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Precalculus 2e
SENIOR CONTRIBUTING AUTHOR
JAY ABRAMSON, ARIZONA STATE UNIVERSITY

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Contents
Preface 1
1 Functions 7
Introduction to Functions 7 1.1 Functions and Function Notation 7 1.2 Domain and Range 34 1.3 Rates of Change and Behavior of Graphs 56 1.4 Composition of Functions 73 1.5 Transformation of Functions 89 1.6 Absolute Value Functions 121 1.7 Inverse Functions 133 Chapter Review 149 Exercises 153
2 Linear Functions 163
Introduction to Linear Functions 163 2.1 Linear Functions 164 2.2 Graphs of Linear Functions 186 2.3 Modeling with Linear Functions 211 2.4 Fitting Linear Models to Data 225 Chapter Review 239 Exercises 240
3 Polynomial and Rational Functions 251
Introduction to Polynomial and Rational Functions 251 3.1 Complex Numbers 252 3.2 Quadratic Functions 262 3.3 Power Functions and Polynomial Functions 283 3.4 Graphs of Polynomial Functions 302 3.5 Dividing Polynomials 325 3.6 Zeros of Polynomial Functions 336 3.7 Rational Functions 350 3.8 Inverses and Radical Functions 376 3.9 Modeling Using Variation 389 Chapter Review 399 Exercises 403
4 Exponential and Logarithmic Functions 409
Introduction to Exponential and Logarithmic Functions 409 4.1 Exponential Functions 410 4.2 Graphs of Exponential Functions 429 4.3 Logarithmic Functions 443 4.4 Graphs of Logarithmic Functions 452 4.5 Logarithmic Properties 474 4.6 Exponential and Logarithmic Equations 486 4.7 Exponential and Logarithmic Models 499 4.8 Fitting Exponential Models to Data 518 Chapter Review 534

Exercises 539
5 Trigonometric Functions 549
Introduction to Trigonometric Functions 549 5.1 Angles 550 5.2 Unit Circle: Sine and Cosine Functions 573 5.3 The Other Trigonometric Functions 594 5.4 Right Triangle Trigonometry 610 Chapter Review 623 Exercises 626
6 Periodic Functions 631
Introduction to Periodic Functions 631 6.1 Graphs of the Sine and Cosine Functions 631 6.2 Graphs of the Other Trigonometric Functions 651 6.3 Inverse Trigonometric Functions 672 Chapter Review 685 Exercises 686
7 Trigonometric Identities and Equations 693
Introduction to Trigonometric Identities and Equations 693 7.1 Solving Trigonometric Equations with Identities 693 7.2 Sum and Difference Identities 704 7.3 Double-Angle, Half-Angle, and Reduction Formulas 719 7.4 Sum-to-Product and Product-to-Sum Formulas 732 7.5 Solving Trigonometric Equations 740 7.6 Modeling with Trigonometric Functions 755 Chapter Review 777 Exercises 781
8 Further Applications of Trigonometry 787
Introduction to Further Applications of Trigonometry 787 8.1 Non-right Triangles: Law of Sines 787 8.2 Non-right Triangles: Law of Cosines 805 8.3 Polar Coordinates 819 8.4 Polar Coordinates: Graphs 833 8.5 Polar Form of Complex Numbers 852 8.6 Parametric Equations 863 8.7 Parametric Equations: Graphs 876 8.8 Vectors 888 Chapter Review 910 Exercises 914
9 Systems of Equations and Inequalities 921
Introduction to Systems of Equations and Inequalities 921 9.1 Systems of Linear Equations: Two Variables 922 9.2 Systems of Linear Equations: Three Variables 941 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables 953
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9.4 Partial Fractions 965 9.5 Matrices and Matrix Operations 975 9.6 Solving Systems with Gaussian Elimination 988 9.7 Solving Systems with Inverses 1002 9.8 Solving Systems with Cramer's Rule 1017 Chapter Review 1030 Exercises 1033

10 Analytic Geometry 1041
Introduction to Analytic Geometry 1041 10.1 The Ellipse 1042 10.2 The Hyperbola 1058 10.3 The Parabola 1075 10.4 Rotation of Axes 1091 10.5 Conic Sections in Polar Coordinates 1106 Chapter Review 1117 Exercises 1119

11 Sequences, Probability and Counting Theory 1125

Introduction to Sequences, Probability and Counting Theory 11.1 Sequences and Their Notations 1125 11.2 Arithmetic Sequences 1140 11.3 Geometric Sequences 1152 11.4 Series and Their Notations 1161 11.5 Counting Principles 1175 11.6 Binomial Theorem 1186 11.7 Probability 1193 Chapter Review 1204 Exercises 1207

1125

12 Introduction to Calculus 1215

Introduction to Calculus 1215 12.1 Finding Limits: Numerical and Graphical Approaches 12.2 Finding Limits: Properties of Limits 1227 12.3 Continuity 1237 12.4 Derivatives 1252 Chapter Review 1275 Exercises 1277

1215

A Basic Functions and Identities 1283 Answer Key 1287 Index 1447

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Preface 1
Preface
About OpenStax
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Format You can access this textbook for free in web view or PDF through openstax.org, and for a low cost in print.
About Precalculus 2e
Precalculus 2e is adaptable and designed to fit the needs of a variety of precalculus courses. It is a comprehensive text that covers more ground than a typical one- or two-semester college-level precalculus course. The content is organized by clearly-defined learning objectives, and includes worked examples that demonstrate problem-solving approaches in an accessible way.
Coverage and Scope Precalculus 2e contains twelve chapters, roughly divided into three groups.
Chapters 1-4 discuss various types of functions, providing a foundation for the remainder of the course.
• Chapter 1: Functions • Chapter 2: Linear Functions • Chapter 3: Polynomial and Rational Functions • Chapter 4: Exponential and Logarithmic Functions
Chapters 5-8 focus on Trigonometry. In Precalculus 2e, we approach trigonometry by first introducing angles and the unit circle, as opposed to the right triangle approach more commonly used in college algebra and trigonometry courses.
• Chapter 5: Trigonometric Functions • Chapter 6: Periodic Functions • Chapter 7: Trigonometric Identities and Equations • Chapter 8: Further Applications of Trigonometry

2 Preface
Chapters 9-12 present some advanced precalculus topics that build on topics introduced in chapters 1-8. Most precalculus syllabi include some of the topics in these chapters, but few include all. Instructors can select material as needed from this group of chapters, since they are not cumulative.
• Chapter 9: Systems of Equations and Inequalities • Chapter 10: Analytic Geometry • Chapter 11: Sequences, Probability and Counting Theory • Chapter 12: Introduction to Calculus
All chapters are broken down into multiple sections, the titles of which can be viewed in the Table of Contents.
Development Overview Precalculus 2e is the product of a collaborative effort by a group of dedicated authors, editors, and instructors whose collective passion for this project has resulted in a text that is remarkably unified in purpose and voice. Special thanks is due to our Lead Author, Jay Abramson of Arizona State University, who provided the overall vision for the book and oversaw the development of each and every chapter, drawing up the initial blueprint, reading numerous drafts, and assimilating field reviews into actionable revision plans for our authors and editors.
The first eight chapters are built on the foundation of Precalculus: An Investigation of Functions by David Lippman and Melonie Rasmussen. Chapters 9-12 were written and developed from by our expert and highly experienced author team. All twelve chapters follow a new and innovative instructional design, and great care has been taken to maintain a consistent voice from cover to cover. New features have been introduced to flesh out the instruction, all of the graphics have been redone in a more contemporary style, and much of the content has been revised, replaced, or supplemented to bring the text more in line with mainstream approaches to teaching precalculus.
Accuracy of the Content We understand that precision and accuracy are imperatives in mathematics, and undertook an dedicated accuracy program led by experienced faculty. Examples, art, problems, and solutions were reviewed by dedicated faculty, with a separate team evaluating the answer key and solutions.
The text also benefits from years of usage by thousands of faculty and students. A core aspect of the second edition revision process included consolidating and ensuring consistency with regard to any errata and corrections that have been implemented during the series' extensive usage and incorporation into homework systems.
Changes to the Second Edition The Precalculus 2e revision focused on mathematical clarity and accuracy as well as inclusivity. Examples, Exercises, and Solutions were reviewed by multiple faculty experts. All improvement suggestions and errata updates, driven by faculty and students from several thousand colleges, were considered and unified across the different formats of the text.
OpenStax and our authors are aware of the difficulties posed by shifting problem and exercise numbers when textbooks are revised. In an effort to make the transition to the 2nd edition as seamless as possible, we have minimized any shifting of exercise numbers.
The revision also focused on supporting inclusive and welcoming learning experiences. The introductory narratives, example and problem contexts, and even many of the names used for fictional people in the text were all reviewed using a diversity, equity, and inclusion framework. Several hundred resulting revisions improve the balance and relevance to the students using the text, while maintaining a variety of applications to diverse careers and academic fields. In particular, explanations of scientific and historical aspects of mathematics have been expanded to include more contributors. For example, the authors added additional historical and multicultural context regarding what is widely known as Pascal’s Triangle, and similarly added details regarding the international process of decoding the Enigma machine (including the role of Polish college students). Several chapter-opening narratives and in-chapter references are completely new, and contexts across all chapters were specifically reviewed for equity in gender representation and connotation.
Pedagogical Foundations and Features
Learning Objectives Each chapter is divided into multiple sections (or modules), each of which is organized around a set of learning objectives. The learning objectives are listed explicitly at the beginning of each section and are the focal point of every instructional element.
Narrative Text Narrative text is used to introduce key concepts, terms, and definitions, to provide real-world context, and to provide transitions between topics and examples. Throughout this book, we rely on a few basic conventions to highlight the
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Preface 3
most important ideas: • Key terms are boldfaced, typically when first introduced and/or when formally defined. • Key concepts and definitions are called out in a blue box for easy reference.
Examples Each learning objective is supported by one or more worked examples that demonstrate the problem-solving approaches that students must master. Typically, we include multiple Examples for each learning objective in order to model different approaches to the same type of problem, or to introduce similar problems of increasing complexity. All told, there are more than 650 Examples, or an average of about 55 per chapter. All Examples follow a simple two- or three-part format. First, we pose a problem or question. Next, we demonstrate the Solution, spelling out the steps along the way. Finally (for select Examples), we conclude with an Analysis reflecting on the broader implications of the Solution just shown. Figures Precalculus 2e contains more than 2000 figures and illustrations, the vast majority of which are graphs and diagrams. Art throughout the text adheres to a clear, understated style, drawing the eye to the most important information in each figure while minimizing visual distractions. Color contrast is employed with discretion to distinguish between the different functions or features of a graph.
Supporting Features Several elements, each marked by a distinctive icon, serve to support Examples.
• A How To is a list of steps necessary to solve a certain type of problem. A How To typically precedes an Example that proceeds to demonstrate the steps in action.
• A Try It exercise immediately follows an Example or a set of related Examples, providing the student with an immediate opportunity to solve a similar problem. In the PDF and the Web View version of the text, answers to the Try It exercises are located in the Answer Key.
• A Q&A may appear at any point in the narrative, but most often follows an Example. This feature pre-empts misconceptions by posing a commonly asked yes/no question, followed by a detailed answer and explanation.
• The Media icon appears at the conclusion of each section, just prior to the Section Exercises. This icon marks a list of links to online video tutorials that reinforce the concepts and skills introduced in the section.
While we have selected tutorials that closely align to our learning objectives, we did not produce these tutorials, nor were they specifically produced or tailored to accompany Precalculus 2e. Section Exercises Each section of every chapter concludes with a well-rounded set of exercises that can be assigned as homework or used selectively for guided practice. With over 5900 exercises across the 12 chapters, instructors should have plenty from which to choose.

4 Preface
Section Exercises are organized by question type, and generally appear in the following order:
• Verbal questions assess conceptual understanding of key terms and concepts. • Algebraic problems require students to apply algebraic manipulations demonstrated in the section. • Graphical problems assess students’ ability to interpret or produce a graph. • Numeric problems require the student to perform calculations or computations. • Technology problems encourage exploration through use of a graphing utility, either to visualize or verify algebraic
results or to solve problems via an alternative to the methods demonstrated in the section. • Extensions pose problems more challenging than the Examples demonstrated in the section. They require students
to synthesize multiple learning objectives or apply critical thinking to solve complex problems. • Real-World Applications present realistic problem scenarios from fields such as physics, geology, biology, finance,
and the social sciences.
Chapter Review Features Each chapter concludes with a review of the most important takeaways, as well as additional practice problems that students can use to prepare for exams.
• Key Terms provides a formal definition for each bold-faced term in the chapter. • Key Equations presents a compilation of formulas, theorems, and standard-form equations. • Key Concepts summarizes the most important ideas introduced in each section, linking back to the relevant
Example(s) in case students need to review. • Chapter Review Exercises include 40-80 practice problems that recall the most important concepts from each
section. • Practice Test includes 25-50 problems assessing the most important learning objectives from the chapter. Note
that the practice test is not organized by section, and may be more heavily weighted toward cumulative objectives as opposed to the foundational objectives covered in the opening sections.
Additional Resources
Student and Instructor Resources
We’ve compiled additional resources for both students and instructors, including Getting Started Guides, instructor solution manual, and PowerPoint slides. Instructor resources require a verified instructor account, which can be requested on your openstax.org log-in. Take advantage of these resources to supplement your OpenStax book.
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Technology partners
As allies in making high-quality learning materials accessible, our technology partners offer optional low-cost tools that are integrated with OpenStax books. To access the technology options for your text, visit your book page on openstax.org.
About the Authors
Senior Contributing Author
Jay Abramson, Arizona State University Jay Abramson has been teaching Precalculus for over 35 years, the last 20 at Arizona State University, where he is a principal lecturer in the School of Mathematics and Statistics. His accomplishments at ASU include co-developing the university’s first hybrid and online math courses as well as an extensive library of video lectures and tutorials. In addition, he has served as a contributing author for two of Pearson Education’s math programs, NovaNet Precalculus and Trigonometry. Prior to coming to ASU, Jay taught at Texas State Technical College and Amarillo College. He received Teacher of the Year awards at both institutions.
Contributing Authors
Valeree Falduto, Palm Beach State College Rachael Gross, Towson University David Lippman, Pierce College
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Melonie Rasmussen, Pierce College Rick Norwood, East Tennessee State University Nicholas Belloit, Florida State College Jacksonville Jean-Marie Magnier, Springfield Technical Community College Harold Whipple Christina Fernandez
Faculty Reviewers and Consultants
Nina Alketa, Cecil College Kiran Bhutani, Catholic University of America Brandie Biddy, Cecil College Lisa Blank, Lyme Central School Bryan Blount, Kentucky Wesleyan College Jessica Bolz, The Bryn Mawr School Sheri Boyd, Rollins College Sarah Brewer, Alabama School of Math and Science Charles Buckley, St. Gregory's University Michael Cohen, Hofstra University Kenneth Crane, Texarkana College Rachel Cywinski, Alamo Colleges Nathan Czuba Srabasti Dutta, Ashford University Kristy Erickson, Cecil College Nicole Fernandez, Georgetown University / Kent State University David French, Tidewater Community College Douglas Furman, SUNY Ulster Lance Hemlow, Raritan Valley Community College Erinn Izzo, Nicaragua Christian Academy John Jaffe Jerry Jared, Blue Ridge School Stan Kopec, Mount Wachusett Community College Kathy Kovacs Cynthia Landrigan, Erie Community College Sara Lenhart, Christopher Newport University Wendy Lightheart, Lane Community College Joanne Manville, Bunker Hill Community College Karla McCavit, Albion College Cynthia McGinnis, Northwest Florida State College Lana Neal, University of Texas at Austin Rhonda Porter, Albany State University Steven Purtee, Valencia College William Radulovich, Florida State College Jacksonville Alice Ramos, Bethel College Nick Reynolds, Montgomery Community College Amanda Ross, A. A. Ross Consulting and Research, LLC Erica Rutter, Arizona State University Sutandra Sarkar, Georgia State University Willy Schild, Wentworth Institute of Technology Todd Stephen, Cleveland State University Scott Sykes, University of West Georgia Linda Tansil, Southeast Missouri State University John Thomas, College of Lake County Diane Valade, Piedmont Virginia Community College Allen Wolmer, Atlanta Jewish Academy

Preface 5

6 Preface Access for free at openstax.org

1 • Introduction to Functions 7

1

FUNCTIONS

Standard and Poor’s Index with dividends reinvested (credit "bull": modification of work by Prayitno Hadinata; credit "graph": modification of work by MeasuringWorth)
Chapter Outline
1.1 Functions and Function Notation 1.2 Domain and Range 1.3 Rates of Change and Behavior of Graphs 1.4 Composition of Functions 1.5 Transformation of Functions 1.6 Absolute Value Functions 1.7 Inverse Functions
Introduction to Functions
Toward the end of the twentieth century, the values of stocks of internet and technology companies rose dramatically. As a result, the Standard and Poor’s stock market average rose as well. The graph above tracks the value of that initial investment of just under $100 over the 40 years. It shows that an investment that was worth less than $500 until about 1995 skyrocketed up to about $1,100 by the beginning of 2000. That five-year period became known as the “dot-com bubble” because so many internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually burst. Many companies grew too fast and then suddenly went out of business. The result caused the sharp decline represented on the graph beginning at the end of 2000.
Notice, as we consider this example, that there is a definite relationship between the year and stock market average. For any year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will explore these kinds of relationships and their properties.
1.1 Functions and Function Notation
Learning Objectives
In this section, you will: Determine whether a relation represents a function. Find the value of a function. Determine whether a function is one-to-one. Use the vertical line test to identify functions. Graph the functions listed in the library of functions.
A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child

8 1 • Functions

increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.
Determining Whether a Relation Represents a Function
A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

The domain is

The range is

Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter

A function is a relation that assigns a single value in the range to each value in the domain. In other words, no x-values

are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is

a function because each element in the domain,

is paired with exactly one element in the range,

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

Notice that each element in the domain,

is not paired with exactly one element in the range,

For example, the term “odd” corresponds to three values from the range,

and the term

“even” corresponds to two values from the range,

This violates the definition of a function, so this relation is not

a function.

Figure 1 compares relations that are functions and not functions.

Figure 1 (a) This relationship is a function because each input is associated with a single output. Note that input and both give output (b) This relationship is also a function. In this case, each input is associated with a single output. (c) This relationship is not a function because input is associated with two different outputs.
Function
A function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.” The input values make up the domain, and the output values make up the range.
...
HOW TO
Given a relationship between two quantities, determine whether the relationship is a function. 1. Identify the input values. 2. Identify the output values. 3. If each input value leads to only one output value, classify the relationship as a function. If any input value leads
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to two or more outputs, do not classify the relationship as a function.
EXAMPLE 1 Determining If Menu Price Lists Are Functions The coffee shop menu, shown in Figure 2 consists of items and their prices. a. ⓐ Is price a function of the item? b. ⓑ Is the item a function of the price?

1.1 • Functions and Function Notation 9

Figure 2
Solution
ⓐ Let’s begin by considering the input as the items on the menu. The output values are then the prices. See Figure
3.

Figure 3 Each item on the menu has only one price, so the price is a function of the item.
ⓑ Two items on the menu have the same price. If we consider the prices to be the input values and the items to be
the output, then the same input value could have more than one output associated with it. See Figure 4.

Therefore, the item is a not a function of price.

Figure 4

10 1 • Functions

EXAMPLE 2
Determining If Class Grade Rules Are Functions In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? Table 1 shows a possible rule for assigning grade points.

Percent grade

0–56 57–61 62–66 67–71 72–77 78–86 87–91 92–100

Grade point average 0.0

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Table 1

Solution For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.
In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.

TRY IT #1 Table 21 lists the five greatest baseball players of all time in order of rank.

Player

Rank

Babe Ruth

1

Willie Mays

2

Ty Cobb

3

Walter Johnson 4

Hank Aaron

5

Table 2

ⓐ Is the rank a function of the player name? ⓑ Is the player name a function of the rank?

Using Function Notation
Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables for height and for age. The

letters

and are often used to represent functions just as we use

and to represent numbers and

and to represent sets.

Remember, we can use any letter to name the function; the notation
1 http://www.baseball-almanac.com/legendary/lisn100.shtml. Accessed 3/24/2014.

shows us that depends on The value

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1.1 • Functions and Function Notation 11

must be put into the function to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example

means “first add a and b, and

the result is the input for the function f.” The operations must be performed in this order to obtain the correct result.

Function Notation

The notation

defines a function named This is read as is a function of The letter represents the

input value, or independent variable. The letter or

represents the output value, or dependent variable.

EXAMPLE 3

Using Function Notation for Days in a Month Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. Assume that the domain does not include leap years.

Solution

The number of days in a month is a function of the name of the month, so if we name the function we write

or

The name of the month is the input to a “rule” that associates a specific number (the

output) with each input.

Figure 5

For example,

because March has 31 days. The notation

(the output), is dependent on the name of the month, (the input).

reminds us that the number of days,

Analysis Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.

EXAMPLE 4

Interpreting Function Notation

A function

gives the number of police officers,

in a town in year What does

represent?

Solution When we read
is 300. Remember, officers in the town.

we see that the input year is 2005. The value for the output, the number of police officers

The statement

tells us that in the year 2005 there were 300 police

TRY IT #2 Use function notation to express the weight of a pig in pounds as a function of its age in days

Q&A

Instead of a notation such as

could we use the same symbol for the output as for the

function, such as

meaning “y is a function of x?”

Yes, this is often done, especially in applied subjects that use higher math, such as physics and

engineering. However, in exploring math itself we like to maintain a distinction between a function such

as which is a rule or procedure, and the output we get by applying to a particular input This is

why we usually use notation such as

and so on.

12 1 • Functions

Representing Functions Using Tables
A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.

Table 3 lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number

of days in that month. This information represents all we know about the months and days for a given year (that is not a

leap year). Note that, in this table, we define a days-in-a-month function where

identifies months by an

integer rather than by name.

Month number, (input) 1 2 3 4 5 6 7 8 9 10 11 12

Days in month, (output) 31 28 31 30 31 30 31 31 30 31 30 31

Table 3

Table 4 defines a function input and gives the output

Remember, this notation tells us that is the name of the function that takes the

12345

86768

Table 4

Table 5 displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.

Age in years, (input)

5 5 6 7 8 9 10

Height in inches, (output) 40 42 44 47 50 52 54

Table 5
...
HOW TO

Given a table of input and output values, determine whether the table represents a function.
1. Identify the input and output values. 2. Check to see if each input value is paired with only one output value. If so, the table represents a function.

EXAMPLE 5
Identifying Tables that Represent Functions Which table, Table 6, Table 7, or Table 8, represents a function (if any)?
Input Output

2

1

5

3

Table 6

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1.1 • Functions and Function Notation 13

Input Output

8

6

Table 6

Input Output

–3

5

0

1

4

5

Table 7

Input Output

1

0

5

2

5

4

Table 8

Solution Table 6 and Table 7 define functions. In both, each input value corresponds to exactly one output value. Table 8 does not define a function because the input value of 5 corresponds to two different output values.
When a table represents a function, corresponding input and output values can also be specified using function notation.
The function represented by Table 6 can be represented by writing

Similarly, the statements

represent the function in Table 7. Table 8 cannot be expressed in a similar way because it does not represent a function.

TRY IT #3 Does Table 9 represent a function?

Input Output

1

10

2

100

3

1000

Table 9

14 1 • Functions

Finding Input and Output Values of a Function

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.

When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

Evaluation of Functions in Algebraic Forms

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the

function

can be evaluated by squaring the input value, multiplying by 3, and then subtracting the

product from 5.

...
HOW TO

Given the formula for a function, evaluate.
1. Substitute the input variable in the formula with the value provided. 2. Calculate the result.

EXAMPLE 6

Evaluating Functions at Specific Values

Evaluate

at:

ⓐⓑⓒ Solution

ⓓ Now evaluate

Replace the in the function with each specified value.

ⓐ Because the input value is a number, 2, we can use simple algebra to simplify.

ⓑ In this case, the input value is a letter so we cannot simplify the answer any further.

ⓒ With an input value of

we must use the distributive property.

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1.1 • Functions and Function Notation 15
ⓓ In this case, we apply the input values to the function more than once, and then perform algebraic operations on
the result. We already found that and we know that Now we combine the results and simplify.

EXAMPLE 7

Evaluating Functions Given the function

evaluate

Solution To evaluate

we substitute the value 4 for the input variable in the given function.

Therefore, for an input of 4, we have an output of 24.

TRY IT #4 Given the function

EXAMPLE 8
Solving Functions Given the function
Solution

solve for

evaluate

If

either

or

(or both of them equal 0). We will set each factor equal to 0 and

solve for in each case.

This gives us two solutions. The output as in Figure 6. The graph verifies that

when the input is either

or

and

We can also verify by graphing

16 1 • Functions

Figure 6

TRY IT #5 Given the function

solve

Evaluating Functions Expressed in Formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express

the function output with a formula involving the input quantity, then we can define a function in algebraic form. For

example, the equation

expresses a functional relationship between and We can rewrite it to decide if

is a function of

...
HOW TO

Given a function in equation form, write its algebraic formula.
1. Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves only the input variable.
2. Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.

EXAMPLE 9

Finding an Equation of a Function

Express the relationship

as a function

if possible.

Solution To express the relationship in this form, we need to be able to write the relationship where is a function of means writing it as

which

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Therefore, as a function of is written as

1.1 • Functions and Function Notation 17

Analysis It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.

EXAMPLE 10

Expressing the Equation of a Circle as a Function

Does the equation

represent a function with

function

as input and

as output? If so, express the relationship as a

Solution First we subtract from both sides.

We now try to solve for in this equation.

We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function

TRY IT #6 If

express as a function of

Q&A

Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?

Yes, this can happen. For example, given the equation

if we want to express as a function of

there is no simple algebraic formula involving only that equals However, each does determine a

unique value for and there are mathematical procedures by which can be found to any desired

accuracy. In this case, we say that the equation gives an implicit (implied) rule for as a function of

even though the formula cannot be written explicitly.

Evaluating a Function Given in Tabular Form
As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.
The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See Table 10.2

Pet

Memory span in hours

Puppy

0.008

Table 10

2 http://www.kgbanswers.com/how-long-is-a-dogs-memory-span/4221590. Accessed 3/24/2014.

18 1 • Functions

Pet

Memory span in hours

Adult dog

0.083

Cat

16

Goldfish

2160

Beta fish

3600

Table 10
At times, evaluating a function in table form may be more useful than using equations. Here let us call the function The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function at the input value of “goldfish.” We would write Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function seems ideally suited to this function, more so than writing it in paragraph or function form.
...
HOW TO

Given a function represented by a table, identify specific output and input values.
1. Find the given input in the row (or column) of input values. 2. Identify the corresponding output value paired with that input value. 3. Find the given output values in the row (or column) of output values, noting every time that output value
appears. 4. Identify the input value(s) corresponding to the given output value.

EXAMPLE 11

Evaluating and Solving a Tabular Function Using Table 11,

ⓐ Evaluate

ⓑ Solve

12345

86768

Table 11

Solution

ⓐ Evaluating

means determining the output value of the function for the input value of

output value corresponding to

is 7, so

The table

ⓑ Solving

means identifying the input values, that produce an output value of 6. The table below shows

two solutions: and

12345

86768

When we input 2 into the function our output is 6. When we input 4 into the function our output is also 6.

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1.1 • Functions and Function Notation 19

TRY IT #7 Using the table from Evaluating and Solving a Tabular Function above, evaluate

Finding Function Values from a Graph
Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

EXAMPLE 12

Reading Function Values from a Graph Given the graph in Figure 7,

ⓐ Evaluate

ⓑ Solve

Solution
ⓐ To evaluate
has coordinates

Figure 7

locate the point on the curve where

so

See Figure 8.

then read the y-coordinate of that point. The point

Figure 8

20 1 • Functions

ⓑ To solve

we find the output value

locate two points of the curve with output value

or This means

and

on the vertical axis. Moving horizontally along the line

we

and

These points represent the two solutions to

or when the input is or the output is See Figure 9.

Figure 9

TRY IT #8 Using Figure 7, solve

Determining Whether a Function is One-to-One
Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in the figure at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.
However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in Table 12.

Letter grade Grade point average

A

4.0

B

3.0

C

2.0

D

1.0

Table 12

This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.
To visualize this concept, let’s look again at the two simple functions sketched in Figure 1(a) and Figure 1(b). The function in part (a) shows a relationship that is not a one-to-one function because inputs and both give output The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

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1.1 • Functions and Function Notation 21

One-to-One Function A one-to-one function is a function in which each output value corresponds to exactly one input value.

EXAMPLE 13

Determining Whether a Relationship Is a One-to-One Function Is the area of a circle a function of its radius? If yes, is the function one-to-one?

Solution A circle of radius has a unique area measure given by is a function of radius

so for any input, there is only one output, The area

If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area

measure is given by the formula

Because areas and radii are positive numbers, there is exactly one solution:

So the area of a circle is a one-to-one function of the circle’s radius.

TRY IT #9

ⓐ Is a balance a function of the bank account number? ⓑ Is a bank account number a function of the balance? ⓒ Is a balance a one-to-one function of the bank account number?

TRY IT #10

Evaluate the following:
ⓐ If each percent grade earned in a course translates to one letter grade, is the letter grade a
function of the percent grade?
ⓑ If so, is the function one-to-one?

Using the Vertical Line Test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.

The most common graphs name the input value and the output value and we say is a function of or

when the function is named The graph of the function is the set of all points

in the plane that satisfies the

equation

If the function is defined for only a few input values, then the graph of the function is only a few

points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding

output value. For example, the black dots on the graph in Figure 10 tell us that

and

However, the set

of all points

satisfying

is a curve. The curve shown includes

and

because the curve passes

through those points.

22 1 • Functions

Figure 10
The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See Figure 11.

...
HOW TO

Figure 11

Given a graph, use the vertical line test to determine if the graph represents a function.
1. Inspect the graph to see if any vertical line drawn would intersect the curve more than once. 2. If there is any such line, determine that the graph does not represent a function.

EXAMPLE 14
Applying the Vertical Line Test Which of the graphs in Figure 12 represent(s) a function

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1.1 • Functions and Function Notation 23
Figure 12 Solution If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of Figure 12. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most x-values, a vertical line would intersect the graph at more than one point, as shown in Figure 13.
Figure 13 TRY IT #11 Does the graph in Figure 14 represent a function?

24 1 • Functions
Figure 14
Using the Horizontal Line Test
Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.
...
HOW TO Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function. 1. Inspect the graph to see if any horizontal line drawn would intersect the curve more than once. 2. If there is any such line, determine that the function is not one-to-one. EXAMPLE 15 Applying the Horizontal Line Test Consider the functions shown in Figure 12(a) and Figure 12(b). Are either of the functions one-to-one?
Solution The function in Figure 12(a) is not one-to-one. The horizontal line shown in Figure 15 intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)
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1.1 • Functions and Function Notation 25

Figure 15 The function in Figure 12(b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.

TRY IT #12 Is the graph shown in Figure 12 one-to-one?

Identifying Basic Toolkit Functions

In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic

formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do

arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-

block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the

graph, formula, and special properties. Some of these functions are programmed to individual buttons on many

calculators. For these definitions we will use as the input variable and

as the output variable.

We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in Table 13.

26 1 • Functions
Name Constant

Toolkit Functions
Function
where is a constant

Graph

Identity

Absolute value

Table 13
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Name Quadratic

Function

Toolkit Functions

1.1 • Functions and Function Notation 27
Graph

Cubic

Reciprocal

Table 13

28 1 • Functions
Name Reciprocal squared

Function

Toolkit Functions

Graph

Square root

Cube root

Table 13
MEDIA Access the following online resources for additional instruction and practice with functions. Determine if a Relation is a Function (http://openstax.org/l/relationfunction) Vertical Line Test (http://openstax.org/l/vertlinetest) Introduction to Functions (http://openstax.org/l/introtofunction) Vertical Line Test on Graph (http://openstax.org/l/vertlinegraph)
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1.1 • Functions and Function Notation 29

One-to-one Functions (http://openstax.org/l/onetoone) Graphs as One-to-one Functions (http://openstax.org/l/graphonetoone)

1.1 SECTION EXERCISES
Verbal

1. What is the difference between a relation and a function?

2. What is the difference between the input and the output of a function?

3. Why does the vertical line test tell us whether the graph of a relation represents a function?

4. How can you determine if a relation is a one-to-one function?

5. Why does the horizontal line test tell us whether the graph of a function is oneto-one?

Algebraic

For the following exercises, determine whether the relation represents a function.

6.

7.

For the following exercises, determine whether the relation represents as a function of

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

For the following exercises, evaluate at the indicated values

27.

28.

30.

31.

29.
32. Given the function evaluate

30 1 • Functions

33. Given the function evaluate
36. Given the function

34. Given the function
ⓐ Evaluate ⓑ Solve
37. Given the function

ⓐ Evaluate ⓑ Solve
39. Consider the relationship

ⓐ Evaluate ⓑ Solve

ⓐ Write the relationship
as a function
ⓑ Evaluate ⓒ Solve

35. Given the function
ⓐ Evaluate ⓑ Solve
38. Given the function
ⓐ Evaluate ⓑ Solve

Graphical

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

40.

41.

42.

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1.1 • Functions and Function Notation 31

43.

44.

45.

46.

47.

48.

49.

50.

51.

32 1 • Functions
52. Given the following graph,
ⓐ Evaluate ⓑ Solve for

53. Given the following graph,
ⓐ Evaluate ⓑ Solve for

54. Given the following graph,
ⓐ Evaluate ⓑ Solve for

For the following exercises, determine if the given graph is a one-to-one function.

55.

56.

57.

58.

59.

Numeric

For the following exercises, determine whether the relation represents a function.

60.

61.

62.

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1.1 • Functions and Function Notation 33

For the following exercises, determine if the relation represented in table form represents as a function of

63. 5 10 15

64. 5 10 15

65. 5 10 10

3 8 14

38 8

3 8 14

For the following exercises, use the function represented in Table 14.

x 0 123 456 7 8 9

f(x) 74 28 1 53 56 3 36 45 14 47

Table 14

66. Evaluate

67. Solve

For the following exercises, evaluate the function at the values

and

68.

69.

70.

71.

72.

73.

For the following exercises, evaluate the expressions, given functions

and

•

•

•

74.

75.

Technology

For the following exercises, graph

on the given domain. Determine the corresponding range. Show each graph.

76.

77.

78.

For the following exercises, graph

79.

80.

on the given domain. Determine the corresponding range. Show each graph. 81.

For the following exercises, graph

82.

83.

on the given domain. Determine the corresponding range. Show each graph. 84.

For the following exercises, graph

85.

86.

on the given domain. Determine the corresponding range. Show each graph. 87.

34 1 • Functions

Real-World Applications

88. The amount of garbage, produced by a city

with population is given by

is

measured in tons per week, and is measured in

thousands of people.

ⓐ The town of Tola has a population of 40,000
and produces 13 tons of garbage each week. Express this information in terms of the function

ⓑ Explain the meaning of the statement

89. The number of cubic yards of dirt, needed to cover a garden with area square feet is given by
ⓐ A garden with area 5000 ft2 requires 50 yd3 of
dirt. Express this information in terms of the function
ⓑ Explain the meaning of the statement

90. Let

be the number of

ducks in a lake years after

1990. Explain the meaning

of each statement:

ⓐ ⓑ

91. Let be the height above ground, in feet, of a rocket seconds after launching. Explain the meaning of each statement:

92. Show that the function is
not one-to-one.

1.2 Domain and Range
Learning Objectives
In this section, you will: Find the domain of a function defined by an equation. Graph piecewise-defined functions.
Horror and thriller movies are both popular and, very often, extremely profitable. When big-budget actors, shooting locations, and special effects are included, however, studios count on even more viewership to be successful. Consider five major thriller/horror entries from the early 2000s—I am Legend, Hannibal, The Ring, The Grudge, and The Conjuring. Figure 1 shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section, we will investigate methods for determining the domain and range of functions such as these.

Figure 1 Based on data compiled by www.the-numbers.com.3
3 The Numbers: Where Data and the Movie Business Meet. “Box Office History for Horror Movies.” http://www.the-numbers.com/market/ Access for free at openstax.org

1.2 • Domain and Range 35
Finding the Domain of a Function Defined by an Equation
In Functions and Function Notation, we were introduced to the concepts of domain and range. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.
We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See Figure 2.

Figure 2

We can write the domain and range in interval notation, which uses values within brackets to describe a set of

numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to

indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to

spend, they would need to express the interval that is more than 0 and less than or equal to 100 and write

We

will discuss interval notation in greater detail later.

Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an odd root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.

Before we begin, let us review the conventions of interval notation:

• The smallest term from the interval is written first. • The largest term in the interval is written second, following a comma. • Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive. • Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.

See Figure 3 for a summary of interval notation.

genre/Horror. Accessed 3/24/2014

36 1 • Functions

Figure 3

EXAMPLE 1

Finding the Domain of a Function as a Set of Ordered Pairs

Find the domain of the following function:

.

Solution First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.

TRY IT #1 Find the domain of the function:

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...
HOW TO
Given a function written in equation form, find the domain.
1. Identify the input values. 2. Identify any restrictions on the input and exclude those values from the domain. 3. Write the domain in interval form, if possible.

1.2 • Domain and Range 37

EXAMPLE 2
Finding the Domain of a Function Find the domain of the function
Solution The input value, shown by the variable in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.
In interval form, the domain of is ∞ ∞

TRY IT #2 Find the domain of the function:
...
HOW TO
Given a function written in an equation form that includes a fraction, find the domain. 1. Identify the input values. 2. Identify any restrictions on the input. If there is a denominator in the function’s formula, set the denominator
equal to zero and solve for . If the function’s formula contains an even root, set the radicand greater than or equal to 0, and then solve. 3. Write the domain in interval form, making sure to exclude any restricted values from the domain.

EXAMPLE 3
Finding the Domain of a Function Involving a Denominator Find the domain of the function
Solution When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for

Now, we will exclude 2 from the domain. The answers are all real numbers where

or

known as the union, to combine the two sets. In interval notation, we write the solution:

We can use a symbol
∞

38 1 • Functions

In interval form, the domain of is ∞

Figure 4
∞

TRY IT #3 Find the domain of the function:
...
HOW TO
Given a function written in equation form including an even root, find the domain. 1. Identify the input values. 2. Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the
radicand greater than or equal to zero and solve for 3. The solution(s) are the domain of the function. If possible, write the answer in interval form.

EXAMPLE 4
Finding the Domain of a Function with an Even Root Find the domain of the function
Solution When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for

Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal
to or ∞

TRY IT #4 Find the domain of the function

Q&A

Can there be functions in which the domain and range do not intersect at all?

Yes. For example, the function

has the set of all positive real numbers as its domain but the

set of all negative real numbers as its range. As a more extreme example, a function’s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.

Using Notations to Specify Domain and Range
In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use

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1.2 • Domain and Range 39

inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-

builder notation. For example,

describes the behavior of in set-builder notation. The braces are

read as “the set of,” and the vertical bar | is read as “such that,” so we would read

as “the set of

x-values such that 10 is less than or equal to and is less than 30.”

Figure 5 compares inequality notation, set-builder notation, and interval notation.

Figure 5

To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier

examples, we use the union symbol, to combine two unconnected intervals. For example, the union of the sets

and

is the set

It is the set of all elements that belong to one or the other (or both) of the

original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending

order of numerical value. If the original two sets have some elements in common, those elements should be listed only

once in the union set. For sets of real numbers on intervals, another example of a union is

∞

∞

Set-Builder Notation and Interval Notation
Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form which is read as, “the set of all such that the statement about is true.” For example,

Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,

40 1 • Functions
...
HOW TO
Given a line graph, describe the set of values using interval notation. 1. Identify the intervals to be included in the set by determining where the heavy line overlays the real line. 2. At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each
excluded end value (open dot). 3. At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each
excluded end value (open dot). 4. Use the union symbol to combine all intervals into one set.
EXAMPLE 5
Describing Sets on the Real-Number Line Describe the intervals of values shown in Figure 6 using inequality notation, set-builder notation, and interval notation.

Figure 6

Solution To describe the values, included in the intervals shown, we would say, “ and less than or equal to 3, or a real number greater than 5.”

is a real number greater than or equal to 1

Inequality

Set-builder notation

Interval notation

∞

Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.

TRY IT #5

Given Figure 7, specify the graphed set in

ⓐ ⓑ ⓒ words

set-builder notation

interval notation

Figure 7
Finding Domain and Range from Graphs
Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. The range is the set of possible output values, which are shown on the y-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See Figure 8.

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1.2 • Domain and Range 41

Figure 8

We can observe that the graph extends horizontally from to the right without bound, so the domain is

∞ The

vertical extent of the graph is all range values and below, so the range is

Note that the domain and range are

always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the

top of the graph for range.

EXAMPLE 6

Finding Domain and Range from a Graph Find the domain and range of the function whose graph is shown in Figure 9.

Figure 9

42 1 • Functions

Solution We can observe that the horizontal extent of the graph is –3 to 1, so the domain of is

The vertical extent of the graph is 0 to –4, so the range is

See Figure 10.

Figure 10
EXAMPLE 7 Finding Domain and Range from a Graph of Oil Production Find the domain and range of the function whose graph is shown in Figure 11.

Figure 11 (credit: modification of work by the U.S. Energy Information Administration)4

Solution

The input quantity along the horizontal axis is “years,” which we represent with the variable for time. The output

quantity is “thousands of barrels of oil per day,” which we represent with the variable for barrels. The graph may

continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can

determine the domain as

and the range as approximately

In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

TRY IT #6 Given Figure 12, identify the domain and range using interval notation.

4 http://www.eia.gov/dnav/pet/hist/LeafHandler.ashx?n=PET&s=MCRFPAK2&f=A. Access for free at openstax.org

1.2 • Domain and Range 43

Figure 12

Q&A

Can a function’s domain and range be the same? Yes. For example, the domain and range of the cube root function are both the set of all real numbers.

Finding Domains and Ranges of the Toolkit Functions
We will now return to our set of toolkit functions to determine the domain and range of each.

Figure 13 For the constant function

the domain consists of all real numbers; there are no restrictions on the

input. The only output value is the constant so the range is the set that contains this single element. In interval

notation, this is written as

the interval that both begins and ends with

44 1 • Functions

Figure 14 For the identity function real numbers.

there is no restriction on Both the domain and range are the set of all

Figure 15 For the absolute value function

there is no restriction on However, because absolute value is

defined as a distance from 0, the output can only be greater than or equal to 0.

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1.2 • Domain and Range 45

Figure 16 For the quadratic function

the domain is all real numbers since the horizontal extent of the

graph is the whole real number line. Because the graph does not include any negative values for the range, the range is

only nonnegative real numbers.

Figure 17 For the cubic function

the domain is all real numbers because the horizontal extent of the graph

is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all

real numbers.

46 1 • Functions

Figure 18 For the reciprocal function

we cannot divide by 0, so we must exclude 0 from the domain.

Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also

write

the set of all real numbers that are not zero.

Figure 19 For the reciprocal squared function

we cannot divide by so we must exclude from the

domain. There is also no that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.

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1.2 • Domain and Range 47

Figure 20 For the square root function

we cannot take the square root of a negative real number, so the

domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number

is defined to be positive, even though the square of the negative number

also gives us

Figure 21 For the cube root function

the domain and range include all real numbers. Note that there is

no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is

an odd function).

...
HOW TO

Given the formula for a function, determine the domain and range.
1. Exclude from the domain any input values that result in division by zero. 2. Exclude from the domain any input values that have nonreal (or undefined) number outputs. 3. Use the valid input values to determine the range of the output values. 4. Look at the function graph and table values to confirm the actual function behavior.

48 1 • Functions
EXAMPLE 8 Finding the Domain and Range Using Toolkit Functions Find the domain and range of
Solution There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.
The domain is ∞ ∞ and the range is also ∞ ∞

EXAMPLE 9

Finding the Domain and Range Find the domain and range of

Solution

We cannot evaluate the function at because division by zero is undefined. The domain is ∞

∞

Because the function is never zero, we exclude 0 from the range. The range is ∞

∞

EXAMPLE 10
Finding the Domain and Range Find the domain and range of
Solution We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.

The domain of

is ∞

We then find the range. We know that
limit. We conclude that the range of is ∞

Analysis Figure 22 represents the function

and the function value increases as increases without any upper

Figure 22 TRY IT #7 Find the domain and range of
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1.2 • Domain and Range 49

Graphing Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For

example, in the toolkit functions, we introduced the absolute value function

With a domain of all real

numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude, or modulus,

of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the

output to be greater than or equal to 0.

If we input 0, or a positive value, the output is the same as the input.

If we input a negative value, the output is the opposite of the input.

Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses

certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain

item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of

piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and

any additional income is taxed at 20%. The tax on a total income would be

if

and

if

Piecewise Function

A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:

In piecewise notation, the absolute value function is

...
HOW TO
Given a piecewise function, write the formula and identify the domain for each interval.
1. Identify the intervals for which different rules apply. 2. Determine formulas that describe how to calculate an output from an input in each interval. 3. Use braces and if-statements to write the function.

EXAMPLE 11

Writing a Piecewise Function A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, to the cost,

Solution Two different formulas will be needed. For n-values under 10,

For values of that are 10 or greater,

50 1 • Functions

Analysis

The function is represented in Figure 23. The graph is a diagonal line from

to

and a constant after that. In

this example, the two formulas agree at the meeting point where

but not all piecewise functions have this

property.

Figure 23
EXAMPLE 12 Working with a Piecewise Function A cell phone company uses the function below to determine the cost, in dollars for gigabytes of data transfer.

Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.

Solution To find the cost of using 1.5 gigabytes of data, Because 1.5 is less than 2, we use the first formula.

we first look to see which part of the domain our input falls in.

To find the cost of using 4 gigabytes of data, formula.

we see that our input of 4 is greater than 2, so we use the second

Analysis

The function is represented in Figure 24. We can see where the function changes from a constant to a shifted and

stretched identity at

We plot the graphs for the different formulas on a common set of axes, making sure each

formula is applied on its proper domain.

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1.2 • Domain and Range 51

Figure 24
...
HOW TO
Given a piecewise function, sketch a graph. 1. Indicate on the x-axis the boundaries defined by the intervals on each piece of the domain. 2. For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece.
Do not graph two functions over one interval because it would violate the criteria of a function.
EXAMPLE 13 Graphing a Piecewise Function Sketch a graph of the function.
Solution Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality. Figure 25 shows the three components of the piecewise function graphed on separate coordinate systems.

Figure 25 (a)

(b)

(c)

52 1 • Functions
Now that we have sketched each piece individually, we combine them in the same coordinate plane. See Figure 26.

Figure 26

Analysis

Note that the graph does pass the vertical line test even at

and

part of the graph of the function, though

and

are.

because the points

and

are not

TRY IT #8 Graph the following piecewise function.

Q&A

Can more than one formula from a piecewise function be applied to a value in the domain? No. Each value corresponds to one equation in a piecewise formula.

MEDIA
Access these online resources for additional instruction and practice with domain and range.
Domain and Range of Square Root Functions (http://openstax.org/l/domainsqroot) Determining Domain and Range (http://openstax.org/l/determinedomain) Find Domain and Range Given the Graph (http://openstax.org/l/drgraph) Find Domain and Range Given a Table (http://openstax.org/l/drtable) Find Domain and Range Given Points on a Coordinate Plane (http://openstax.org/l/drcoordinate)

1.2 SECTION EXERCISES
Verbal

1. Why does the domain differ for different functions?

2. How do we determine the domain of a function defined by an equation?

3. Explain why the domain of is different from
the domain of

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4. When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?

5. How do you graph a piecewise function?

1.2 • Domain and Range 53

Algebraic

For the following exercises, find the domain of each function using interval notation.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26. Find the domain of the

function

by:

ⓐ using algebra. ⓑ graphing the function
in the radicand and determining intervals on the x-axis for which the radicand is nonnegative.

Graphical

For the following exercises, write the domain and range of each function using interval notation.

27.

28.

29.

54 1 • Functions

30.

31.

32.

33.

34.

35.

36.

37.

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

38.

39.

40.

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41.

42.

44.

45.

Numeric

For the following exercises, given each function evaluate

46.

47.

For the following exercises, given each function evaluate

49.

50.

43.
and 48.
and 51.

1.2 • Domain and Range 55

For the following exercises, write the domain for the piecewise function in interval notation.

52.

53.

54.

Technology

55. Graph

on the

viewing window and
Determine the corresponding range for the viewing window. Show the graphs.

56. Graph

on the

viewing window

and

Determine the

corresponding range for

the viewing window. Show

the graphs.

Extension

57. Suppose the range of a

function is

What

is the range of

58. Create a function in which the range is all nonnegative real numbers.

59. Create a function in which the domain is

56 1 • Functions

Real-World Applications

60. The height of a projectile is a function of the time it is in the air. The height in feet for seconds is given by the function What is the domain of the function? What does the domain mean in the context of the problem?

61. The cost in dollars of making items is given by the function
ⓐ The fixed cost is
determined when zero items are produced. Find the fixed cost for this item.
ⓑ What is the cost of
making 25 items?
ⓒ Suppose the maximum
cost allowed is $1500. What are the domain and range of the cost function,

1.3 Rates of Change and Behavior of Graphs
Learning Objectives
In this section, you will: Find the average rate of change of a function. Use a graph to determine where a function is increasing, decreasing, or constant. Use a graph to locate local maxima and local minima. Use a graph to locate the absolute maximum and absolute minimum.
Gasoline costs have experienced some wild fluctuations over the last several decades. Table 15 lists the average cost, in dollars, of a gallon of gasoline for the years 2005–2012. The cost of gasoline can be considered as a function of year.
2005 2006 2007 2008 2009 2010 2011 2012
2.31 2.62 2.84 3.30 2.41 2.84 3.58 3.68
Table 1
If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed per year. In this section, we will investigate changes such as these.
Finding the Average Rate of Change of a Function
The price change per year is a rate of change because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in Table 1 did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the average rate of change over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.

The Greek letter (delta) signifies the change in a quantity; we read the ratio as “delta-y over delta-x” or “the change in
5 http://www.eia.gov/totalenergy/data/annual/showtext.cfm?t=ptb0524. Accessed 3/5/2014.
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1.3 • Rates of Change and Behavior of Graphs 57
divided by the change in ” Occasionally we write instead of which still represents the change in the function’s output value resulting from a change to its input value. It does not mean we are changing the function into some other function. In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was
On average, the price of gas increased by about 19.6¢ each year. Other examples of rates of change include:
• A population of rats increasing by 40 rats per week • A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes) • A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon) • The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage • The amount of money in a college account decreasing by $4,000 per quarter Rate of Change A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are “output units per input units.” The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.
...
HOW TO
Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values and 1. Calculate the difference 2. Calculate the difference 3. Find the ratio
EXAMPLE 1 Computing an Average Rate of Change Using the data in Table 1, find the average rate of change of the price of gasoline between 2007 and 2009.
Solution In 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The average rate of change is
Analysis Note that a decrease is expressed by a negative change or “negative increase.” A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.

58 1 • Functions
TRY IT #1 Using the data in Table 1, find the average rate of change between 2005 and 2010.
EXAMPLE 2 Computing Average Rate of Change from a Graph Given the function shown in Figure 1, find the average rate of change on the interval

Solution

At

Figure 2 shows

Figure 1

At

the graph shows

Figure 2

The horizontal change

is shown by the red arrow, and the vertical change

is shown by the turquoise

arrow. The output changes by –3 while the input changes by 3, giving an average rate of change of

Analysis

Note that the order we choose is very important. If, for example, we use

we will not get the correct answer.

Decide which point will be 1 and which point will be 2, and keep the coordinates fixed as

and

EXAMPLE 3

Computing Average Rate of Change from a Table After picking up a friend who lives 10 miles away, Anna records her distance from home over time. The values are shown in Table 2. Find her average speed over the first 6 hours.

t (hours)

012 3

4

5

6

7

Table 2

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1.3 • Rates of Change and Behavior of Graphs 59
D(t) (miles) 10 55 90 153 214 240 292 300 Table 2 Solution Here, the average speed is the average rate of change. She traveled 282 miles in 6 hours, for an average speed of

The average speed is 47 miles per hour.
Analysis Because the speed is not constant, the average speed depends on the interval chosen. For the interval [2,3], the average speed is 63 miles per hour.

EXAMPLE 4

Computing Average Rate of Change for a Function Expressed as a Formula

Compute the average rate of change of

on the interval

Solution We can start by computing the function values at each endpoint of the interval.

Now we compute the average rate of change.

TRY IT #2 Find the average rate of change of

on the interval

EXAMPLE 5

Finding the Average Rate of Change of a Force

The electrostatic force measured in newtons, between two charged particles can be related to the distance between

the particles in centimeters, by the formula

Find the average rate of change of force if the distance

between the particles is increased from 2 cm to 6 cm.

Solution We are computing the average rate of change of

on the interval

60 1 • Functions

The average rate of change is newton per centimeter.

EXAMPLE 6

Finding an Average Rate of Change as an Expression

Find the average rate of change of

on the interval

Solution We use the average rate of change formula.

The answer will be an expression involving

This result tells us the average rate of change in terms of between

interval

the average rate of change would be

and any other point

For example, on the

TRY IT #3 Find the average rate of change of

on the interval

Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant
As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. Figure 3 shows examples of increasing and decreasing intervals on a function.

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1.3 • Rates of Change and Behavior of Graphs 61

Figure 3 The function

is increasing on ∞

∞ and is decreasing on

While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is the location of a local maximum. The function value at that point is the local maximum. If a function has more than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is the location of a local minimum. The function value at that point is the local minimum. The plural form is “local minima.” Together, local maxima and minima are called local extrema, or local extreme values, of the function. (The singular form is “extremum.”) Often, the term local is replaced by the term relative. In this text, we will use the term local.

Clearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of local extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function’s entire domain.

For the function whose graph is shown in Figure 4, the local maximum is 16, and it occurs at is and it occurs at

The local minimum

62 1 • Functions

Figure 4
To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. Figure 5 illustrates these ideas for a local maximum.

Figure 5 Definition of a local maximum These observations lead us to a formal definition of local extrema.
Local Minima and Local Maxima

A function is an increasing function on an open interval if interval where

for every two input values and in the

A function is a decreasing function on an open interval if interval where

for every two input values and in the

A function has a local maximum at a point in an open interval

if

equal ) in the interval. has a local minimum at a point in

if

) in the interval.

for every point ( does not for every point ( does not equal

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1.3 • Rates of Change and Behavior of Graphs 63
EXAMPLE 7 Finding Increasing and Decreasing Intervals on a Graph Given the function in Figure 6, identify the intervals on which the function appears to be increasing.

Figure 6

Solution

We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to

the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from

to

and from

on.

In interval notation, we would say the function appears to be increasing on the interval (1,3) and the interval ∞

Analysis

Notice in this example that we used open intervals (intervals that do not include the endpoints), because the function is

neither increasing nor decreasing at

,

, and

. These points are the local extrema (two minima and a

maximum).

EXAMPLE 8

Finding Local Extrema from a Graph

Graph the function

Then use the graph to estimate the local extrema of the function and to determine

the intervals on which the function is increasing.

Solution

Using technology, we find that the graph of the function looks like that in Figure 7. It appears there is a low point, or

local minimum, between

and

and a mirror-image high point, or local maximum, somewhere between

and

64 1 • Functions

Figure 7
Analysis Most graphing calculators and graphing utilities can estimate the location of maxima and minima. Figure 8 provides screen images from two different technologies, showing the estimate for the local maximum and minimum.

Figure 8

Based on these estimates, the function is increasing on the interval ∞

and

∞ Notice that, while we

expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing

approximation algorithms used by each. (The exact location of the extrema is at

but determining this requires

calculus.)

TRY IT #4

Graph the function

to estimate the local extrema of the function.

Use these to determine the intervals on which the function is increasing and decreasing.

EXAMPLE 9
Finding Local Maxima and Minima from a Graph For the function whose graph is shown in Figure 9, find all local maxima and minima.

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1.3 • Rates of Change and Behavior of Graphs 65

Figure 9

Solution

Observe the graph of The graph attains a local maximum at

because it is the highest point in an open interval

around

The local maximum is the -coordinate at

which is

The graph attains a local minimum at

because it is the lowest point in an open interval around

minimum is the y-coordinate at

which is

The local

Analyzing the Toolkit Functions for Increasing or Decreasing Intervals
We will now return to our toolkit functions and discuss their graphical behavior in Figure 10, Figure 11, and Figure 12.

Figure 10

66 1 • Functions
Figure 11
Figure 12
Use A Graph to Locate the Absolute Maximum and Absolute Minimum
There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The coordinates (output) at the highest and lowest points are called the absolute maximum and absolute minimum, respectively. To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See Figure 13.
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1.3 • Rates of Change and Behavior of Graphs 67

Figure 13 Not every function has an absolute maximum or minimum value. The toolkit function
Absolute Maxima and Minima

is one such function.

The absolute maximum of at The absolute minimum of at

is

where

is

where

for all in the domain of for all in the domain of

EXAMPLE 10
Finding Absolute Maxima and Minima from a Graph For the function shown in Figure 14, find all absolute maxima and minima.

Figure 14

Solution

Observe the graph of The graph attains an absolute maximum in two locations,

and

because at these

locations, the graph attains its highest point on the domain of the function. The absolute maximum is the y-coordinate

at

and

which is

The graph attains an absolute minimum at The absolute minimum is the y-coordinate at

because it is the lowest point on the domain of the function’s graph. which is

MEDIA Access this online resource for additional instruction and practice with rates of change.

68 1 • Functions

Average Rate of Change (http://openstax.org/l/aroc)

1.3 SECTION EXERCISES
Verbal

1. Can the average rate of change of a function be constant?

2. If a function is increasing

on

and decreasing on

then what can be said

about the local extremum of

on

3. How are the absolute maximum and minimum similar to and different from the local extrema?

4. How does the graph of the absolute value function compare to the graph of the quadratic function, in terms of increasing and decreasing intervals?

Algebraic

For the following exercises, find the average rate of change of each function on the interval specified for real numbers or

5.

on

6.

on

7.

on

8.

on

9.

on

10.

on

11.

on

14.

on

12.

on

13.

on

15.

on

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Graphical
For the following exercises, consider the graph of shown in Figure 15.

1.3 • Rates of Change and Behavior of Graphs 69

16. Estimate the average rate

of change from

to

Figure 15

17. Estimate the average rate

of change from

to

For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.

18.

19.

20.

21.

70 1 • Functions
For the following exercises, consider the graph shown in Figure 16.

22. Estimate the intervals where the function is increasing or decreasing.

Figure 16
23. Estimate the point(s) at which the graph of has a local maximum or a local minimum.

For the following exercises, consider the graph in Figure 17.

24. If the complete graph of the function is shown, estimate the intervals where the function is increasing or decreasing.

Figure 17
25. If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum.

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1.3 • Rates of Change and Behavior of Graphs 71

Numeric

26. Table 3 gives the annual sales (in millions of dollars) of a product from 1998 to 2006. What was the average rate of change of annual sales (a) between 2001 and 2002, and (b) between 2001 and 2004?

Year Sales (millions of dollars)

1998

201

1999

219

2000

233

2001

243

2002

249

2003

251

2004

249

2005

243

2006

233

Table 3

27. Table 4 gives the population of a town (in thousands) from 2000 to 2008. What was the average rate of change of population (a) between 2002 and 2004, and (b) between 2002 and 2006?

Year Population (thousands)

2000

87

2001

84

2002

83

2003

80

2004

77

2005

76

2006

78

2007

81

2008

85

Table 4

For the following exercises, find the average rate of change of each function on the interval specified.

28.

on

29.

on

30.

on

31.

on

32.

on

33.

on

34.

on

Technology

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

35.

36.

37.

38.

39.

40.

72 1 • Functions

Extension

41. The graph of the function is shown in Figure 18.

42. Let

Find a number such that the

average rate of change of the function on the

interval

is

Figure 18
Based on the calculator screen shot, the point is which of the following?
ⓐ a relative (local) maximum of the function ⓑ the vertex of the function ⓒ the absolute maximum of the function ⓓ a zero of the function

43. Let

. Find the

number such that the

average rate of change of

on the interval

is

Real-World Applications

44. At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?

45. A driver of a car stopped at a gas station to fill up their gas tank. They looked at their watch, and the time read exactly 3:40 p.m. At this time, they started pumping gas into the tank. At exactly 3:44, the tank was full and they noticed that they had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?

46. Near the surface of the

moon, the distance that an

object falls is a function of

time. It is given by

where is

in seconds and is in

feet. If an object is dropped

from a certain height, find

the average velocity of the

object from

to

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47. The graph in Figure 19 illustrates the decay of a radioactive substance over days.

1.4 • Composition of Functions 73

Figure 19

Use the graph to estimate the

average decay rate from

to

1.4 Composition of Functions

Learning Objectives
In this section, you will: Combine functions using algebraic operations. Create a new function by composition of functions. Evaluate composite functions. Find the domain of a composite function. Decompose a composite function into its component functions.

Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day.

Using descriptive variables, we can notate these two functions. The function

gives the cost of heating a house

for a given average daily temperature in degrees Celsius. The function

gives the average daily temperature on

day of the year. For any given day,

means that the cost depends on the temperature, which in turns

depends on the day of the year. Thus, we can evaluate the cost function at the temperature

For example, we could

evaluate

to determine the average daily temperature on the 5th day of the year. Then, we could evaluate the cost

function at that temperature. We would write

By combining these two relationships into one function, we have performed function composition, which is the focus of this section.

Combining Functions Using Algebraic Operations

Function composition is only one way to combine existing functions. Another way is to carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division. We do this by performing the operations with the function outputs, defining the result as the output of our new function.

Suppose we need to add two columns of numbers that represent a husband and wife’s separate annual incomes over a

period of years, with the result being their total household income. We want to do this for every year, adding only that

year’s incomes and then collecting all the data in a new column. If

is the wife’s income and is the husband’s

74 1 • Functions

income in year and we want to represent the total income, then we can define a new function.

If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write

Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.

For two functions relations

and

with real number outputs, we define new functions

and by the

EXAMPLE 1

Performing Algebraic Operations on Functions

Find and simplify the functions

and

given

and

function?

Solution Begin by writing the general form, and then substitute the given functions.

Are they the same

No, the functions are not the same.

Note: For

the condition

the function undefined.

is necessary because when

the denominator is equal to 0, which makes

TRY IT #1 Find and simplify the functions

and

Are they the same function?
Create a Function by Composition of Functions
Performing algebraic operations on functions combines them into a new function, but we can also create functions by

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1.4 • Composition of Functions 75

composing functions. When we wanted to compute a heating cost from a day of the year, we created a new function that takes a day as input and yields a cost as output. The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions. The resulting function is known as a composite function. We represent this combination by the following notation:

We read the left-hand side as composed with at and the right-hand side as of of The two sides of the equation have the same mathematical meaning and are equal. The open circle symbol is called the composition operator. We use this operator mainly when we wish to emphasize the relationship between the functions themselves without referring to any particular input value. Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication takes two numbers and gives a new number. However, it is important not to confuse function composition with multiplication because, as we learned above, in most cases

It is also important to understand the order of operations in evaluating a composite function. We follow the usual

convention with parentheses by starting with the innermost parentheses first, and then working to the outside. In the

equation above, the function takes the input first and yields an output

Then the function takes

as an

input and yields an output

In general,

and

are different functions. In other words, in many cases

see that sometimes two functions can be composed only in one specific order.

For example, if

and

then

for all We will also

but

These expressions are not equal for all values of so the two functions are not equal. It is irrelevant that the expressions happen to be equal for the single input value
Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function. Less formally, the composition has to make sense in terms of inputs and outputs.
Composition of Functions
When the output of one function is used as the input of another, we call the entire operation a composition of functions. For any input and functions and this action defines a composite function, which we write as such that

The domain of the composite function

is all such that is in the domain of and

is in the domain of

It is important to realize that the product of functions because, in general,

is not the same as the function composition

76 1 • Functions

EXAMPLE 2

Determining whether Composition of Functions is Commutative

Using the functions provided, find

and

Determine whether the composition of the functions is

commutative.

Solution

Let’s begin by substituting

into

Now we can substitute

into

We find that

so the operation of function composition is not commutative.

EXAMPLE 3

Interpreting Composite Functions The function gives the number of calories burned completing sit-ups, and can complete in minutes. Interpret

gives the number of sit-ups a person

Solution

The inside expression in the composition is

Because the input to the s-function is time,

and is the number of sit-ups completed in 3 minutes.

represents 3 minutes,

Using as the input to the function gives us the number of calories burned during the number of sit-ups that can be completed in 3 minutes, or simply the number of calories burned in 3 minutes (by doing sit-ups).

EXAMPLE 4

Investigating the Order of Function Composition

Suppose

gives miles that can be driven in hours and

these expressions is meaningful:

or

gives the gallons of gas used in driving miles. Which of

Solution The function driven.

is a function whose output is the number of miles driven corresponding to the number of hours

The function

is a function whose output is the number of gallons used corresponding to the number of miles

driven. This means:

The expression takes miles as the input and a number of gallons as the output. The function

requires a

number of hours as the input. Trying to input a number of gallons does not make sense. The expression

is

meaningless.

The expression

takes hours as input and a number of miles driven as the output. The function requires a

number of miles as the input. Using

(miles driven) as an input value for

where gallons of gas depends on

miles driven, does make sense. The expression

makes sense, and will yield the number of gallons of gas used,

driving a certain number of miles,

in hours.

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1.4 • Composition of Functions 77

Q&A

Are there any situations where

and

would both be meaningful or useful expressions?

Yes. For many pure mathematical functions, both compositions make sense, even though they usually produce different new functions. In real-world problems, functions whose inputs and outputs have the same units also may give compositions that are meaningful in either order.

TRY IT #2

The gravitational force on a planet a distance r from the sun is given by the function

The

acceleration of a planet subjected to any force is given by the function

Form a meaningful

composition of these two functions, and explain what it means.

Evaluating Composite Functions
Once we compose a new function from two existing functions, we need to be able to evaluate it for any input in its domain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables as inputs to functions expressed as formulas. In each case, we evaluate the inner function using the starting input and then use the inner function’s output as the input for the outer function.
Evaluating Composite Functions Using Tables
When working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the input to the outside function.

EXAMPLE 5

Using a Table to Evaluate a Composite Function

Using Table 1, evaluate

and

16

3

28

5

33

2

41

7

Table 1

Solution

To evaluate

we start from the inside with the input value 3. We then evaluate the inside expression

the table that defines the function

We can then use that result as the input to the function so

replaced by 2 and we get

Then, using the table that defines the function we find that

using is

To evaluate

we first evaluate the inside expression

  we can evaluate

using the first table:

Table 2 shows the composite functions

and

as tables.

Then, using the table for

Table 2

78 1 • Functions

32

8

3

2

Table 2

TRY IT #3 Using Table 1, evaluate

and

Evaluating Composite Functions Using Graphs
When we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the process we use for evaluating tables. We read the input and output values, but this time, from the and axes of the graphs.
...
HOW TO
Given a composite function and graphs of its individual functions, evaluate it using the information provided by the graphs.
1. Locate the given input to the inner function on the axis of its graph. 2. Read off the output of the inner function from the axis of its graph. 3. Locate the inner function output on the axis of the graph of the outer function. 4. Read the output of the outer function from the axis of its graph. This is the output of the composite function.

EXAMPLE 6
Using a Graph to Evaluate a Composite Function Using Figure 1, evaluate

Solution To evaluate

Figure 1 we start with the inside evaluation. See Figure 2.

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1.4 • Composition of Functions 79

Figure 2

We evaluate using the graph of

finding the input of 1 on the axis and finding the output value of the graph

at that input. Here,

We use this value as the input to the function

We can then evaluate the composite function by looking to the graph of

reading the output value of the graph at this input. Here,

so

finding the input of 3 on the axis and

Analysis Figure 3 shows how we can mark the graphs with arrows to trace the path from the input value to the output value.

Figure 3

TRY IT #4 Using Figure 1, evaluate

Evaluating Composite Functions Using Formulas
When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.

While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that

will calculate the result of a composition

To do this, we will extend our idea of function evaluation. Recall that,

when we evaluate a function like

we substitute the value inside the parentheses into the formula wherever

80 1 • Functions
we see the input variable.
...
HOW TO
Given a formula for a composite function, evaluate the function. 1. Evaluate the inside function using the input value or variable provided. 2. Use the resulting output as the input to the outside function.

EXAMPLE 7

Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input

Given

and

evaluate

Solution Because the inside expression is

we start by evaluating

at 1.

Then

so we evaluate at an input of 5.

Analysis It makes no difference what the input variables and numerical values.

were called in this problem because we evaluated for specific

TRY IT #5

Given
ⓐ

and
ⓑ

evaluate

Finding the Domain of a Composite Function

As we discussed previously, the domain of a composite function such as

is dependent on the domain of and the

domain of It is important to know when we can apply a composite function and when we cannot, that is, to know the

domain of a function such as

Let us assume we know the domains of the functions and separately. If we write

the composite function for an input as

we can see right away that must be a member of the domain of in

order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation.

However, we also see that

must be a member of the domain of otherwise the second function evaluation in

cannot be completed, and the expression is still undefined. Thus the domain of

consists of only those

inputs in the domain of that produce outputs from belonging to the domain of Note that the domain of

composed with is the set of all such that is in the domain of and

is in the domain of

Domain of a Composite Function

The domain of a composite function domain of

is the set of those inputs in the domain of for which

is in the

...
HOW TO
Given a function composition

determine its domain.

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1.4 • Composition of Functions 81

1. Find the domain of

2. Find the domain of

3. Find those inputs in the domain of for which

is in the domain of That is, exclude those inputs

the domain of for which

is not in the domain of The resulting set is the domain of

from

EXAMPLE 8
Finding the Domain of a Composite Function Find the domain of

Solution

The domain of

consists of all real numbers except

since that input value would cause us to divide by 0.

Likewise, the domain of consists of all real numbers except 1. So we need to exclude from the domain of

that

value of for which

So the domain of

is the set of all real numbers except and This means that

We can write this in interval notation as

∞

∞

EXAMPLE 9
Finding the Domain of a Composite Function Involving Radicals Find the domain of

Solution
Because we cannot take the square root of a negative number, the domain of is ∞
of the composite function

Now we check the domain

For are positive,

since the radicand of a square root must be positive. Since square roots

or,

which gives a domain of

.

Analysis

This example shows that knowledge of the range of functions (specifically the inner function) can also be helpful in

finding the domain of a composite function. It also shows that the domain of

can contain values that are not in the

domain of though they must be in the domain of

TRY IT #6 Find the domain of

82 1 • Functions

Decomposing a Composite Function into its Component Functions
In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function, so we may choose the decomposition that appears to be most expedient.

EXAMPLE 10

Decomposing a Function

Write

as the composition of two functions.

Solution

We are looking for two functions, and so

To do this, we look for a function inside a function in the

formula for

As one possibility, we might notice that the expression

is the inside of the square root. We could

then decompose the function as

We can check our answer by recomposing the functions.

TRY IT #7 Write

as the composition of two functions.

MEDIA
Access these online resources for additional instruction and practice with composite functions.
Composite Functions (http://openstax.org/l/compfunction) Composite Function Notation Application (http://openstax.org/l/compfuncnot) Composite Functions Using Graphs (http://openstax.org/l/compfuncgraph) Decompose Functions (http://openstax.org/l/decompfunction) Composite Function Values (http://openstax.org/l/compfuncvalue)

1.4 SECTION EXERCISES
Verbal

1. How does one find the domain of the quotient of two functions,

2. What is the composition of two functions,

3. If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.

4. How do you find the domain for the composition of two functions,

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1.4 • Composition of Functions 83

Algebraic

5. Given

and find

and

Determine the domain for each function in interval notation.

6. Given

and

find

and

Determine the domain for each function in interval notation.

7. Given

and

find

and
Determine the domain for each function in interval notation.

8. Given

and find

and
Determine the domain for each function in interval notation.

9. Given

and

find

and

Determine the domain for each function in interval notation.

10. Given

and find

Determine the domain of the function in interval notation.

11. Given following:

and find the

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

For the following exercises, use each pair of functions to find

12.

13.

and

Simplify your answers.

14.

15.

16.

17.

For the following exercises, use each set of functions to find

18.

19.

and

and

Simplify your answers.

20. Given following:

and find the

ⓐ ⓑ the domain of
in interval notation
ⓒ ⓓ the domain of

ⓔ

84 1 • Functions

21. Given

and

find the

following:

ⓐ ⓑ the domain of
in interval notation

22. Given the functions

find the following:

ⓐ

ⓑ

24. Given functions

and

state

the domain of each of the

following functions using

interval notation.

ⓐⓑ ⓒ

25. For

and

write the

domain of

in

interval notation.

23. Given functions

and

state

the domain of each of the

following functions using

interval notation:

ⓐⓑ ⓒ

For the following exercises, find functions

and

so the given function can be expressed as

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

Graphical
For the following exercises, use the graphs of shown in Figure 4, and shown in Figure 5, to evaluate the expressions.

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Figure 4

1.4 • Composition of Functions 85

Figure 5

42.

43.

44.

45.

46.

47.

48.

49.

For the following exercises, use graphs of to evaluate the expressions.

shown in Figure 6,

shown in Figure 7, and

shown in Figure 8,

Figure 6 Figure 7

86 1 • Functions

Figure 8

50.

51.

52.

53.

54.

55.

56.

57.

Numeric
For the following exercises, use the function values for

shown in Table 3 to evaluate each expression.

07

9

16

5

25

6

38

2

44

1

50

8

62

7

71

3

89

4

93

0

Table 3

58.

59.

60.

61.

62.

63.

64.

65.

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For the following exercises, use the function values for

1.4 • Composition of Functions 87
shown in Table 4 to evaluate the expressions.

-3 11

-8

-2 9

-3

-1 7

0

0

5

1

1

3

0

2

1

-3

3 -1

-8

Table 4

66.

67.

68.

69.

70.

71.

For the following exercises, use each pair of functions to find

72.

73.

75.

and 74.

For the following exercises, use the functions function as indicated.

76.

77.

79.

and

to evaluate or find the composite

78.

Extensions

For the following exercises, use

and

80. Find

and

81. Find

and

Compare the

two answers.

82. What is the domain of

88 1 • Functions

83. What is the domain of

84. Let

ⓐ Find ⓑ Is
function

for any the same result

as the answer to part (a)

for any function? Explain.

For the following exercises, let

and

85. True or False:

86. True or False:

For the following exercises, find the composition when

87.

88.

for all

and

89.

Real-World Applications

90. The function

gives the

number of items that will

be demanded when the

price is The production

cost

is the cost of

producing items. To

determine the cost of

production when the price

is $6, you would do which

of the following?

ⓐ Evaluate ⓑ Evaluate ⓒ Solve ⓓ Solve

91. The function

gives the

pain level on a scale of 0 to

10 experienced by a patient

with milligrams of a pain-

reducing drug in her

system. The milligrams of

the drug in the patient’s

system after minutes is

modeled by

Which of

the following would you do

in order to determine

when the patient will be at

a pain level of 4?

ⓐ Evaluate ⓑ Evaluate ⓒ Solve ⓓ Solve

92. A store offers customers a

30% discount on the price

of selected items. Then,

the store takes off an

additional 15% at the cash

register. Write a price

function

that

computes the final price of

the item in terms of the

original price (Hint: Use

function composition to

find your answer.)

93. A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to
find the area of the ripple as a function of time. Find the area of the ripple at

94. A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula express the area burned as a function of time, (minutes).

95. Use the function you found in the previous exercise to find the total area burned after 5 minutes.

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96. The radius in inches, of a spherical balloon is related to the volume, by Air is
pumped into the balloon, so the volume after seconds is given by
ⓐ Find the composite
function
ⓑ Find the exact time
when the radius reaches 10 inches.

97. The number of bacteria in a refrigerated food product is given by
where is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by
where is the time in hours.
ⓐ Find the composite
function
ⓑ Find the time (round to
two decimal places) when the bacteria count reaches 6752.

1.5 Transformation of Functions
Learning Objectives
In this section, you will: Graph functions using vertical and horizontal shifts. Graph functions using reflections about the -axis and the -axis. Determine whether a function is even, odd, or neither from its graph. Graph functions using compressions and stretches. Combine transformations.

1.5 • Transformation of Functions 89

Figure 1 (credit: "Misko"/Flickr)
We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.
Graphing Functions Using Vertical and Horizontal Shifts
Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.
Identifying Vertical Shifts
One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest

90 1 • Functions

shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative

constant to the function. In other words, we add the same constant to the output value of the function regardless of the

input. For a function

the function

is shifted vertically units. See Figure 2 for an example.

Figure 2 Vertical shift by

of the cube root function

To help you visualize the concept of a vertical shift, consider that

Therefore,

is equivalent to

Every unit of is replaced by

so the value increases or decreases depending on the value of The result is a

shift upward or downward.

Vertical Shift

Given a function

a new function

All the output values change by units. If

down.

where is a constant, is a vertical shift of the function is positive, the graph will shift up. If is negative, the graph will shift

EXAMPLE 1
Adding a Constant to a Function To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. Figure 3 shows the area of open vents (in square feet) throughout the day in hours after midnight, During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.

Figure 3 Solution We can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will have the effect of shifting the graph vertically up, as shown in Figure 4.
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