Second Edition Intermediate Algebra Connecting Concepts Through Applications Mark Clark Palomar College Cynthia Anfinson Palomar College Australia • Brazil • Mexico • Singapore • United Kingdom • United States Copyright 2019 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it This is an electronic version of the print textbook Due to electronic rights restrictions, some third party content may be suppressed Editorial review has deemed that any suppressed content does not materially affect the overall learning experience The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest Important Notice: Media content referenced within the product description or the product text may not be available in the eBook version Copyright 2019 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it To my wife Christine for her love and support throughout our lives together and to our children Will and rosemary MC To my husband Fred and son Sean, thank you for your love and support CA extra thanks go to Jim and mary eninger for opening up their family cabin for me to spend many a day and night writing also to Dedad and mimi, who built the cabin in the 1940s MC Intermediate Algebra: Connecting Concepts Through Applications Second Edition Mark Clark, Cynthia Anfinson © 2019, 2012 Cengage Learning, Inc WCN: 02-300 Unless otherwise noted, all content is © Cengage ALL RIGHTS RESERVED No part of this work covered by the copyright herein Product Director: Mark Santee may be reproduced or distributed in any form or by any means, except as Product Manager: Frank Snyder permitted by U.S copyright law, without the prior written permission of the Content Developers: Samantha Gomez, Alison copyright owner Duncan For product information and technology assistance, Product Assistant: Jaime Manz contact us at Cengage Customer & Sales Support, 1-800-354-9706 Marketing Manager: Pamela Polk For permission to use material from this text or product, Senior Content Project Manager: submit all requests online at www.cengage.com/permissions Further permissions questions can be emailed to Michael Lepera permissionrequest@cengage.com Manufacturing Planner: Becky Cross IP Analyst: Reba Frederics IP Project Manager: Sarah Shainwald Production Service and Compositor: Lumina Datamatics, Inc Senior Art Director: Vernon Boes Text Designer: Diane Beasley Library of Congress Control Number: 2017951382 Student Edition: ISBN: 978-1-337-61558-7 Loose-leaf Edition: ISBN: 978-1-337-61563-1 Cover Designer: Irene Morris Cover Image: WAYHOME Studio/Creative Market Cengage 20 Channel Center Street Boston, MA 02210 USA Cengage is a leading provider of customized learning solutions with employees residing in nearly 40 different countries and sales in more than 125 countries around the world Find your local representative at www.cengage.com Cengage products are represented in Canada by Nelson Education, Ltd Printed in the United States of America Print Number: 01 Print Year: 2017 To learn more about Cengage platforms and services, visit www.cengage.com Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com Copyright 2019 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Brief Contents ChApTEr ChApTEr SySTemS oF linear equaTionS anD inequaliTieS 149 ChApTEr exPonenTS, PolynomialS, anD FunCTionS 253 ChApTEr quaDraTiC FunCTionS 331 ChApTEr exPonenTial FunCTionS 469 ChApTEr logariThmiC FunCTionS 547 ChApTEr raTional FunCTionS 621 ChApTEr raDiCal FunCTionS 691 ChApTEr ConiC SeCTionS 769 ChApTEr linear FunCTionS 10 SequenCeS anD SerieS (ONLINE ONLY) 815 AppEndIx A BaSiC algeBra revieW a-1 AppEndIx B maTriCeS B-1 AppEndIx C uSing The graPhing CalCulaTor C-1 AppEndIx D anSWerS To PraCTiCe ProBlemS D-1 AppEndIx E anSWerS To SeleCTeD exerCiSeS e-1 index i-1 unit Conversions reF-2 geometric Formulas reF-3 equation Solving Toolbox reF-4 expression Simplifying Toolbox reF-5 modeling Toolbox reF-6 Factoring Toolbox reF-6 Copyright 2019 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it iii Contents 1.1 Linear Functions Solving Linear Equations Linear equations • Applications • Literal equations 1.2 Fundamentals of Graphing and Slope 19 Introduction to Graphing Equations • Linear Equations in Two Variables • The Meaning of Slope in an Application • Graphing Lines Using Slope and Intercept 1.3 Intercepts and Graphing 41 The General Form of Lines • Intercepts and Their Meaning • Graphing Lines Using Intercepts • Horizontal and Vertical Lines 1.4 Finding Equations of Lines 58 Equations of Lines • Parallel and Perpendicular Lines • Interpreting the Characteristics of a Line: A Review 1.5 Functions and Function Notation 74 Relations and Functions • Vertical line test • Function Notation • Domain and Range of Functions 1.6 Using Data to Create Scatterplots 95 Using Data to Create Scatterplots • Adjusting Data • Graphical Models • Domain and Range of a Model • Applications 1.7 Finding Linear Models 114 Using a Calculator to Create Scatterplots • Linear Models • Applications Chapter Chapter Chapter Chapter 2.1 Summary 133 Review Exercises Test 145 Projects 146 142 Systems of Linear Equations and Inequalities 149 Systems of Linear Equations 150 Definition of Systems • Graphical and Numerical Solutions • Types of Systems • Applications 2.2 Solving Systems of Equations Using the Substitution Method 166 Substitution Method • Consistent and Inconsistent Systems • Applications 2.3 iv Solving Systems of Equations Using the Elimination Method 179 Elimination Method • Applications of Systems Copyright 2019 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ntents 2.4 Solving Linear Inequalities 189 Introduction to Inequalities • Solving Inequalities • Systems as Inequalities • Solving Inequalities Numerically and Graphically • Solving Compound Inequalities • Applications 2.5 Absolute Value Equations and Inequalities 205 Absolute Value Equations • Absolute Value Inequalities Involving “Less Than” or “Less Than or Equal To” • Absolute Value Inequalities Involving “Greater Than” or “Greater Than or Equal To” • Applications 2.6 Solving Systems of Linear Inequalities 220 Graphing Linear Inequalities with Two Variables • Solving Systems of Linear Inequalities • Applications Chapter Summary 235 Chapter Review Exercises 243 Chapter Test 246 Chapter Projects 248 Cumulative Review Chapters 1-2 250 Exponents, polynomials, and Functions 253 3.1 Rules for Exponents 254 Rules for Exponents • Negative Exponents and Zero as an Exponent • Using Scientific Notation • Rational exponents • Applications 3.2 Combining Functions 270 The Terminology of Polynomials • Degree • Adding and Subtracting Functions • Multiplying and Dividing Functions • Applications 3.3 Composing Functions 288 Combining functions using composition • Applications 3.4 Factoring Polynomials 299 Factoring Out the GCF • Factoring by Grouping • Factoring Using the AC Method • Factoring Using Trial and Error • Prime Polynomials 3.5 Special Factoring Techniques 312 Perfect Square Trinomials • Difference of Squares • Difference and Sum of Cubes • Multistep Factorizations • Trinomials in Quadratic Form Chapter Summary 320 Chapter Review Exercises 325 Chapter Test 328 Chapter Projects 330 Quadratic Functions 331 4.1 Quadratic Functions and Parabolas 332 Introduction to Quadratic Functions and Identifying the Vertex • Identifying a Quadratic Function • Recognizing Graphs of Quadratic Functions and Identifying the Vertex • Applications Copyright 2019 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it v vi CoNTENTS 4.2 Graphing Quadratic Functions from Vertex Form 345 Axis of Symmetry • Vertex Form • Graphing Quadratic Functions from Vertex Form • Domain and Range • Applications 4.3 Finding Quadratic Models 363 Quadratic Models • Domain and Range • Applications 4.4 Solving Quadratic Equations by the Square Root Property and Completing the Square 378 Solving from Vertex Form • Square Root Property • Using the Pythagorean Theorem • The Distance Formula • Completing the Square • Converting to Vertex Form • Graphing from Vertex Form with x-Intercepts • Applications 4.5 Solving Equations by Factoring 399 The Product Property of Zero • Solving by Factoring • Finding a Quadratic Function from the Graph • Solving Nonlinear Polynomial Inequalities in One Variable • Applications 4.6 Solving Quadratic Equations by Using the Quadratic Formula 417 Solving by the Quadratic Formula • Determining Which Algebraic Method to Use When Solving a Quadratic Equation • Solving Systems of Equations Involving Quadratic Functions 4.7 Graphing Quadratic Functions from Standard Form 430 Graphing from Standard Form • Graphing Quadratic Inequalities in Two Variables • Applications Chapter Summary 444 Chapter Review Exercises 457 Chapter Test 461 Chapter Projects 462 Cumulative Review Chapters 1-4 465 Exponential Functions 5.1 Exponential Functions: Patterns of Growth and Decay 470 469 Exploring Exponential Growth and Decay • Recognizing Exponential Patterns • Applications 5.2 Solving Equations Using Exponent Rules 489 Recap of the Rules for Exponents • Solving Power Equations • Solving Exponential Equations by Inspection • Identifying Exponential Equations and Power Equations • Applications 5.3 Graphing Exponential Functions 5.4 Exploring Graphs of Exponentials • Domain and Range of Exponential Functions • Exponentials of the Form f 1x2 a # bx c Finding Exponential Models 498 510 Exponential Functions • Exponential Models • Domain and Range for Exponential Models • Applications 5.5 Exponential Growth and Decay Rates and Compounding Interest 523 Exponential Growth and Decay Rates • Compounding Interest • Growth Rates and Exponential Functions • Applications Copyright 2019 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ntents Chapter Chapter Chapter Chapter Summary 534 Review Exercises Test 541 Projects 543 539 Logarithmic Functions 6.1 Functions and Their Inverses 548 547 Introduction to Inverse Functions • One-to-One Functions • Applications 6.2 Logarithmic Functions 561 Definition of Logarithms • Properties of Logarithms • Change of Base Formula • Inverses • Equivalent Logarithm and Exponential Forms • Solving Logarithmic Equations 6.3 Graphing Logarithmic Functions 570 Graphing Logarithmic Functions • Domain and Range of Logarithmic Functions 6.4 Properties of Logarithms 578 Properties of Logarithms • Simplifying and Expanding Logarithm Expressions 6.5 Solving Exponential Equations 584 Solving Exponential Equations • Compounding Interest • Applications 6.6 Solving Logarithmic Equations 596 Solving Logarithmic Equations • Applications Chapter Summary 605 Chapter Review Exercises 611 Chapter Test 613 Chapter Projects 614 Cumulative Review Chapters 1-6 616 7.1 rational Functions 621 Rational Functions and Variation 622 Rational Functions • Direct and Inverse Variation • Domain of a Rational Function • Applications • Vertical Asymptotes and Holes in Graphs 7.2 Simplifying Rational Expressions 638 Simplifying Rational Expressions • Long Division of Polynomials • Synthetic Division • Relationship between Division and Factoring 7.3 Multiplying and Dividing Rational Expressions 651 Multiplying Rational Expressions • Dividing Rational Expressions 7.4 Adding and Subtracting Rational Expressions 657 Least Common Denominator • Adding Rational Expressions • Subtracting Rational Expressions • Simplifying Complex Fractions Copyright 2019 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it vii viii C o ntents 7.5 Solving Rational Equations 668 Solving Rational Equations • Applications Chapter 7 Chapter 7 Chapter 7 Chapter 7 Radical Functions 691 Summary 679 Review Exercises 684 Test 687 Projects 688 8.1 Radical Functions 692 Relationships between Radicals and Rational Exponents • Radical Functions That Model Data • Square Roots and Higher Roots • Simplifying Radicals • Applications 8.2 Graphing Radical Functions 705 Domain and Range of Radical Functions • Graphing Radical Functions • Odd and Even Indexes 8.3 Adding and Subtracting Radicals 715 Adding and Subtracting Radicals 8.4 Multiplying and Dividing Radicals 718 Multiplying Radicals • Dividing Radicals and Rationalizing the Denominator • Conjugates 8.5 Solving Radical Equations 729 Solving Radical Equations • Solving Radical Equations Involving more than One Square Root • Extraneous Solution(s) • Solving Radical Equations Involving Higher-order Roots • Applications 8.6 Complex Numbers 741 Definition of Imaginary and Complex Numbers • Operations with Complex Numbers • Solving Equations with Complex Solutions Chapter 8 Summary 751 Chapter 8 Review Exercises 756 Chapter 8 Test 759 Chapter 8 Projects 760 Cumulative Review Chapters 1-8 763 Conic Sections 769 9.1 Parabolas and Circles 770 Introduction to Conic Sections • Revisiting Parabolas • A Geometric Approach to Parabolas • Circles • Applications 9.2 Ellipses and Hyperbolas 791 Ellipses • Hyperbolas • Recognizing the Equations for Conic Sections • Applications Chapter 9 Chapter 9 Chapter 9 Chapter 9 Summary 805 Review Exercises 809 Test 811 Projects 812 70 Chapter 1 Linear Functions 27 Write the equation of the line with slope and passing through the point (12, 20) 28 Write the equation of the line with slope and passing through the point (22, 12) 42 29 Write the equation of the line that passes through the points (1, 3) and (4, 12) 30 Write the equation of the line that passes through the points (2, 4) and (7, 24) 43 31 Write the equation of the line that passes through the points (7, 6) and (21, 21) 32 Write the equation of the line that passes through the points (25, 22) and (23, 210) 33 Write the equation of the line that passes through the points (24, 25) and (21, 17) 44 34 Write the equation of the line that passes through the points (28, 23) and (212, 215) 35 Write the equation of the line that passes through the points (7, 23) and (7, 9) 36 Write the equation of the line that passes through the points (4, 5) and (4, 12) x y 23 21 14.25 10.75 3.75 21.5 x y 11 218 26 18 x y 14 19 25.5 218 230.5 37 Write the equation of the line that passes through the points (2, 8) and (4, 8) For Exercises 45 through 58, determine, without graphing, whether the two given lines are parallel, perpendicular, or neither Check your answer by graphing both lines on a calculator or by hand 38 Write the equation of the line that passes through the points (28, 215) and (10, 215) 45 y 3x y 3x 46 y 4x y 4x For Exercises 39 through 44, write the equation for the line passing through the points given in the table 47 2x 3y 15 y x 48 y 2x y x 49 2x 5y 40 24y 10x 10 50 y 0.25x x 4y 51 4x 3y 20 12x 9y 30 52 y 4(x 3) x y x 53 x y 54 x 23 x 55 2x 5y 20 10x 4y 20 56 3x 2y 26y 9x 12 57 5x y 2y 10x 58 x 210 y 39 40 41 x y 11 15 19 x y 24 218 24 10 24 x y 22 21 29 22 213 59 Write the equation of the line that goes through the point (2, 8) and is parallel to the line y 4x 13 60 Write the equation of the line that goes through the point (4, 6) and is parallel to the line y 23x 24 61 Write the equation of the line that goes through the point (26, 8) and is parallel to the line 10x 15y 212 section 1.4 Finding Equations of Lines 71 62 Write the equation of the line that goes through the point (24, 29) and is parallel to the line 2x 2y 20.5 b Find the vertical intercept for the equation and give its meaning in this situation 63 Write the equation of the line that goes through the point (1, 7) and is perpendicular to the line y 2x c Find the horizontal intercept for the equation and give its meaning in this situation 64 Write the equation of the line that goes through the point (6, 5) and is perpendicular to the line y 3x 65 Write the equation of the line that goes through the point (5, 1) and is perpendicular to the line 4x y 5 66 Write the equation of the line that goes through the point (24, 23) and is perpendicular to the line 6x y 211 67 Write the equation of the line that goes through the point (2, 3) and is perpendicular to the line y x 68 Write the equation of the line that goes through the point (29, 3) and is perpendicular to the line 3x 7y 221 69 Write the equation of the line that goes through the point (4, 3) and is perpendicular to the line y 70 Write the equation of the line that goes through the point (3, 7) and is perpendicular to the line x 24 71 A student says that a line perpendicular to the line y x has slope m Explain what is wrong with this statement Give the correct slope 72 A student says that a line perpendicular to the line y 25x 12 has slope m 5 Explain what is wrong with this statement Give the correct slope 73 A student says that a line parallel to the line 2x 5y 14 has slope m Explain what is wrong with this statement Give the correct slope 74 A student says that a line parallel to the line 6x 3y 10 has slope m Explain what is wrong with this statement Give the correct slope 75 A student says that the line with slope passing through the point (2, 6) is y 4x Explain what is wrong with this equation Give the correct equation 76 A student says that the line with slope 23 passing through the point (7, 0) is y 23x Explain what is wrong with this equation Give the correct equation 77 The population of Washington state can be estimated by the equation P 85.18t 6744.5, where P represents the population of Washington in thousands of people t years since 2010 78 The population of New York state can be estimated by the equation P 80.7t 19392.3, where P represents the population of New York in thousands of people t years since 2010 Source: Model derived from data from the U.S Census Bureau a Find the slope of the equation and give its meaning in regard to the population of New York b Find the vertical intercept for the equation and give its meaning in regard to the population of New York c Find the horizontal intercept for the equation and give its meaning in regard to the population of New York 79 The percent of women who have coronary heart disease can be estimated by the equation C 0.3a 9, where C represents the percent of women who have coronary heart disease at a years of age Source: Model derived from data from The American Heart Association a Find the slope of the equation and give its meaning in this example b Find the vertical intercept for the equation and give its meaning in this example c Find the horizontal intercept for the equation and give its meaning in this example 80 The amount of wind energy produced in Texas can be estimated by the equation W 4464.5t 22361.7, where W represents the thousands of megawatt hours of wind energy produced in Texas t years since 2010 Source: U.S Energy Information Administration a Find the slope of the equation and interpret what it means in the terms of the situation b Find the vertical intercept for the equation and interpret what it means in the terms of the situation c Find the horizontal intercept for the equation and interpret what it means in the terms of the situation 81 The monthly profit for a small used car lot can be estimated by the equation P 500c 6000, where P represents the profit in dollars for the car lot when c cars are sold in a month a Find the slope of the equation and give its meaning in regard to the profit Source: Model derived from data from the U.S Census Bureau b Find the vertical intercept for the equation and give its meaning in regard to the profit a Find the slope of the equation and give its meaning in this situation c Find the horizontal intercept for the equation and give its meaning in regard to the profit 72 Chapter 1 Linear Functions 82 The monthly profit for a coffee shop can be estimated by the equation P 0.75c 3500, where P represents the profit in dollars for the coffee shop when c customers visit in a month a Find the slope of the equation and give its meaning in regard to the profit b Find the vertical intercept for the equation and give its meaning in regard to the profit c Find the horizontal intercept for the equation and give its meaning in regard to the profit 83 Dan gives surfing lessons over the summer and earns $30 for each hour lesson given His surfboards and supplies for the summer cost him $700 a Write an equation for the profit in dollars that Dan makes from giving s surf lessons b Find the slope of the equation in part a and give its meaning in this situation c Find the vertical intercept of the equation and give its meaning in this situation d Find the horizontal intercept of the equation and give its meaning in this situation 84 Janell tutors math students and earns $20 an hour for each session She spends about $100 a month on transportation and other costs related to tutoring a Write an equation for the monthly profit in dollars that Janell makes from tutoring for h hours a month b Find the slope of the equation in part a and give its meaning in terms of her monthly profit c Find the vertical intercept of the equation and give its meaning in terms of her monthly profit d Find the horizontal intercept of the equation and give its meaning in terms of her monthly profit 85 The Parent Teacher Association (PTA) at Mission Meadows Elementary School is starting a recycling program to help raise money for a new running track to be installed on the campus For each pound of aluminum cans recycled, they earn $1.24 The PTA started their fund raising with $2000 donated from the parents d Find the horizontal intercept of the equation and give its meaning in this situation 86 Through the year 2060, the population of the United States is expected to grow by about 2.1 million people per year In 2016, the population of the United States was estimated to be 324 million Source: U.S Census Bureau a Write an equation to represent the population of the United States t years since 2010 b Use the equation, from part a to estimate the population of the United States in 2030 c When will the population of the United States reach 400 million? d What is the vertical intercept of the equation you found? Explain what it means using the terms of the problem 87 The percentage of Americans who have been diagnosed with diabetes has been growing steadily over the years In 2012, 8.9% of Americans had been diagnosed with diabetes In 2014, 9.1% of Americans had been diagnosed with diabetes Source: CDC U.S Diabetes Surveillance System a Assuming that the percentage of Americans diagnosed with diabetes continues to grow at a constant rate, write an equation to represent this situation b Use the equation from part a to estimate the percentage of Americans who will have been diagnosed with diabetes in 2020 c What is the slope of the equation you found? Explain what it means using the terms of the problem 88 A toy manufacturer finds that if they produce 1000 toy cars an hour, 1% of the cars are defective If production is increased to 1500 toys an hour, 1.5% of the cars are defective a Assuming that the percentage of cars that are defective is linearly related to the number of cars produced an hour, write an equation for the percentage of cars that are defective if t toys are produced an hour a Write an equation for the total amount the PTA has raised for the track depending on how many pounds of aluminum cans they recycle b Use the equation from part a to find the percentage of cars that are defective if 2500 cars are produced an hour b What is the slope of the equation you found? Give its meaning in this situation c What is the slope of the equation you wrote? What does it mean in regard to the percentage of cars that are defective? c Find the vertical intercept of the equation and give its meaning in this situation section 1.4 Finding Equations of Lines 89 Use the equation for the cost of shirts that you wrote in Exercise 17 on page 69 to answer the following questions a Use the equation to find how much 50 shirts will cost b What is the slope of the equation? What does it mean in regard to the cost of shirts? c Find the vertical intercept of the equation and give its meaning in regard to the cost of shirts d Find the horizontal intercept of the equation and give its meaning in regard to the cost of shirts 90 Use the equation for the cost of sports jerseys that you wrote in Exercise 18 on page 69 to answer the following questions a Use the equation to find how much 40 jerseys will cost 73 93 Use the equation for the optimal weight of a woman that you wrote in Exercise 21 on page 69 to answer the following questions a What does the equation give as the optimal weight of a woman who is feet tall? b What is the slope of the equation? What does it mean in this situation? c Find the vertical intercept of the equation and give its meaning in this situation d Find the horizontal intercept of the equation and give its meaning in this situation 94 Use the equation for the optimal weight of a man that you wrote in Exercise 22 on page 69 to answer the following questions b What is the slope of the equation? What does it mean in this situation? a What does the equation give as the optimal weight of a man who is feet tall? c Find the vertical intercept of the equation and give its meaning in this situation b What is the slope of the equation? Interpret its meaning in terms of the problem d Find the horizontal intercept of the equation and give its meaning in this situation c Find the vertical intercept of the equation and interpret its meaning in terms of the problem 91 Use the equation for the population of Washington that you wrote in Exercise 19 on page 69 to answer the following questions a What does the equation predict the population of Washington will be in 2030? d Find the horizontal intercept of the equation and interpret its meaning in terms of the problem 95 Use the equation for the national healthcare expenditures that you wrote in Exercise 23 on page 69 to answer the following questions b What is the slope of the equation? Interpret its meaning in terms of the problem a What does the equation predict as the national healthcare expenditures in 2022? c Find the vertical intercept of the equation and interpret its meaning in terms of the problem b What is the slope of the equation? What does it mean in this situation? d Find the horizontal intercept of the equation and interpret its meaning in terms of the problem c Find the vertical intercept of the equation and give its meaning in this situation 92 Use the equation for the population of Nevada that you wrote in Exercise 20 on page 69 to answer the following questions a What does the equation predict the population of Nevada will be in 2030? d Find the horizontal intercept of the equation and give its meaning in this situation 96 Use the equation for the number of hospitals in the United States that you wrote in Exercise 24 on page 69 to answer the following questions b What is the slope of the equation? What does it mean in regard to the population of Nevada? a What does the equation predict as the number of hospitals in the United States in 2020? c Find the vertical intercept of the equation and give its meaning in regard to the population of Nevada b What is the slope of the equation? What does it mean in regard to the number of hospitals? d Find the horizontal intercept of the equation and give its meaning in regard to the population of Nevada c Find the vertical intercept of the equation and give its meaning in regard to the number of hospitals d Find the horizontal intercept of the equation and give its meaning in regard to the number of hospitals 74 Chapter 1 Linear Functions 1.5 Functions and Function Notation LEARNING OBJECTIVES Identify a function Apply the vertical line test for functions Use function notation Identify the domain and range of a function Relations and Functions The way things in life are related to one another is important to understand In mathematics, we are interested in how different sets are related These relationships can be as simple as the relationship between a person and his or her height, age, or weight A manager might want to examine the day of the week and the number of work absences at their company A relationship that a college administrator might consider is the number of credits each student is taking in a semester All of these represent what we call a relation A relation is any connection between the elements of a set of input(s) (domain) and the elements of a set of output(s) (range) It is typically represented by a set of ordered pairs or by using an equation The equation y 5x represents a relation that relates the x-values with a corresponding y-value using arithmetic operations Using this equation, we can calculate an infinite number of ordered pairs that represent the inputs and outputs for this relation Definition Relation A set of ordered pairs Relation A: (1, 5), (3, 7), (9, 4), (22, 4), (3, 21) Domain The set of the first coordinates of all the ordered pairs of a relation The set of all input(s) Range The set of the second coordinates of all the ordered pairs of a relation The set of all output(s) A relation represents a correspondence between the elements of one set of quantities (domain) and the elements of another set of quantities (range) Relation B: Domain Range 11 11 19 30 A special type of relation is one in which each input is paired with only one output That is, when you put in one value, you only get out exactly one value This type of relation is called a function in mathematics In determining whether a relation is a function, it is important to consider whether each and every input has exactly one output associated with it In the following example, we will examine which relations are functions and which are not section 1.5 Functions and Function Notation Definition Function A relation in which each input is related to only one output For each input value in the domain, there is one and only one output value in the range Function Domain Range 19 30 11 Example Determining whether sets and word descriptions are functions Determine whether the following descriptions of relations are functions or not Explain your reasoning a The set S {(1, 3), (5, 7), (7, 9), (15, 17)} b The set B {(2, 8), (2, 7), (3, 16), (4, 11)} c d Day 5 7 Height of Plant (in cm) 12 17 Age of Student Grade Level 6 10 2nd 3rd 3rd 1st 1st 4th 4th e The relationship between Monique’s age, in days, and her height f The advertised prices of Sony 32-inch TVs in this Sunday’s newspaper Solution In each part, consider whether for each input value, there is exactly one output a The set S is a function, since each input has exactly one output value b The set B is not a function, since the input has two different output values c This table is a function because each day has one plant height associated with it d In this table, the age of the student could be associated with more than one grade level The 7-year-olds in this table go to either second or third grade, so this relation is not a function e If we consider just one age in days, Monique will have only one height, so this is a function f Sony 32-inch TVs would be advertised for several different prices at different stores, so this is a relation but not a function PRACTICE PROBLEM for example Determine whether the following descriptions of relations are functions or not Explain your reasoning a Units Produced 100 150 200 250 300 350 400 Total Cost (in $) 789 1565 2037 2589 3604 4568 5598 75 76 Chapter 1 Linear Functions b First Name of Student John Mary Mark Fred Juan Karla John Number of Credits 10 12 15 16 21 c The amount of profit that a company makes each month of a year d The population of California each year In Example 1, functions can be represented by sets of data or words Any set of data can be considered a function if it satisfies the requirement that each input has only one output associated with it Functions can also be represented as equations or by using graphs Most of the equations that we work with in this book are functions When given an equation, look for anything that is out of the ordinary, such as a symbol that might result in two answers for any one input One way to get an idea whether an equation represents a function is to pick an arbitrary input and substitute it into the equation to see whether one or more outputs come out If the equation gives only one output, it may be a function We will consider whether there are any values that could be used as inputs that would result in more than one output If we consider the linear equation y 2x 5, any x-value (input) that we substitute into this equation will give us only one y-value (output) For example, x results in y 2(3) y 11 The input x results in the output y 11 If we select any other x-value, it also results in only one output Because each input results in only one output, the equation represents a function An equation such as x y2 does not satisfy the condition that each input have only one output y2 x (22) x (2)2 y 22 y 2 In this example, the input x results in two outputs: y and y 22 Therefore, this equation is not a function Another way to determine whether or not an equation is a function is to look at its graph Consider the following graph y x Every point that lies on this curve represents an input (x) and an output (y) Using this graph, we can examine whether each input (x) has exactly one output (y) associated with it To determine whether an input (x) has only one output (y), select an x-value on section 1.5 Functions and Function Notation the horizontal axis, draw an imaginary vertical line at that location, and see how many times the vertical line crosses the curve y x (1.4, ]2) In this example, we chose the input x 1.4 and drew a vertical line, which crosses the graph in only one place, (1.4, 22) If any vertical line drawn crosses the graph only once, then we have the graph of a function In the graph above, all possible vertical lines cross the graph only once This is the graph of a function y (]3.1, 2.1) (2.8, 1) x (0.9, ]2.6) (]0.3, ]2.9) This process of testing vertical lines through a graph is called the vertical line test This test can be used with equations graphed by hand or on a calculator When using the vertical line test on a calculator, be sure to get a viewing window that shows the overall characteristics of the graph, or we might incorrectly decide that the graph passes the vertical line test Although the graphing calculator is a great tool, it can show us only what we ask it to Definition The vertical line test for a function If any vertical line intersects a graph in at most one point, the graph represents a function An example of the vertical line test proving that a curve is not a function is the following graph y x 77 78 Chapter 1 Linear Functions Drawing an arbitrary vertical line through the graph shows that it intersects the curve more than once Therefore, this is not the graph of a function y (3, 0.2) x (3, ]2.2) Example Determining whether equations and graphs are functions Consider the following equations and graphs and determine whether or not they are functions a P 2.57t 65 Using Your TI Graphing Calculator Entering Equations With a Plus Minus (±) Symbol When an equation has the ± symbol, it cannot be entered into the calculator as one equation For example, in part c of Example 2, the equation y 2x (6x 9) has to be entered into the calculator as two separate equations One equation will use the plus, and the other will use the minus Graphing these two equations gives b W 2g2 5g c y 2x (6x 9) d e. y y x Solution a This is a linear equation, and for each input t, there is a single output P Therefore, this equation represents a function b This equation is not linear, yet it still has only one output W associated with each input value g Therefore, it is a function c This equation has a symbol, which means that almost all inputs will result in more than one output For example, if x 5, Considering the two lines as one graph shows us that this equation will not pass the vertical line test Therefore, the equation is not a function, as demonstrated algebraically in Example y 2(5) (6(5) 9) Substitute x 5 y 10 (30 9) y 10 21 The symbol means that y 10 21 y 10 21 we write two equations y 31 y 211 We get two results This equation gives two outputs for an input, so this equation does not represent a function x section 1.5 Functions and Function Notation d This graph passes the vertical line test, e This graph does not pass the vertical line so it does represent a function test, so it does not represent a function y y (2, 3) (]2, 0.9) x x (2, ]2.6) (]2, ]3.3) PRACTICE PROBLEM FOR EXAMPLE Determine whether or not the following tables, graphs, and equations represent functions Explain your reasoning Connecting the Concepts When working with equations, the variable x is assumed to be the input (independent) variable, and y is assumed to be the output (dependent) variable If other variables are used, their placement in the equation can help us determine which is the input variable and which is the output variable The variable that is isolated on one side of the equation is often considered to be the output (dependent) variable The variable that is more involved in the calculation is considered the input (independent) variable p 3t a Name of Student Gender b Mary Mark Karla Fred Mark Female Male Female Male Male c. y In this equation, since p is isolated, it would be considered the output variable The input variable would be t, since it is part of the calculation for this equation y x x d C 3.59u 1359.56 e H 17.125 !3.5m T his can be entered in the calculator as two equations: one using the plus and the other using the minus Function Notation Function notation was developed as a shorthand way of providing a great deal of information in a very compact form If variables are defined properly with units and clear definitions, then function notation can be used to communicate what we want to with the function and what input and/or output values are being considered Let’s define the following variables: P Population of Hawaii (in millions) t Years since 2000 P(t) Population of Hawaii in millions, t years since 2000 Then the population of Hawaii at time t can be represented by the following function P(t) 0.015t 1.21 P(t) is read “P of t” and represents a function named P that depends on the variable t In real-world applications, the variable in the same position as P represents the output 79 80 Chapter 1 Linear Functions variable, and the variable in the same position as t represents the input variable The variable P, outside the parentheses, is the output, and t, inside the parentheses, is the input There is not much of a difference between this function notation and the equation P 0.015t 1.21 Using function notation, P(t), emphasizes that the population of Hawaii is changing with respect to time Given the variable definitions above, we can make the following statements What’s That Mean? Function Notation The parts of function notation tell us a lot of information about the function input output f(3) = 15 name of function a In words: i Use the given equation to determine what the population of Hawaii was in 2005 ii Use the given equation to determine when the population of Hawaii will be 4 million b Using function notation, these same statements can be written as follows: i Find P(5). ii Find t such that P(t) Using the function notation, allows us to communicate what the input variable or output variable is equal to without words P(5) is asking to substitute for the input variable t and determine the value of the function P(t) P(t) indicates that the output variable P is equal to and directs us to determine the value of the input variable t that results in a population of million people Function notation can be a simple way to communicate information in a short way, but be careful when interpreting the information Be sure to know how the variables are defined and use these definitions as a basis for interpreting any results Example Interpreting function notation in an application Given the following definitions, write sentences interpreting the following mathematical statements G(t) Number of guests at a local beach resort during year t P(b) Profit, in millions of dollars, from the sale of b boats a G(2010) 1600 b P(10) Solution a It is important to consider the location of each number The 2010 is inside the parentheses, so it is the value of t Therefore, 2010 represents the year Because 1600 is what G(t) is equal to, this represents the number of guests at the beach resort The final interpretation might say, In 2010, there were 1600 guests at this local beach resort b The number of boats must be 10, since it is in the parentheses, and must be the profit in millions The profit from the sale of 10 boats is $7 million PRACTICE PROBLEM FOR EXAMPLE Given the following definitions, write sentences interpreting the following mathematical statements C(m) Cost, in hundreds of dollars, for producing m Miracle Mops P(t) Population of Michigan, in millions, t years since 2000 a C(2500) 189 b P(10) 10.4 section 1.5 Functions and Function Notation We can interpret the meaning of the results by referring back to the definitions of each variable Pay close attention to the units involved in each problem If the profit from making boats were measured in dollars and not millions of dollars, the boats probably would not be made The most common letters used to name functions in mathematics are f, g, and h The functions f (x), g(x), and h (x) often replace the output variable y Using different letters helps to distinguish between different equations When the following equations are written in slope-intercept form, we cannot easily distinguish between them y 2x 7 y 24.7x 8.6 y 3x If these equations are written using function notation, it is easier to distinguish between them f (x) 2x 7 g(x) 24.7x 8.6 h(x) 3x To evaluate f (x) at x 4, we write f (4), and the reader will know which function to use and what to substitute in for x f (4) 2(4) f (4) 15 We use the y notation for graphing and solving, and the f (x) notation for evaluating and almost everything else Example Using function notation Let f (x) 23x 5 g(x) 2.5x 9.7 h(x) 4x2 19 Find the following a f (7) b. h(3) c x such that g(x) 12.3 d. x such that f (x) 11 Solution a The input value is Substitute for x and solve f (7) 23(7) f (7) 221 f (7) 216 b Using the function h, substitute for the input variable x h(3) 4(3)2 19 h(3) 4(9) 19 h(3) 36 19 h(3) 17 c We are given that the output value of the function is 12.3, so set g(x) equal to 12.3 and solve for the input variable 12.3 2.5x 9.7 19.7 9.7 22 2.5x 22 2.5x 2.5 2.5 8.8 x g(8.8) 2.5(8.8) 9.7 g(8.8) 12.3 Set g (x) 12.3 on the left Check the answer The answer checks 81 82 Chapter 1 Linear Functions d We are given that the output value of the function is 11, so set f (x) equal to 11 and solve for the input variable 11 23x 25 25 23x 23x 23 23 22 x f (22) 23(22) Set f (x) 11 on the left f (22) 11 The answer checks Check the answer PRACTICE PROBLEM FOR EXAMPLE Let f (x) 24.25x 5.75 g(x) 4x Find the following a f (3) b x such that g(x) 20 Example Using function notation Use the graph to estimate the following y ]5 ]4 ]3 ]2 ]1 ]2 ]4 ]6 f(x) ]8 ]10 x a f (24) b x such that f (x) 22 Solution a 24 is inside the parentheses, so it is an input or x-value, and we want to find the output, or y-value According to the graph, when x 24, the line has a y-value of 26 Therefore, f (24) 26 b The line has an output of 22 when x 21 section 1.5 Functions and Function Notation 83 PRACTICE PROBLEM FOR EXAMPLE Use the graph to estimate the following y ]5 ]4 ]3 ]2 ]1 ]2 ]4 f(x) ]6 ]8 ]10 x a f (4) b x such that f (x) Domain and Range of Functions What’s That Mean? When finding the domain of a function that is given as an equation or graph, start with all real numbers and look for any restrictions that limit the domain The only restriction to the domain of a function is any real number that results in the function being non-real or undefined All real numbers can be expressed using interval notation: (2`, `) When using the graph to determine the domain of a function, read from left to right and consider all input values that are used on the graph If the curve extends forever in each direction and there are no undefined or missing input values, the domain is all real numbers When using the graph to determine the range of a function, look from the bottom to the top of the graph Again, if the graph extends forever down and forever up, the range is all real numbers In some cases, the graph will not extend in both directions forever; this may restrict the domain, range or both The domain will always be written from left to right, and the range will always be written from bottom (low) to top (high) Example Domain and range of functions from a graph Find the domain and range of the given functions a. b. y ]20 ]10 y 25 12 20 10 15 10 ]5 ]10 ]15 ]20 ]25 10 20 x ]4 ]3 ]2 ]1 ]2 ]4 ]6 ]8 ]10 ]12 x Infinity To infinity and beyond! This wellknown saying has helped many people to know the word infinity In mathematics, infinity is a quantity that is unlimited We use the infinity symbol, `, or the negative infinity symbol, 2`, in interval notation to indicate that an interval is unlimited 84 Chapter 1 Linear Functions c 30 25 20 20 15 10 15 10 5 x ]5 ]4 ]3 ]2 ]1 ]5 ]10 ]15 ]20 25 20 Range: all real numbers 15 10 ]5 ]10 10 x 20 Domain To find the domain, look at the x-values from left to right Since the line extends forever to the left and right and includes all x-values, the domain of this function is all real numbers Domain: all real numbers or (2`, `) ]15 ]20 ]25 Range To find the range, we look at the y-values from low to high This line extends down and up forever and includes all y-values, so the range of this function is all real numbers Domain: all real numbers y Range: all real numbers or (2`, `) 12 b Domain The arrowheads on each end of this curve indicates that it extends to the left and right forever The curve also includes all x-values, so the domain of this function is all real numbers Range: y # 10 Highest output value y = 8 ]4 ]3 ]2 ]1 ]2 ]4 ]6 x Domain: all real numbers or (2`, `) Range The arrowheads on this curve indicate that it extends down forever, so the lowest value is negative infinity The curve only goes up to a value of 8, so it has a highest output value of y Therefore, the range of this function is y # ]8 ]10 ]12 Range: y # or (2`, 8] Domain: all real numbers c The solid dots on each end of this curve indicate that the curve does not extend forever but stops at a specific point This will limit both the domain and range of the function y 30 25 Range: ]15 # y # 25 Domain This curve starts on the left at x 24 and stops on the right at x The solid dots indicate that these endpoints are part of the domain, so the domain will be all x-values between and including 24 and 20 15 10 ]5 ]4 ]3 ]2 ]1 ]5 ]10 ]15 x SOLUTION a The arrowheads on each end of this line indicate that the line extends in both directions forever y ]10 y 30 25 ]5 ]4 ]3 ]2 ]1 ]5 ]10 ]15 ]20 ]20 d y ]20 Domain: ] # x # 3 x Domain: 24 # x # or [24, 3] Range The lowest output value on this curve is y 215, and the highest output value is y 25, so the range is all y-values between and including 215 and 25 Range: 215 # y # 25 or [215, 25] ... 29 215 5 23 2 (23) Second and third points (2, 14 ) (11 , 213 ) 213 14 227 5 23 11 2 First and third points (23, 29) (11 , 213 ) 213 229 242 5 23 11 (23) 14 All of the slope calculations are equal to... at a time: 14 20 26 5 23 220 x y (x, y) 23 11 29 14 213 (23, 29) (2, 14 ) (11 , 213 ) (x, y) As one calculation: 14 20 23 220 Slope First and second points (23, 29) (2, 14 ) 14 29 215 5 23 2... for y and find values for x x 22y 10 y (x, y) x 22( 21) 10 12 x 22(0) 10 10 x 22(2) 10 21 (12 , 21) (10 , 0) (6, 2) x 22(4) 10 (2, 4) 21 section 1. 2 Fundamentals of Graphing and Slope Now plot these