Inverse Functions and

Relations

LESSON 6–2

Over Lesson 6–1

5-Minute Check 6

A. f(1)

B. g(1)

C. (g ○ f)(1)

D. f(0)

Let f(x) = x – 3 and g(x) = x2. Which of the following is equivalent to (f ○ g)(1)?

TEKS

Targeted TEKS

A2.2(D) Use the composition of two

functions, including the necessary

restrictions on the domain, to determine if the

functions are inverses of each other.

Also addresses A2.2(C).

Mathematical Processes

A2.1(E), A2.1(G)

Then/Now

• Find the inverse of a function or relation.

• Determine whether two functions or relations are inverses.

Vocabulary

• inverse relation

• inverse function

Example 1

Find an Inverse Relation

GEOMETRY The ordered pairs of the relation {(1, 3), (6, 3), (6, 0), (1, 0)} are the coordinates of the vertices of a rectangle. Find the inverse of this relation. Describe the graph of the inverse.

To find the inverse of this relation, reverse the coordinates of the ordered pairs. The inverse of the relation is {(3, 1), (3, 6), (0, 6), (0, 1)}.

Example 1

Find an Inverse Relation

Answer: Plotting the points shows that the ordered pairs also describe the vertices of a rectangle. Notice that the graph of the relation and the inverse are reflections over the graph of y = x.

Example 1

A. cannot be determined

B. {(–3, 4), (–1, 5), (2, 3), (1, –2)}

C. {(–3, 4), (–1, 5), (2, 3), (1, 1), (–2, 1)}

D. {(4, –3), (5, –1), (3, 2), (1, 1), (1, –2)}

GEOMETRY The ordered pairs of the relation {(–3, 4), (–1, 5), (2, 3), (1, 1), (–2, 1)} are the coordinates of the vertices of a pentagon. What is the inverse of this relation?

Concept

Example 2

Find and Graph an Inverse

Step 1 Replace f(x) with y in the original equation.

Then graph the

function and its inverse.

Step 2 Interchange x and y.

Example 2

Find and Graph an Inverse

Step 3 Solve for y.

Inverse

Step 4 Replace y with f –1(x).

y = –2x + 2 f –1(x) = –2x + 2

Multiply each side by –2.

Add 2 to each side.

Example 2

Find and Graph an Inverse

Example 2

Find and Graph an Inverse

Answer:

Example 2

A.

B.

C.

D.

Graph the function

and its inverse.

Concept

Example 2

Step 1 Replace f(x) with y in the original equation.

Step 2 Interchange x and y.

Find the inverse of f(x) = x2 – 4x + 1. Then

graph the function and its inverse. If

necessary, restrict the domain of the inverse

to that it is a function.

Inverses with Restricted Domains

Example 2

Find and Graph an Inverse

Step 3 Solve for y.

Inverse

Subtract 1 from each side.

Complete the square

Simplify

Take the square root of each side.

Add 2 to each side.

Example 2

Find and Graph an Inverse

If the domain is restricted to [4, ∞), find the inverse of

f(x) = x2 – 8x + 10.

Check to see if the compositions of f(x) and g(x) are identity functions.

Verify that Two Functions are Inverses

Answer: The functions are inverses since both [f ○ g](x) and [g ○ f](x) equal x.

A. They are not inverses since [f ○ g](x) = x + 1.

B. They are not inverses since both compositions equal x.

C. They are inverses since both compositions equal x.

D. They are inverses since both compositions equal x + 1.

Inverse Functions and

Relations

LESSON 6–2