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Intermediate Algebra 2e

10.1 Finding Composite and Inverse Functions

Intermediate Algebra 2e10.1 Finding Composite and Inverse Functions
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:
  • Find and evaluate composite functions
  • Determine whether a function is one-to-one
  • Find the inverse of a function
Be Prepared 10.1

Before you get started, take this readiness quiz.

If f(x)=2x3f(x)=2x3 and g(x)=x2+2x3,g(x)=x2+2x3, find f(4).f(4).
If you missed this problem, review Example 3.48.

Be Prepared 10.2

Solve for x,x, 3x+2y=12.3x+2y=12.
If you missed this problem, review Example 2.31.

Be Prepared 10.3

Simplify: 5(x+4)54.5(x+4)54.
If you missed this problem, review Example 1.25.

In this chapter, we will introduce two new types of functions, exponential functions and logarithmic functions. These functions are used extensively in business and the sciences as we will see.

Find and Evaluate Composite Functions

Before we introduce the functions, we need to look at another operation on functions called composition. In composition, the output of one function is the input of a second function. For functions ff and g,g, the composition is written fgfg and is defined by (fg)(x)=f(g(x)).(fg)(x)=f(g(x)).

We read f(g(x))f(g(x)) as ff of gg of x.”x.”

This figure shows x as the input to a box denoted as function g with g of x as the output of the box. Then, g of x is the input to a box denoted as function f with f of g of x as the output of the box.

To do a composition, the output of the first function, g(x),g(x), becomes the input of the second function, f, and so we must be sure that it is part of the domain of f.

Composition of Functions

The composition of functions f and g is written fgfg and is defined by

(fg)(x)=f(g(x))(fg)(x)=f(g(x))

We read f(g(x))f(g(x)) as ff of gg of x.

We have actually used composition without using the notation many times before. When we graphed quadratic functions using translations, we were composing functions. For example, if we first graphed g(x)=x2g(x)=x2 as a parabola and then shifted it down vertically four units, we were using the composition defined by (fg)(x)=f(g(x))(fg)(x)=f(g(x)) where f(x)=x4.f(x)=x4.

This figure shows x as the input to a box denoted as g of x equals x squared with x squared as the output of the box. Then, x squared is the input to a box denoted as f of x equals x minus 4 with f of g of x equals x squared minus 4 as the output of the box.

The next example will demonstrate that (fg)(x),(fg)(x), (gf)(x)(gf)(x) and (f·g)(x)(f·g)(x) usually result in different outputs.

Example 10.1

For functions f(x)=4x5f(x)=4x5 and g(x)=2x+3,g(x)=2x+3, find: (fg)(x),(fg)(x), (gf)(x),(gf)(x), and (f·g)(x).(f·g)(x).

Try It 10.1

For functions f(x)=3x2f(x)=3x2 and g(x)=5x+1,g(x)=5x+1, find (fg)(x)(fg)(x) (gf)(x)(gf)(x) (f·g)(x)(f·g)(x).

Try It 10.2

For functions f(x)=4x3,f(x)=4x3, and g(x)=6x5,g(x)=6x5, find (fg)(x),(fg)(x), (gf)(x),(gf)(x), and (f·g)(x).(f·g)(x).

In the next example we will evaluate a composition for a specific value.

Example 10.2

For functions f(x)=x24,f(x)=x24, and g(x)=3x+2,g(x)=3x+2, find: (fg)(−3),(fg)(−3), (gf)(−1),(gf)(−1), and (ff)(2).(ff)(2).

Try It 10.3

For functions f(x)=x29,f(x)=x29, and g(x)=2x+5,g(x)=2x+5, find (fg)(−2),(fg)(−2), (gf)(−3),(gf)(−3), and (ff)(4).(ff)(4).

Try It 10.4

For functions f(x)=x2+1,f(x)=x2+1, and g(x)=3x5,g(x)=3x5, find (fg)(−1),(fg)(−1), (gf)(2),(gf)(2), and (ff)(−1).(ff)(−1).

Determine Whether a Function is One-to-One

When we first introduced functions, we said a function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.

We used the birthday example to help us understand the definition. Every person has a birthday, but no one has two birthdays and it is okay for two people to share a birthday. Since each person has exactly one birthday, that relation is a function.

This figure shows two tables. To the left is the table labeled Name, which from top to bottom reads Alison, Penelope, June, Gregory, Geoffrey, Lauren, Stephen, Alice, Liz, and Danny. The table on the right is labeled Birthday, which from top to bottom reads January 12, February 3, April 25, May 10, May 23, July 24, August 2, and September 15. There are arrows going from Alison to April 25, Penelope to May 23, June to August 2, Gregory to September 15, Geoffrey to January 12, Lauren to May 10, Stephen to July 24, Alice to February 3, Liz to July 24, and Danny to no birthday.

A function is one-to-one if each value in the range has exactly one element in the domain. For each ordered pair in the function, each y-value is matched with only one x-value.

Our example of the birthday relation is not a one-to-one function. Two people can share the same birthday. The range value August 2 is the birthday of Liz and June, and so one range value has two domain values. Therefore, the function is not one-to-one.

One-to-One Function

A function is one-to-one if each value in the range corresponds to one element in the domain. For each ordered pair in the function, each y-value is matched with only one x-value. There are no repeated y-values.

Example 10.3

For each set of ordered pairs, determine if it represents a function and, if so, if the function is one-to-one.

{(−3,27),(−2,8),(−1,1),(0,0),(1,1),(2,8),(3,27)}{(−3,27),(−2,8),(−1,1),(0,0),(1,1),(2,8),(3,27)} and {(0,0),(1,1),(4,2),(9,3),(16,4)}.{(0,0),(1,1),(4,2),(9,3),(16,4)}.

Try It 10.5

For each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

{(−3,−6),(−2,−4),(−1,−2),(0,0),(1,2),(2,4),(3,6)}{(−3,−6),(−2,−4),(−1,−2),(0,0),(1,2),(2,4),(3,6)} {(−4,8),(−2,4),(−1,2),(0,0),(1,2),(2,4),(4,8)}{(−4,8),(−2,4),(−1,2),(0,0),(1,2),(2,4),(4,8)}

Try It 10.6

For each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

{(27,−3),(8,−2),(1,−1),(0,0),(1,1),(8,2),(27,3)}{(27,−3),(8,−2),(1,−1),(0,0),(1,1),(8,2),(27,3)} {(7,−3),(−5,−4),(8,0),(0,0),(−6,4),(−2,2),(−1,3)}{(7,−3),(−5,−4),(8,0),(0,0),(−6,4),(−2,2),(−1,3)}

To help us determine whether a relation is a function, we use the vertical line test. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. Also, if any vertical line intersects the graph in more than one point, the graph does not represent a function.

The vertical line is representing an x-value and we check that it intersects the graph in only one y-value. Then it is a function.

To check if a function is one-to-one, we use a similar process. We use a horizontal line and check that each horizontal line intersects the graph in only one point. The horizontal line is representing a y-value and we check that it intersects the graph in only one x-value. If every horizontal line intersects the graph of a function in at most one point, it is a one-to-one function. This is the horizontal line test.

Horizontal Line Test

If every horizontal line intersects the graph of a function in at most one point, it is a one-to-one function.

We can test whether a graph of a relation is a function by using the vertical line test. We can then tell if the function is one-to-one by applying the horizontal line test.

Example 10.4

Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.

This first graph shows a straight line passing through (0, 2) and (3, 0). This second shows a parabola opening up with vertex at (0, negative 1).
Try It 10.7

Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.

Graph a shows a parabola opening to the right with vertex at (negative 1, 0). Graph b shows an exponential function that does not cross the x axis and that passes through (0, 1) before increasing rapidly.
Try It 10.8

Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.

Graph a shows a parabola opening up with vertex at (0, 3). Graph b shows a straight line passing through (0, negative 2) and (2, 0).

Find the Inverse of a Function

Let’s look at a one-to one function, ff, represented by the ordered pairs {(0,5),(1,6),(2,7),(3,8)}.{(0,5),(1,6),(2,7),(3,8)}. For each xx-value, ff adds 5 to get the yy-value. To ‘undo’ the addition of 5, we subtract 5 from each yy-value and get back to the original xx-value. We can call this “taking the inverse of ff” and name the function f−1.f−1.

This figure shows the set (0, 5), (1, 6), (2, 7) and (3, 8) on the left side of an oval. The oval contains the numbers 0, 1, 2, and 3. There are black arrows from these numbers that point to the numbers 5, 6, 7, and 8, respectively in a second oval to the right of the first. Above this, there is a black arrow labeled “f add 5” coming from the left oval to the right oval. There are red arrows from the numbers 5, 6, 7, and 8 in the right oval to the numbers 0, 1, 2, and 3, respectively, in the left oval. Below this, we have a red arrow labeled “f with a superscript negative 1” and “subtract 5”. To the right of this, we have the set (5, 0), (6, 1), (7, 2) and (8, 3).

Notice that that the ordered pairs of ff and f−1f−1 have their xx-values and yy-values reversed. The domain of ff is the range of f−1f−1 and the domain of f−1f−1 is the range of f.f.

Inverse of a Function Defined by Ordered Pairs

If f(x)f(x) is a one-to-one function whose ordered pairs are of the form (x,y),(x,y), then its inverse function f−1(x)f−1(x) is the set of ordered pairs (y,x).(y,x).

In the next example we will find the inverse of a function defined by ordered pairs.

Example 10.5

Find the inverse of the function {(0,3),(1,5),(2,7),(3,9)}.{(0,3),(1,5),(2,7),(3,9)}. Determine the domain and range of the inverse function.

Try It 10.9

Find the inverse of {(0,4),(1,7),(2,10),(3,13)}.{(0,4),(1,7),(2,10),(3,13)}. Determine the domain and range of the inverse function.

Try It 10.10

Find the inverse of {(−1,4),(−2,1),(−3,0),(−4,2)}.{(−1,4),(−2,1),(−3,0),(−4,2)}. Determine the domain and range of the inverse function.

We just noted that if f(x)f(x) is a one-to-one function whose ordered pairs are of the form (x,y),(x,y), then its inverse function f−1(x)f−1(x) is the set of ordered pairs (y,x).(y,x).

So if a point (a,b)(a,b) is on the graph of a function f(x),f(x), then the ordered pair (b,a)(b,a) is on the graph of f−1(x).f−1(x). See Figure 10.2.

This figure shows the line y equals x with points (3,1) and (1,3) on either side of the line. These two points are connected by a dashed blue line segment.
Figure 10.2

The distance between any two pairs (a,b)(a,b) and (b,a)(b,a) is cut in half by the line y=x.y=x. So we say the points are mirror images of each other through the line y=x.y=x.

Since every point on the graph of a function f(x)f(x) is a mirror image of a point on the graph of f−1(x),f−1(x), we say the graphs are mirror images of each other through the line y=x.y=x. We will use this concept to graph the inverse of a function in the next example.

Example 10.6

Graph, on the same coordinate system, the inverse of the one-to one function shown.

This figure shows a line from (negative 5, negative 3) to (negative 3, negative 1) then to (negative 1,0) then to (0,2) and then to (3, 4).
Try It 10.11

Graph, on the same coordinate system, the inverse of the one-to one function.

The graph shows a line from (negative 3, negative 4) to (negative 2, negative 2) then to (0, negative 1), then to (1, 2) and then to (4, 3). The graph shows a line from (negative 3, 4) to (0, 3) then to (1, 2) and then to (4, 1).
Try It 10.12

Graph, on the same coordinate system, the inverse of the one-to one function.

.

When we began our discussion of an inverse function, we talked about how the inverse function ‘undoes’ what the original function did to a value in its domain in order to get back to the original x-value.

This figure shows x as the input to a box denoted as function f with f of x as the output of the box. Then, f of x is the input to a box denoted as function f superscript negative 1 with f superscript negative 1 of f of x equals x as the output of the box.

Inverse Functions

f−1(f(x))=x,for allxin the domain offf(f−1(x))=x,for allxin the domain off−1f−1(f(x))=x,for allxin the domain offf(f−1(x))=x,for allxin the domain off−1

We can use this property to verify that two functions are inverses of each other.

Example 10.7

Verify that f(x)=5x1f(x)=5x1 and g(x)=x+15g(x)=x+15 are inverse functions.

Try It 10.13

Verify that the functions are inverse functions.

f(x)=4x3f(x)=4x3 and g(x)=x+34.g(x)=x+34.

Try It 10.14

Verify that the functions are inverse functions.

f(x)=2x+6f(x)=2x+6 and g(x)=x62.g(x)=x62.

We have found inverses of function defined by ordered pairs and from a graph. We will now look at how to find an inverse using an algebraic equation. The method uses the idea that if f(x)f(x) is a one-to-one function with ordered pairs (x,y),(x,y), then its inverse function f−1(x)f−1(x) is the set of ordered pairs (y,x).(y,x).

If we reverse the x and y in the function and then solve for y, we get our inverse function.

Example 10.8 How to Find the inverse of a One-to-One Function

Find the inverse of f(x)=4x+7.f(x)=4x+7.

Try It 10.15

Find the inverse of the function f(x)=5x3.f(x)=5x3.

Try It 10.16

Find the inverse of the function f(x)=8x+5.f(x)=8x+5.

We summarize the steps below.

How To

How to Find the inverse of a One-to-One Function

  1. Step 1. Substitute y for f(x).f(x).
  2. Step 2. Interchange the variables x and y.
  3. Step 3. Solve for y.
  4. Step 4. Substitute f−1(x)f−1(x) for y.
  5. Step 5. Verify that the functions are inverses.

Example 10.9 How to Find the Inverse of a One-to-One Function

Find the inverse of f(x)=2x35.f(x)=2x35.

Try It 10.17

Find the inverse of the function f(x)=3x25.f(x)=3x25.

Try It 10.18

Find the inverse of the function f(x)=6x74.f(x)=6x74.

Section 10.1 Exercises

Practice Makes Perfect

Find and Evaluate Composite Functions

In the following exercises, find (fg)(x), (gf)(x), and (f · g)(x).

1.

f(x)=4x+3f(x)=4x+3 and g(x)=2x+5g(x)=2x+5

2.

f(x)=3x1f(x)=3x1 and g(x)=5x3g(x)=5x3

3.

f(x)=6x5f(x)=6x5 and g(x)=4x+1g(x)=4x+1

4.

f(x)=2x+7f(x)=2x+7 and g(x)=3x4g(x)=3x4

5.

f(x)=3xf(x)=3x and g(x)=2x23xg(x)=2x23x

6.

f(x)=2xf(x)=2x and g(x)=3x21g(x)=3x21

7.

f(x)=2x1f(x)=2x1 and g(x)=x2+2g(x)=x2+2

8.

f(x)=4x+3f(x)=4x+3 and g(x)=x24g(x)=x24

In the following exercises, find the values described.

9.

For functions f(x)=2x2+3f(x)=2x2+3 and g(x)=5x1,g(x)=5x1, find (fg)(−2)(fg)(−2) (gf)(−3)(gf)(−3) (ff)(−1)(ff)(−1)

10.

For functions f(x)=5x21f(x)=5x21 and g(x)=4x1,g(x)=4x1, find (fg)(1)(fg)(1) (gf)(−1)(gf)(−1) (ff)(2)(ff)(2)

11.

For functions f(x)=2x3f(x)=2x3 and g(x)=3x2+2,g(x)=3x2+2, find (fg)(−1)(fg)(−1) (gf)(1)(gf)(1) (gg)(1)(gg)(1)

12.

For functions f(x)=3x3+1f(x)=3x3+1 and g(x)=2x23,g(x)=2x23, find (fg)(−2)(fg)(−2) (gf)(−1)(gf)(−1) (gg)(1)(gg)(1)

Determine Whether a Function is One-to-One

In the following exercises, determine if the set of ordered pairs represents a function and if so, is the function one-to-one.

13.

{(−3,9),(−2,4),(−1,1),(0,0){(−3,9),(−2,4),(−1,1),(0,0),
(1,1),(2,4),(3,9)}(1,1),(2,4),(3,9)}

14.

{(9,−3),(4,−2),(1,−1),(0,0){(9,−3),(4,−2),(1,−1),(0,0),
(1,1),(4,2),(9,3)}(1,1),(4,2),(9,3)}

15.

{(−3,−5),(−2,−3),(−1,−1){(−3,−5),(−2,−3),(−1,−1),
(0,1),(1,3),(2,5),(3,7)}(0,1),(1,3),(2,5),(3,7)}

16.

{(5,3),(4,2),(3,1),(2,0){(5,3),(4,2),(3,1),(2,0),
(1,−1),(0,−2),(−1,−3)}(1,−1),(0,−2),(−1,−3)}

In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.

17.


This figure shows a graph of a circle with center at the origin and radius 3.



This figure shows a graph of a parabola opening upward with vertex at (0k, 2).
18.


This figure shows a parabola opening to the right with vertex at (negative 2, 0).



This figure shows a graph of a polynomial with odd order, so that it starts in the third quadrant, increases to the origin and then continues increasing through the first quadrant.
19.


This figure shows a graph of a curve that starts at (negative 6 negative 2) increases to the origin and then continues increasing slowly to (6, 2).



This figure shows a parabola opening upward with vertex at (0, negative 4).
20.


This figure shows a straight line segment decreasing from (negative 4, 6) to (2, 0), after which it increases from (2, 0) to (6, 4).



This figure shows a circle with radius 4 and center at the origin.

In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.

21.

{(2,1),(4,2),(6,3),(8,4)}{(2,1),(4,2),(6,3),(8,4)}

22.

{(6,2),(9,5),(12,8),(15,11)}{(6,2),(9,5),(12,8),(15,11)}

23.

{(0,−2),(1,3),(2,7),(3,12)}{(0,−2),(1,3),(2,7),(3,12)}

24.

{(0,0),(1,1),(2,4),(3,9)}{(0,0),(1,1),(2,4),(3,9)}

25.

{(−2,−3),(−1,−1),(0,1),(1,3)}{(−2,−3),(−1,−1),(0,1),(1,3)}

26.

{(5,3),(4,2),(3,1),(2,0)}{(5,3),(4,2),(3,1),(2,0)}

In the following exercises, graph, on the same coordinate system, the inverse of the one-to-one function shown.

27.


This figure shows a series of line segments from (negative 4, negative 3) to (negative 3, 0) then to (negative 1, 2) and then to (3, 4).
28.


This figure shows a series of line segments from (negative 4, negative 4) to (negative 3, 1) then to (0, 2) and then to (2, 4).
29.


This figure shows a series of line segments from (negative 4, 4) to (0, 3) then to (3, 2) and then to (4, negative 1).
30.


This figure shows a series of line segments from (negative 4, negative 4) to (negative 1, negative 3) then to (0, 1), then to (1, 3), and then to (4, 4).

In the following exercises, determine whether or not the given functions are inverses.

31.

f(x)=x+8f(x)=x+8 and g(x)=x8g(x)=x8

32.

f(x)=x9f(x)=x9 and g(x)=x+9g(x)=x+9

33.

f(x)=7xf(x)=7x and g(x)=x7g(x)=x7

34.

f(x)=x11f(x)=x11 and g(x)=11xg(x)=11x

35.

f(x)=7x+3f(x)=7x+3 and g(x)=x37g(x)=x37

36.

f(x)=5x4f(x)=5x4 and g(x)=x45g(x)=x45

37.

f(x)=x+2f(x)=x+2 and g(x)=x22g(x)=x22

38.

f(x)=x43f(x)=x43 and g(x)=x3+4g(x)=x3+4

In the following exercises, find the inverse of each function.

39.

f(x)=x12f(x)=x12

40.

f(x)=x+17f(x)=x+17

41.

f(x)=9xf(x)=9x

42.

f(x)=8xf(x)=8x

43.

f(x)=x6f(x)=x6

44.

f(x)=x4f(x)=x4

45.

f(x)=6x7f(x)=6x7

46.

f(x)=7x1f(x)=7x1

47.

f(x)=−2x+5f(x)=−2x+5

48.

f(x)=−5x4f(x)=−5x4

49.

f(x)=x2+6,f(x)=x2+6, x0x0

50.

f(x)=x29,f(x)=x29, x0x0

51.

f(x)=x34f(x)=x34

52.

f(x)=x3+6f(x)=x3+6

53.

f(x)=1x+2f(x)=1x+2

54.

f(x)=1x6f(x)=1x6

55.

f(x)=x2,f(x)=x2, x2x2

56.

f(x)=x+8,f(x)=x+8, x−8x−8

57.

f(x)=x33f(x)=x33

58.

f(x)=x+53f(x)=x+53

59.

f(x)=9x54,f(x)=9x54, x59x59

60.

f(x)=8x34,f(x)=8x34, x38x38

61.

f(x)=−3x+55f(x)=−3x+55

62.

f(x)=−4x35f(x)=−4x35

Writing Exercises

63.

Explain how the graph of the inverse of a function is related to the graph of the function.

64.

Explain how to find the inverse of a function from its equation. Use an example to demonstrate the steps.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four rows and four columns. The first row, which serves as a header, reads I can…, Confidently, With some help, and No—I don’t get it. The first column below the header row reads Find and evaluate composite functions, determine whether a function is one-to-one, and find the inverse of a function. The rest of the cells are blank.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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