean Theorem ■

Three-Dimensional Objects ■ Similar Triangles

Common Factors ■ Factoring by Grouping ■ Factoring ■

Factoring Trinomials by Grouping ■ Factoring Trinomials with FOIL ■

Difference of Two Squares ■ Perfect Square Trinomials ■ Sum andDifference of Two Cubes

x2 + bx + c

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R.5 Rational Expressions R-30

Simplifying Rational Expressions ■ Review of Multiplication and Division ofFractions ■ Multiplication and Division of Rational Expressions ■ LeastCommon Multiples and Denominators ■ Review of Addition andSubtraction of Fractions ■ Addition and Subtraction of Rational Expressions ■ Clearing Fractions ■ Complex Fractions

Radical Notation ■ Rational Exponents ■ Properties of Rational Exponents

Product Rule for Radical Expressions ■ Quotient Rule for RadicalExpressions ■ Addition and Subtraction ■ Multiplication ■ Rationalizingthe Denominator

Appendix A: Using the Graphing Calculator AP-1 Appendix B: A Library of Functions AP-19 Appendix C: Partial Fractions AP-22 Bibliography B-1

Answers to Selected Exercises A-1 Photo Credits P-1

Index of Applications I-1 Index I-5

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Essentials of College Algebra with Modeling & Visualization, Fourth Edition, offers an

innovative approach that consistently links mathematical concepts to real-world tions by moving from the concrete to the abstract It demonstrates the relevance of mathe-matics and answers the question, “When will I ever need to know this?” This text suc-cinctly provides a comprehensive curriculum with the balance and flexibility necessary fortoday’s college mathematics courses The early introduction of functions and graphs allowsthe instructor to use applications and visualization to present mathematical topics Realdata, graphs, and tables play an important role in the course, giving meaning to the num-bers and equations that students encounter This approach increases students’ interest andmotivation and their likelihood of success This streamlined text covers linear, quadratic,nonlinear, exponential, and logarithmic functions and systems of equations and inequali-ties, to get to the heart of what students need from this course

applica-The vast majority of students who study mathematics in college will not become fessional mathematicians Because the importance of mathematics and its applications isaccelerating rapidly, however, essentially all of these students will use mathematics duringtheir lifetimes to a much greater extent than they might anticipate Mathematics courses mustprepare students with a variety of skills and the understanding needed to be productive andinformed members of society

pro-Approach

Instructors are free to strike their own balance of skills, rule of four, applications, modeling,and technology With a flexible approach to the rule of four (verbal, graphical, numerical, andsymbolic methods), instructors can easily emphasize one rule more than another to meet theirstudents’ needs This approach also extends to modeling and applications The use of tech-

nology, which helps students visualize mathematical concepts, is optional and not a

require-ment for students to benefit from this approach Nevertheless, the text still provides a strongoption for instructors who wish to implement graphing calculator technology The textcontains numerous applications, including models of real-world data with functions andproblem-solving strategies It is not necessary for an instructor to discuss any particularapplication; rather, an instructor has the option to choose from a wide variety of topics.The concept of a function is the unifying theme of the text It is common for students

to examine a type of function and its graph and then use this knowledge to solve ated equations and inequalities For example, students apply their knowledge of quadraticfunctions and parabolas to solve quadratic equations and inequalities This visual approachcomplements the traditional symbolic approach to solving equations and inequalities andallows students to solve a problem by using more than one method Functions and theirgraphs are frequently used to solve applications and model real-world data, and studentsare often asked to interpret their results Mathematical skills also play an important role inthis text Numerous exercises have been included so that students can practice their skills.When students arrive in a college-level mathematics class, they often lack a full mas-tery of intermediate algebra Rather than reviewing all of the necessary intermediate alge-bra skills in the first chapter in hopes that “saying it louder and faster” will help studentsremember it better, this book integrates them seamlessly throughout the early chapters Inaddition, review notes and geometry notes appearing in the margins refer students “just intime” to extra help found in Chapter R: Basic Concepts from Algebra and Geometry.Preface

associ-xiii

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Content Changes to the Fourth Edition

The fourth edition contains several important changes, which are the result of the manycomments and suggestions made by instructors, students, and reviewers This text containssix chapters plus a review chapter at the end of the text The breadth and depth of severaltopics have been increased Some highlights of this revision include the following:

■Hundreds of examples and exercises have been added and revised to meet the needs

■Arrows have been added to graphs whenever appropriate

■Calculator graphs have a new, smooth look, and equation labels have been added tomake these graphs easier to read

■Greater use has been made of headings and subheadings to add clarity to the text

■Each exercise set has been carefully revised to ensure that there are sufficient cises of various types for each example and mathematical concept Exercise sets arecarefully graded with several levels of difficulty

exer-■Chapter 1 has been expanded from four sections to five Increasing and decreasingfunctions, average rates of change, and difference quotients are now discussed inSection 1.5 Circles now appear in Section 1.2 instead of Chapter R

■In Chapter 2, piecewise-linear functions are now discussed earlier, in Section 2.1

■Complex numbers have been moved to Chapter 3 as a new section

■Chapter 4 has been expanded and reorganized from seven to eight sections, making iteasier to cover one section per class Division of polynomials and real zeros of poly-nomial functions are now in separate sections Coverage of several topics, such asrational functions, has been revised and enhanced

■In Chapter 5, the change of base formula is now presented in Section 5.4 and tional modeling has been included in Section 5.7

addi-■The first two sections in Chapter 6 have been reorganized so that systems of equationsare discussed in Section 6.1 and systems of inequalities and linear programming arediscussed in Section 6.2 Substitution and elimination are both presented in Section 6.1

■In Chapter R, additional problems involving factoring and rational expressions havebeen included

■Appendix C: Partial Fractions is a new appendix, included for students who need thistopic for calculus

■The following is a partial list of important topics on which discussion has been added

or enhanced:

Rational functions and expressions Graphing rational functions by hand

Interpolation and extrapolation

■MathXL®coverage has been significantly increased from the third edition, addingover 700 new exercises “Why Math Matters” animations, guided-discovery projects,and interactive discovery exercises have been added to this course

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■Chapter and Section Introductions

Many college algebra students have little or no understanding of mathematics beyondbasic computation To motivate students, chapter and section introductions explainsome of the reasons for studying mathematics (See pages 1, 76, 146, and 249.)

■Now Try

This feature occurs after each example It suggests a similar exercise students can work tosee if they understand the concept presented in the example (See pages 5, 115, and 189.)

■Algebra and Geometry Review Notes

Throughout the text, Algebra and Geometry Review Notes, located in the margins,direct students “just in time” to Chapter R, where important topics in algebra andgeometry are reviewed Instructors can use this chapter for extra review or refer stu-

dents to it as needed This feature frees instructors from having to frequently review

material from intermediate algebra and geometry (See pages 124 and 174.)

■Calculator Help Notes

The Calculator Help Notes in the margins direct students “just in time” to AppendixA: Using the Graphing Calculator This appendix shows students the keystrokes nec-

essary to complete specific examples from the text This feature frees instructors from

having to teach the specifics of the graphing calculator and gives students a ient reference written specifically for this text (See pages 6, 22, and 119.)

pre-■Putting It All Together

This helpful feature at the end of each section summarizes techniques and reinforcesthe mathematical concepts presented in the section It is given in an easy-to-followgrid (See pages 125–126, 348–349, and 394–396.)

■Checking Basic Concepts

This feature, included after every two sections, provides a small set of exercises thatcan be used for review These exercises require about 15 or 20 minutes to completeand can be used for collaborative learning exercises if time permits (See pages 113,

204, and 276–277.)

■Exercise Sets

The exercise sets are the heart of any mathematics text, and this text includes a largevariety of instructive exercises Each set of exercises covers skill building, mathemat-ical concepts, and applications Graphical interpretation and tables of data are oftenused to extend students’ understanding of mathematical concepts The exercise setsare graded carefully and categorized according to topic, making it easy for an instruc-tor to select appropriate assignments (See pages 107–113 and 200–204.)

■Chapter Summaries

Chapter summaries are presented in an easy-to-read grid They allow students toquickly review key concepts from the chapter (See pages 240–243 and 354–358.)

■Chapter Review Exercises

This exercise set contains both skill-building and applied exercises These exercisesstress different techniques for solving problems and provide students with the reviewnecessary to pass a chapter test (See pages 71–74 and 358–361)

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■Extended and Discovery Exercises

Extended and Discovery Exercises occur at the end of selected sections and at the end

of every chapter These exercises are usually more complex and challenging than therest of the exercises and often require extension of a topic presented or exploration of

a new topic They can be used for either collaborative learning or extra homeworkassignments (See pages 74–75, 246–247, and 362.)

■Cumulative Review Exercises

These comprehensive exercise sets, which occur after every two chapters, give dents an opportunity to review previous material (See pages 165–168 and 363–366.)

stu-ANNOTATED INSTRUCTOR’S EDITION

■ A special edition of the text

■ Includes new Teaching Examples, an extra set of

examples for instructors to present in class, doubling

the number of examples available for instructors

■ Sample homework assignments, selected by the author,

are now indicated by a red underline in each

end-of-section exercise set and may be assigned in MathXL

and MyMathLab®

■ Includes new Teaching Tips, helpful ideas about

presenting topics or teaching from the text

■ Includes all of the answers to the exercise sets, usually

right on the page where the exercise appears

ISBN: 0-321-71564-0 / 978-0-321-71564-7

INSTRUCTOR’S SOLUTIONS MANUAL

■ By David Atwood and Terry Krieger, Rochester

Community and Technical College

■ Provides complete solutions to all text exercises,

excluding Writing about Mathematics

ISBN: 0-321-57697-7 / 978-0-321-57697-2

INSTRUCTOR’S TESTING MANUAL

(DOWNLOAD ONLY)

■ By Vincent McGarry, Austin Community College

■ Provides prepared tests for each chapter of the text, as

well as answers

■ Available for download at

www.pearsonhighered.com/irc

TESTGEN® (DOWNLOAD ONLY)

■ Enables instructors to build, edit, print, and administer

tests using a computerized bank of questions that cover

all the objectives of the text

Instructor Supplements

■ Using algorithmically based questions, allowsinstructors to create multiple but equivalent versions ofthe same question or test with the click of a button

■ Lets instructors modify test bank questions or add newquestions

■ Provides printable or online tests

■ Available for download from Pearson Education’sonline catalog (www.pearsonhighered.com/irc)

INSIDER’S GUIDE

■ Includes resources to help faculty with coursepreparation and classroom management

■ Provides helpful teaching tips correlated to the sections

of text, as well as general teaching adviceISBN: 0-321-57706-X / 978-0-321-57706-1

POWERPOINT PRESENTATION (DOWNLOAD ONLY)

■ Classroom presentation software correlated specifically

to this textbook sequence

■ Available for download within MyMathLab or at

www.pearsonhighered.com/irc

PEARSON MATH ADJUNCT SUPPORT CENTER

The Pearson Math Adjunct Support Center

(www.pearsontutorservices.com/math-adjunct.html)

is staffed by qualified instructors with more than 50 years

of combined experience at both the community collegeand the university level Assistance is provided for faculty

in the following areas:

■ Suggested syllabus consultation

■ Tips on using materials packed with your book

■ Book-specific content assistance

■ Teaching suggestions, including advice on classroomstrategies

NEW!

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Student Supplements

■ Affordable and portable for students

■ Ideal for distance learning or supplemental instruction

■ Include optional subtitles in English and SpanishISBN: 0-321-57703-5 / 978-0-321-57703-0

MathXL ® Online Course (access code required)

MathXL®is a powerful online homework, tutorial, and assessment system that nies Pearson Education’s textbooks in mathematics or statistics With MathXL, instructorscan:

accompa-■ Create, edit, and assign online homework and tests using algorithmically generatedexercises correlated at the objective level to the textbook

■ Create and assign their own online exercises and import TestGen tests for addedflexibility

■ Maintain records of all student work tracked in MathXL’s online gradebook

With MathXL, students can:

■ Take chapter tests in MathXL and receive personalized study plans and/or personalizedhomework assignments based on their test results

■ Use the study plan and/or the homework to link directly to tutorial exercises for theobjectives they need to study

■ Access supplemental animations and video clips directly from selected exercises

MathXL is available to qualified adopters For more information, visit our website at

www.mathxl.com, or contact your Pearson representative.

MyMathLab ® Online Course (access code required)

MyMathLab®is a series of text-specific, easily customizable online courses for PearsonEducation’s textbooks in mathematics and statistics MyMathLab gives you the tools youneed to deliver all or a portion of your course online, whether your students are in a lab

or working from home MyMathLab provides a rich and flexible set of course materials,featuring free-response exercises that are algorithmically generated for unlimited prac-tice and mastery Students can also use online tools, such as video lectures, animations,and a multimedia textbook, to independently improve their understanding and perform-ance Instructors can use MyMathLab’s homework and test managers to select and assign

STUDENT’S SOLUTIONS MANUAL

■ By David Atwood and Terry Krieger, Rochester

Community and Technical College

■ Provides complete solutions to all odd-numbered text

exercises, excluding Writing about Mathematics and

Extended and Discovery Exercises

ISBN: 0-321-57702-7 / 978-0-321-57702-3

VIDEOS ON DVD WITH OPTIONAL SUBTITLES

■ DVD format enables students to watch the videos at

home or on campus

Technology Resources

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online exercises correlated directly to the textbook, as well as media related to that book, and they can also create and assign their own online exercises and import TestGen®tests for added flexibility MyMathLab’s online gradebook—designed specifically formathematics and statistics—automatically tracks students’ homework and test resultsand gives you control over how to calculate final grades You can also add offline (paper-

text-and-pencil) grades to the gradebook MyMathLab also includes access to the Pearson

Tutor Center ( www.pearsontutorservices.com) The Tutor Center is staffed by

quali-fied mathematics instructors who provide textbook-specific tutoring for students viatoll-free phone, fax, email, and interactive Web sessions MyMathLab is available toqualified adopters For more information, visit our website at www.mymathlab.comorcontact your Pearson representative

MathXL and MyMathLab have been expanded to provide more support to instructors andstudents

■ MathXL and MyMathLab have 65% more exercises than the previous edition,

giving instructors more flexibility when assigning homework and practice

■ “Why Math Matters” Animations accompany select applications to help illustrate

these concepts for students

■ Explorations are guided-discovery projects (in PDF), created by Nolan Mitchell

(Chemeketa Community College), and can be assigned as class work or homework.These projects can be submitted via digital dropbox and graded by the instructor

Answers to the Explorations are posted under Instructor Resources.

■ Animations were also created by Nolan Mitchell and use GeoGebra These animations

are a collection of interactive discovery exercises designed to help students to betterunderstand functions and their graphs

InterAct Math Tutorial Web Site: www.interactmath.com

Get practice and tutorial help online! This interactive tutorial website provides algorithmicallygenerated practice exercises that correlate directly to the exercises in the textbook Studentscan retry an exercise as many times as they like, with new values each time, for unlimited prac-tice and mastery Every exercise is accompanied by an interactive guided solution that pro-vides helpful feedback for incorrect answers, and students can also view a worked-out sampleproblem that steps them through an exercise similar to the one they’re working on

Acknowledgments

Many individuals contributed to the development of this textbook I would like to thankthe following reviewers, whose comments and suggestions were invaluable in preparingthis and previous editions of the text

John C Hake St Louis Community College at Florissant Valley

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Libby Higgins Greenville Technical College

Tuesday J Johnson New Mexico State University

Nancy Kolakowsky Paradise Valley Community College

Canda Mueller-Engheta University of Kansas

Stephen Nicoloff Paradise Valley Community College

Judith D Smalling St Petersburg Junior College

Christine Wise University of Louisiana at Lafayette

A special thank you is due Terry Krieger at Rochester Community and TechnicalCollege for his exceptional work with the manuscript and answers I would also like to thankPaul Lorczak, Christine Nguyen, Marcia Nermoe, Kathleen Pellissier, and David Atwood

at Rochester Community and Technical College and Janis Cimperman at St Cloud StateUniversity for their superb work with proofreading and accuracy checking Thanks also toJessica Rockswold for preparing the art manuscript and proofreading the text

Without the excellent cooperation from the professional staff at Addison-Wesley, thisproject would have been impossible They are, without a doubt, the best Thanks go to GregTobin for his support of this project Particular recognition is due Anne Kelly, Dana JonesBettez, and Leah Goldberg, who gave advice, support, assistance, and encouragement Theoutstanding contributions of Beth Houston, Brenden Berger, Heather Scott, Beth Paquin,Roxanne McCarley, Katherine Minton, and Joe Vetere are much appreciated

Thanks go to Wendy Rockswold, who gave invaluable assistance and encouragementthroughout the project

A special thank you goes to the many students and instructors who used the first three editions Their suggestions were insightful Please feel free to contact me at

gary.rockswold@mnsu.edu or Department of Mathematics, Minnesota State University,

Mankato, MN 56001 with your comments Your opinion is important

Gary Rockswold

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Have you ever thought

about how we “live by the

numbers”? Money, digital

tel-evisions, speed limits, grade

point averages, gas mileages,

and temperatures are all

based on numbers When we

are told what our weight,

blood pressure, body mass

index, and cholesterol level

are, these numbers can even

affect how we feel about

our-selves Numbers permeate

our society

People are concerned

about our environment and

how it is changing Do cars

and their carbon dioxide

emissions contribute to global

warming? Conventional cars are inherently inefficient because they burn

gaso-line when they are not moving Hybrid vehicles may be a viable option, but no

doubt numbers will be used to make a decision Rates of change, consumption,

efficiency, and pollution levels are all described by numbers

Numbers are part of mathematics, but mathematics is much more than

numbers Mathematics also includes techniques to analyze these numbers and

to guide our decisions about the future Mathematics is used not only in

sci-ence and technology; it is also used to describe almost every facet of life,

including consumer behavior and the Internet

In this chapter we discuss numbers and how functions are used to perform

computations with these numbers Understanding numbers and mathematical

concepts is essential to understanding and dealing with the many changes that

will occur in our lifetimes Mathematics makes life easier!

Source: Andrew Frank, “Plug-in Hybrid Vehicles for a Sustainable Future,” American Scientist,

March–April, 2007.

1

The essence of mathematics is not to make simple things complicated, but to make complicated things simple.

—Stanley Gudder Introduction to

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Because society is becoming more complex and diverse, our need for mathematics is ing dramatically each year Numbers are essential to our everyday lives For example, theiPhone is 4.5 inches in height, 2.4 inches in width, and 0.46 inch in thickness It has an8-gigabyte flash drive, a 2-megapixel camera, and 480-by-320-pixel screen resolution, and

increas-it can operate at temperatures between and (Source: Apple Corporation.)

Mathematics not only provides numbers to describe new products, but also gives usproblem-solving strategies This section discusses basic sets of numbers and introducessome essential problem-solving strategies

Sets of Numbers

One important set of numbers is the set of natural numbers This set comprises the

posi-tive and not presented in fractional parts

con-tains the natural numbers, their additive inverses (negatives), and 0 Historically, negativenumbers were not readily accepted Today, however, when a person overdraws a personalchecking account for the first time, negative numbers quickly take on meaning There is asignificant difference between a positive and a negative balance

A rational number can be expressed as the ratio of two integers where Rational numbers include the integers Examples of rational numbers are

Note that 0 and 1.2 are both rational numbers They can be represented by the fractions and Because two fractions that look different can be equivalent, rational numbers havemore than one form A rational number can always be expressed in a decimal form that

either repeats or terminates For example, , a repeating decimal, and , a minating decimal The overbar indicates that

ter-Real numbers can be represented by decimal numbers Since every rational number

has a decimal form, real numbers include rational numbers However, some real numbers

cannot be expressed as a ratio of two integers These numbers are called irrational

num-bers The numbers , , and are examples of irrational numbers They can be

represented by nonrepeating, nonterminating decimals Note that for any positive integer a,

if is not an integer, then is an irrational number

Real numbers include both rational and irrational numbers and can be approximated

by a terminating decimal Examples of real numbers include

The symbol means approximately equal This symbol is used in place of an

equals sign whenever two unequal quantities are close in value For example, ,whereas 13 L 0.3333

p21522

0 1

• Learn scientific notation

and use it in applications

• Apply problem-solving

strategies

1.1 Numbers, Data, and Problem Solving

CLASS DISCUSSION

The number 0 was invented well

after the natural numbers Many

societies did not have a zero—for

example, there is no Roman

numeral for 0 Discuss some

pos-sible reasons for this

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Order of Operations

Does equal 0 or 6? Does equal 25 or ? Figure 1.1 correctly shows that

and that Because multiplication is performed before tion, Similarly, because exponents are evaluated before performing nega-tion, It is essential that algebraic expressions be evaluated consistently, so thefollowing rules have been established

parenthe-1 Evaluate all exponents Then do any negation after evaluating exponents.

2 Do all multiplication and division from left to right.

3 Do all addition and subtraction from left to right.

6 3ⴱ2

0– 52

– 2 5

Figure 1.1

Evaluating arithmetic expressions

Evaluate each expression by hand

SOLUTION (a)

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Scientific Notation

Numbers that are large or small in absolute value are often expressed in scientific

nota-tion Table 1.1 lists examples of numbers in standard (decimal) form and in scientific

notation.

Calculator Help

To display numbers in scientific

notation, see Appendix A

(page AP-2).

Table 1.1

Standard Form Scientific Notation Application

5.38 * 10-6

9 * 1091.3517 * 1049.3 * 107

To write 0.00000538 in scientific notation, start by moving the decimal point to theright of the first nonzero digit, 5, to obtain 5.38 Since the decimal point was moved six

decimal point is moved to the left, the exponent of 10 is positive, rather than negative Here

is a formal definition of scientific notation

0.00000538 = 5.38 * 10-6-6

this technology may use the movement of the human body to power tiny devices such aspacemakers The next example demonstrates how scientific notation appears in thedescription of this new technology

Analyzing the energy produced by your body

Nanotechnology is a technology of the very small: on the order of one billionth of a meter.Researchers are looking to power tiny devices with energy generated by the human body

(Source: Z Wang, “Self-Powered Nanotech,” Scientific American, January 2008.)

(a) Write one billionth in scientific notation.

(b) While typing, a person’s fingers generate about watt of electrical energy.Write this number in standard (decimal) form

SOLUTION (a) One billionth can be written as (b) Move the decimal point in 2.2 three places to the left:

10 9 = 1 * 10 -9.2.2 * 10-3

EXAMPLE 3

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Evaluating expressions by hand

Evaluate each expression Write your result in scientific notation and standard form

Add exponents

Scientific notation Standard form

Subtract exponents

Scientific notationStandard form

䉳

might be displayed as 4.2E–3 On some calculators, numbers can be entered in scientificnotation with the key, which you can find by pressing

Computing in scientific notation with a calculator

Approximate each expression Write your answer in scientific notation

SOLUTION (a) The given expression is entered in two ways in Figure 1.2 Note that in both cases

(b) Be sure to insert parentheses around and around the numerator, , inthe ratio From Figure 1.3 we can see that the result is approximately1.59 * 1012

(3 * 103)(2 * 104)

EXAMPLE 4

( 6ⴱ1 0 ^ 3 ) / ( 4ⴱ1 0 ^ 6)ⴱ( 1 2ⴱ1 0 ^ 2 )

1 8( 6E3 ) / ( 4E6 )ⴱ( 1 2

E2 ) 1 8

√ (4500)ⴱ((103450) / 233)^3

To enter numbers in scientific

notation, see Appendix A

(page AP-2).

Algebra Review

To review exponents, see Chapter R

(page R-7).

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Computing with a calculator

Use a calculator to evaluate each expression Round answers to the nearest thousandth

SOLUTION (a) On some calculators the cube root can be found by using the MATH menu If your cal-

culator does not have a cube root key, enter 131^(1/3) From the first two lines in

(b) Do not use 3.14 for the value of Instead, use the built-in key to obtain a more

accu-rate value of From the bottom two lines in Figure 1.4,

(c) When evaluating this expression be sure to include parentheses around the numerator

and around the denominator Most calculators have a special square root key that can

be used to evaluate From the first three lines in Figure 1.5,

(d) The absolute value can be found on some calculators by using the MATH NUM

menus From the bottom two lines in Figure 1.5, ƒ 23 - 6 ƒ L 4.268

3.7 + 9.8 L 0.179.22

32.44627668

( 1 √ (2)) / (3.79

8 ) 1 7 8 8 3 0 6 3 4 2

Many problem-solving strategies are used in algebra However, in this subsection we focus

on two important strategies that are used frequently: making a sketch and applying one ormore formulas These strategies are illustrated in the next three examples

Finding the speed of Earth

Earth travels around the sun in an approximately circular orbit with an average radius of

93 million miles If Earth takes 1 year, or about 365 days, to complete one orbit, estimatethe orbital speed of Earth in miles per hour

SOLUTION Getting StartedSpeed S equals distance D divided by time T, We need to find thenumber of miles Earth travels in 1 year and then divide it by the number of hours in

1 year.䉴

Distance Traveled A sketch of Earth orbiting the sun is shown in Figure 1.6 In 1 yearEarth travels the circumference of a circle with a radius of 93 million miles Thecircumference of a circle is , where r is the radius, so the distance D is

To find the circumference of a circle,

see Chapter R (page R-2).

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Hours in 1 Year The number of hours H in 1 year, or 365 days, equals

䉳

Many times in geometry we evaluate formulas to determine quantities, such asperimeter, area, and volume In the next example we use a formula to determine the num-ber of fluid ounces in a soda can

Finding the volume of a soda can

The volume V of the cylindrical soda can in Figure 1.7 is given by where r is its radius and h is its height.

(a) If inches and inches, find the volume of the can in cubic inches

(b) Could this can hold 16 fluid ounces? (Hint: 1 cubic inch equals 0.55 fluid ounce.)

SOLUTION (a)

(b) To find the number of fluid ounces, multiply the number of cubic inches by 0.55.

Measuring the thickness of a very thin layer of material can be difficult to do directly.For example, it would be difficult to measure the thickness of a sheet of aluminum foil or

a coat of paint with a ruler However, it can be done indirectly using the following formula

That is, the thickness of a thin layer equals the volume of the substance divided by the areathat it covers For example, if a volume of 1 cubic inch of paint is spread over an area of

100 square inches, then the thickness of the paint equals inch This formula is illustrated

in the next example

Calculating the thickness of aluminum foil

A rectangular sheet of aluminum foil is 15 centimeters by 35 centimeters and weighs5.4 grams If 1 cubic centimeter of aluminum weighs 2.7 grams, find the thickness of thealuminum foil (Source: U Haber-Schaim, Introductory Physical Science.)

SOLUTION Getting StartedStart by making a sketch of a rectangular sheet of aluminum, as shown in

Figure 1.8 To complete this problem we need to find the volume V of the aluminum foil and its area A Then we can determine the thickness T by using the formula T = V 䉴

A

EXAMPLE 9

1 100

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For the rectangular box shape shown in Figure 1.8 on the previous page,

Area

It follows that

Volume Because the aluminum foil weighs 5.4 grams and each 2.7 grams equals 1 cubiccentimeter, the volume of the aluminum foil is

Divide weight by density

Area The aluminum foil is rectangular with an area of square centimeters

Thickness The thickness of 2cubic centimeters of aluminum foil with an area of 525

To find the area of a rectangle, see

Chapter R (page R-1) To find the

volume of a box, see Chapter R

(page R-3).

1.1 Putting

It All

Together

Numbers play a central role in our society Without numbers, data could bedescribed qualitatively but not quantitatively For example, we could say thatthe day seems hot but would not be able to give an actual number for the tem-perature Problem-solving strategies are used in almost every facet of our lives,providing the procedures needed to systematically complete tasks and per-form computations

The following table summarizes some of the concepts in this section

Integers Include the natural numbers, their opposites, and 0 , -2, -1, 0, 1, 2, Rational numbers Include integers; all fractions where p and q are integers p q,

Irrational numbers Can be written as nonrepeating, nonterminating decimals;

cannot be a rational number; if a square root of a positive ger is not an integer, it is an irrational number

inte-p, 22, - 25, 23 7, p4with q Z 0;all repeating and all terminating decimals

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Concept Comments Examples

Real numbers Any number that can be expressed in

standard (decimal) formInclude the rational numbers and

Then perform any remaining calculations

1 Evaluate all exponents Then do any

negation after evaluating exponents.

2 Do all multiplication and division

from left to right.

3 Do all addition and subtraction from

left to right.

= -32

Scientific notation A number in the form where

and n is an integer

Used to represent numbers that are large

or small in absolute value

Classifying Numbers

Exercises 1–6: Classify the number as one or more of the

follow-ing: natural number, integer, rational number, or real number.

1. (Fraction of people in the United States completing at

least 4 years of high school)

2 20,082 (Average cost in dollars of tuition and fees at a

private college in 2004)

3 7.5 (Average number of gallons of water used each minute

while taking a shower)

4 25.8 (Nielsen rating of the TV show Grey’s Anatomy the

11 Shoe sizes 12 Populations of states

13 Gallons of gasoline 14 Speed limits

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15 Temperatures in a winter weather forecast in Montana

16 Numbers of compact disc sales

Exercises 29–40: Write the number in scientific notation.

29 184,800 (New lung cancer cases reported in 2005)

30 29,285,000 (People worldwide living with HIV)

31 0.04361 (Proportion of U.S deaths attributed to

Exercises 41–52: Write the number in standard form.

41. (Wavelength in meters of visible light)

43. (Years required for the sun to orbit our galaxy)

44. (Federal debt in dollars in 2007)

47. 5 * 105 48 3.5 * 103

-5.68 * 10-11.567 * 102

1.53

12 + p - 5

3.2(1.1)2 - 4(1.1) + 2

0.3 + 1.55.5 - 1.2

1.72 - 5.9835.6 + 1.02

ƒ p - 3.2 ƒ

2(32 + p3)

23 192

2p(4.56 * 104) + (3.1 * 10-2)(8.5 * 10-5)(-9.5 * 107)2

23 (2.5 * 10-8) + 10-7

a101 + 230.42 b2 + 23.4 * 10-2(9.87 * 106)(34 * 1011)

8.947 * 1070.00095 (4.5 * 108)

2.4 * 10-54.8 * 10-7

(3 * 101)(3 * 104)(4 * 103)(2 * 105)

0.0032 * 10-1

67 * 103

-5.4 * 10-50.045 * 105

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Calculate the percent change for the given A and B Round your

answer to the nearest tenth of a percent when appropriate.

81. Percent Change Suppose that tuition is initially $100

per credit and increases by 6% from the first year to the

second year What is the cost of tuition the second year?

Now suppose that tuition decreases by 6% from the

sec-ond to the third year Is tuition equal to $100 per credit

the third year? Explain

82. Tuition Increases From 1976 to 2004, average annual

tuition and fees at public colleges and universities

increased from $433 to $5132 Calculate the percent

change over this time period

83. Nanotechnology (Refer to Example 3.) During

inhala-tion, the typical body generates 0.14 watt of electrical

power, which could be used to power tiny electrical

cir-cuits Write this number in scientific notation (Source:

Scientific American, January 2008.)

84. Movement of the Pacific Plate The Pacific plate (the

floor of the Pacific Ocean) near Hawaii is moving at

about 0.000071 kilometer per year This is about the

speed at which a fingernail grows Use scientific

nota-tion to determine how many kilometers the Pacific plate

travels in one million years

85. Orbital Speed (Refer to Example 7.) The planet Mars

travels around the sun in a nearly circular orbit with a

radius of 141 million miles If it takes 1.88 years for

Mars to complete one orbit, estimate the orbital speed of

Mars in miles per hour

86. Size of the Milky Way The speed of light is about

186,000 miles per second The Milky Way galaxy has an

approximate diameter of miles Estimate, to

the nearest thousand, the number of years it takes for

light to travel across the Milky Way (Source: C Ronan,

The Natural History of the Universe.)

6 * 1017

Mars

Sun 141,000,000 mi

dra-(Sources: Department of the Treasury, Bureau of the Census.)

(a) In 1970 the population of the United States was

203,000,000 and the federal debt was $370 billion.Find the debt per person

(b) In 2000 the population of the United States was

approximately 281,000,000 and the federal debt was

$5.54 trillion Find the debt per person

88. Discharge of Water The Amazon River dischargeswater into the Atlantic Ocean at an average rate of4,200,000 cubic feet per second, the highest rate of anyriver in the world Is this more or less than 1 cubic mile

of water per day? Explain your calculations (Source:

The Guinness Book of Records 1993.)

89. Thickness of an Oil Film (Refer to Example 9.) A drop

of oil measuring 0.12 cubic centimeter is spilled onto alake The oil spreads out in a circular shape having a

diameter of 23 centimeters Approximate the thickness

of the oil film

90. Thickness of Gold Foil (Refer to Example 9.) A flat,rectangular sheet of gold foil measures 20 centimeters

by 30 centimeters and has a mass of 23.16 grams If

1 cubic centimeter of gold has a mass of 19.3 grams,find the thickness of the gold foil (Source: U Haber-

Schaim, Introductory Physical Science.)

91. Analyzing Debt A 1-inch-high stack of $100 bills tains about 250 bills In 2000 the federal debt wasapproximately 5.54 trillion dollars

con-(a) If the entire federal debt were converted into a stack

of $100 bills, how many feet high would it be?

(b) The distance between Los Angeles and New York is

approximately 2500 miles Could this stack of $100bills reach between these two cities?

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92. Volume of a Cone The volume V of a cone is given by

where r is its radius and h is its height Find

answer to the nearest hundredth

93. Size of a Soda Can (Refer to Example 8.) The volume

r is its radius and h is its height.

(a) If inches and inches, find the

vol-ume of the can in cubic inches

(b) Could this can hold 12 fluid ounces? (Hint: 1 cubic

inch equals about 0.55 fluid ounce.)

94. Volume of a Sphere The volume of a sphere is given by

where r is the radius of the sphere Calculate

the volume if the radius is 3 feet Approximate your

answer to the nearest tenth

100-96. Depth of a Lake (Refer to Example 9.) A lake covers

square feet and contains cubic feet

of water Find the average depth of the lake

Writing about Mathematics

97 Describe some basic sets of numbers that are used in

mathematics

98 Suppose that a positive number a is written in scientific

Explain what n indicates about the size of a.

EXTENDED AND DISCOVERY EXERCISE

1 If you have access to a scale that weighs in grams, find

the thickness of regular and heavy-duty aluminum foil

Is heavy-duty foil worth the price difference? (Hint:

Each 2.7 grams of aluminum equals 1 cubic centimeter.)

1 … b … 10 a = b * 10

n,

7.5 * 1082.5 * 107

Introduction

Technology is giving us access to huge amounts of data For example, space telescopes,such as the Hubble telescope, are providing a wealth of information about the universe.The challenge is to convert the data into meaningful information that can be used to solveimportant problems Before conclusions can be drawn, data must be analyzed A powerfultool in this step is visualization, as pictures and graphs are often easier to understand thanwords This section discusses how different types of data can be visualized by using vari-ous mathematical techniques

One-Variable Data

Data often occur in the form of a list A list of test scores without names is an example;

the only variable is the score Data of this type are referred to as one-variable data If the

values in a list are unique, they can be represented visually on a number line

Means and medians can be found for one-variable data sets To calculate the mean (or

average) of a set of n numbers, we add the n numbers and then divide the sum by n The median is equal to the value that is located in the middle of a sorted list If there is an odd

• Analyze one-variable data

• Find the domain and range

of a relation

• Graph in the xy-plane

• Calculate distance

• Find the midpoint

• Learn the standard

equation of a circle

• Learn to graph equations

with a calculator (optional)

1.2 Visualizing and Graphing Data

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number of data items, the median is the middle data item If there is an even number ofdata items, the median is the average of the two middle items.

Analyzing a list of data

Table 1.2 lists the monthly average temperatures in degrees Fahrenheit at Mould Bay,Canada

EXAMPLE 1

CLASS DISCUSSION

In Example 1(c), the mean of the

temperatures is approximately

Interpret this temperature

Explain your reasoning

0.6°F

Source: A Miller and J Thompson, Elements of Meteorology.

-24-17-9

-26-31-27

°F

Table 1.2 Monthly Average Temperatures at Mould Bay, Canada

Precipitation (inches) 6.2 3.9 3.6 2.3 2.0 1.5 0.5 1.1 1.6 3.1 5.2 6.4

Table 1.3 Average Precipitation for Portland, Oregon

(a) Plot these temperatures on a number line.

(b) Find the maximum and minimum temperatures.

(c) Determine the mean of these 12 temperatures.

(d) Find the median and interpret the result.

SOLUTION (a) In Figure 1.9 the numbers in Table 1.2 are plotted on a number line.

Figure 1.9 Monthly Average Temperatures

(d) Because there is an even number of data items, the median is the average of the

mid-dle two values From the number line we see that the midmid-dle two values are andThus the median is This result means that half the months have

an average temperature that is greater than and half the months have an average

Now Try Exercises 1 and 5

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If x is the month and y is the precipitation, then the ordered pair (x, y) represents the

average amount of precipitation y during month x For example, the ordered pair (5, 2.0)

indicates that the average precipitation in May is 2.0 inches, whereas the ordered pair

(2, 3.9) indicates that the average precipitation in February is 3.9 inches Order is important

in an ordered pair

Since the data in Table 1.3 involve two variables, the month and precipitation, we refer

to them as variable data It is important to realize that a relation established by

two-variable data is between two lists rather than within a single list January is not related toAugust, and 6.2 inches of precipitation is not associated with 1.1 inches of precipitation.Instead, January is paired with 6.2 inches, and August is paired with 1.1 inches We nowdefine the mathematical concept of a relation

If we denote the ordered pairs in a relation by (x, y), then the set of all x-values is

called the domain of the relation and the set of all y-values is called the range The

rela-tion shown in Table 1.3 has domain

x-valuesand range

y-values

Finding the domain and range of a relation

A physics class measured the time y that it takes for an object to fall x feet, as shown in

Table 1.4 The object was dropped twice from each height

EXAMPLE 2

R = {0.5, 1.1, 1.5, 1.6, 2.0, 2.3, 3.1, 3.6, 3.9, 5.2, 6.2, 6.4}

D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Relation

A relation is a set of ordered pairs.

(a) Express the data as a relation S.

(b) Find the domain and range of S.

SOLUTION (a) A relation is a set of ordered pairs, so we can write

(b) The domain is the set of x-values of the ordered pairs, or The range is

the set of y-values of the ordered pairs, or

䉳

To visualize a relation, we often use the Cartesian (rectangular) coordinate plane,

Now Try Exercise 47

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intersect at the origin and determine four regions called quadrants, numbered I, II, III,

and IV, counterclockwise, as shown in Figure 1.10 We can plot the ordered pair (x, y) using the x- and y-axes The point (1, 2) is located in quadrant I, ( , 3) in quadrant II, ( , )

in quadrant III, and (1, ) in quadrant IV A point lying on a coordinate axis does notbelong to any quadrant The point ( , 0) is located on the x-axis, whereas the point (0, )

lies on the y-axis.

The term scatterplot is given to a graph in the xy-plane where distinct points are

plot-ted Figure 1.10 is an example of a scatterplot

Graphing a relation

Complete the following for the relation

(a) Find the domain and range of the relation.

(b) Determine the maximum and minimum of the x-values and then of the y-values (c) Label appropriate scales on the xy-axes.

(d) Plot the relation.

SOLUTION (a) The elements of the domain correspond to the first number in each ordered pair Thus

Similarly, the elements of the range correspond to the second number in each orderedpair Thus

(b) x-minimum: ; x-maximum: 5; y-minimum: ; y-maximum: 15

(c) An appropriate scale for both the x-axis and the y-axis might be to 20, with eachtick mark representing a distance of 5 This scale is shown in Figure 1.11

Figure 1.10 The xy-plane

–15–10 –5 5 10 15

–15 –10 –5 5 10 15

x y

Figure 1.11

–15–10 –5 5 10 15 –5

5

(0, 15) (–10, 10)

(–15, –10)

(5, 10)

(5, –5)

x y

is called a line graph.

Now Try Exercise 51

-10-15-10

-5

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