Holt Algebra 2

9-5 Functions and Their Inverses

Determine whether the inverse of a function is a function.

Write rules for the inverses of functions.

Objectives

Holt Algebra 2

9-5 Functions and Their Inverses

one-to-one function

Vocabulary

When both a relation and its inverses are functions, the relation is called a one-to-one function.

In a one-to-one function, each y-value is paired with exactly one x-value.

Holt Algebra 2

9-5 Functions and Their InversesRecall that the vertical-line test can help you determine whether a relation is a function.

Vertical Line Test: a relation is a function if a vertical line drawn through its graph, passes through only one point.

Similarly, the horizontal-line test can help you determine whether the inverse of a function is a function.

Holt Algebra 2

9-5 Functions and Their Inverses

04/21/23 03:24

1-6 Relations and Functions

4

Vertical Line Test

Would this graph be a function?YES

Holt Algebra 2

9-5 Functions and Their Inverses

04/21/23 03:24

1-6 Relations and Functions

5

Vertical Line Test

Would this graph be a function?

NO

Holt Algebra 2

9-5 Functions and Their Inverses

Use the horizontal line test to decide if the inverse is a function.

Holt Algebra 2

9-5 Functions and Their Inverses

Use the horizontal-line test to determine whether the inverse of the blue relation is a function.

Example 1A: Using the Horizontal-Line Test

The inverse is a function because no horizontal line passes through two points on the graph.

Holt Algebra 2

9-5 Functions and Their Inverses

Use the horizontal-line test to determine whether the inverse of the red relation is a function.

Example 1B: Using the Horizontal-Line Test

The inverse is a not a function because a horizontal line passes through more than one point on the graph.

Holt Algebra 2

9-5 Functions and Their Inverses

Use the horizontal-line test to determine whether the inverse of each relation is a function.

The inverse is a function because no horizontal line passes through two points on the graph.

Check It Out! Example 1

Holt Algebra 2

9-5 Functions and Their Inverses

Recall from Lesson 7-2 that to write the rule for the inverse of a function, you can exchange x and y and solve the equation for y. Because the value of x and y are switched, the domain of the function will be the range of its inverse and vice versa.

Holt Algebra 2

9-5 Functions and Their Inverses

Example 2: Writing Rules for inverses

Find the inverse of . Determine whether it is a function, and state its domain and range.

Step 1 The horizontal-line test shows that the inverse is a function. Note that the domain and range of f are all real numbers.

Holt Algebra 2

9-5 Functions and Their Inverses

Example 2 Continued

Rewrite the function using y instead of f(x).

Step 1 Find the inverse.

Simplify.

Switch x and y in the equation.

Cube both sides.

Isolate y.

Holt Algebra 2

9-5 Functions and Their Inverses

Example 2 Continued

Because the inverse is a function, .

The domain of the inverse is the range of f(x):{x|x R}.

The range is the domain of f(x):{y|y R}.

Check Graph both relations to see that they are symmetric about y = x.

Holt Algebra 2

9-5 Functions and Their Inverses

Find the inverse of f(x) = x3 – 2. Determine whether it is a function, and state its domain and range.

Step 1 The horizontal-line test shows that the inverse is a function. Note that the domain and range of f are all real numbers.

Check It Out! Example 2

Holt Algebra 2

9-5 Functions and Their Inverses

Rewrite the function using y instead of f(x).

Step 1 Find the inverse.

Take the cube root of both sides.

Switch x and y in the equation.

Add 2 to both sides of the equation.

Simplify.

Check It Out! Example 2 Continued

y = x3 – 2

x = y3 – 2

x + 2 = y3

3 x + 2 = y

33 3x + 2 = y

Holt Algebra 2

9-5 Functions and Their Inverses

The domain of the inverse is the range of f(x): R.

The range is the domain of f(x): R.

Check Graph both relations to see that they are symmetric about y = x.

Check It Out! Example 2 Continued

Because the inverse is a function, .

Holt Algebra 2

9-5 Functions and Their Inverses

You have seen that the inverses of functions are not necessarily functions. When both a relation and its inverses are functions, the relation is called a one-to-one function. In a one-to-one function, each y-value is paired with exactly one x-value.

You can use composition of functions to verify that two functions are inverses. Because inverse functions “undo” each other, when you compose two inverses the result is the input value x.

Holt Algebra 2

9-5 Functions and Their Inverses

Holt Algebra 2

9-5 Functions and Their Inverses

Determine by composition whether each pair of functions are inverses.

Example 3: Determining Whether Functions Are Inverses

Find the composition f(g(x)).

f(g(x)) = 3( x + 1) – 1 1 3

Use the Distributive Property.

Simplify.

f(x) = 3x – 1 and g(x) = x + 1 1 3

Substitute x + 1 for x in f.

1 3

= (x + 3) – 1

= x + 2

Holt Algebra 2

9-5 Functions and Their Inverses

Because f(g(x)) ≠ x, f and g are not inverses. There is no need to check g(f(x)).

Example 3 Continued

Check The graphs are not symmetric about the line y = x.

Holt Algebra 2

9-5 Functions and Their Inverses

Example 3B: Determining Whether Functions Are Inverses

Find the compositions f(g(x)) and g(f (x)).

For x ≠ 1 or 0, f(x) = and g(x) = + 1. 1 x

1x – 1

Because f(g(x)) = g(f (x)) = x for all x but 0 and 1, f and g are inverses.

= x

= (x – 1) + 1

= x

Holt Algebra 2

9-5 Functions and Their Inverses

Example 3B Continued

Check The graphs are symmetric about the line y = x for all x but 0 and 1.

Holt Algebra 2

9-5 Functions and Their Inverses

Check It Out! Example 3a

Determine by composition whether each pair of functions are inverses.

Find the composition f(g(x)) and g(f(x)).

3 2

f(x) = x + 6 and g(x) = x – 9 2 3

= x – 6 + 6

= x

f(g(x)) = ( x – 9) + 6 3 2

2 3

g(f(x)) = ( x + 6) – 9 2 3

3 2

= x + 9 – 9

= x

Because f(g(x)) = g(f(x)) = x, they are inverses.

Holt Algebra 2

9-5 Functions and Their Inverses

Check The graphs are symmetric about the line y = x for all x.

Check It Out! Example 3a Continued

Holt Algebra 2

9-5 Functions and Their Inverses

Find the compositions f(g(x)) and g(f(x)).

Simplify.

Check It Out! Example 3b

Substitute for x in f.

f(g(x)) = + 5

f(x) = x2 + 5 and for x ≥ 0

= x + 25 +5 10 x

= x – 10 x + 30

Holt Algebra 2

9-5 Functions and Their Inverses

Because f(g(x)) ≠ x, f and g are not inverses. There is no need to check g(f(x)).

Check The graphs are not symmetric about the line y = x.

Check It Out! Example 3b Continued

Holt Algebra 2

9-5 Functions and Their Inverses

Lesson Quiz: Part I

A: yes; B: no

1. Use the horizontal-line test to determine whether the inverse of each relation is a function.

Holt Algebra 2

9-5 Functions and Their Inverses

Lesson Quiz: Part II

D: {x|x ≥ 4}; R: {all Real Numbers}

2. Find the inverse f(x) = x2 – 4. Determine whether it is a function, and state its domain and range.

not a function

Holt Algebra 2

9-5 Functions and Their Inverses

3. Determine by composition whether f(x) =

3(x – 1)2 and g(x) = +1 are inverses

for x ≥ 0.

Lesson Quiz: Part III

yes