Inverse Functions

Section 7.4

WHAT YOU WILL LEARN:

1. How to find the inverses of linear functions.

2. How to find inverses of nonlinear functions.

A Question

• If I ask you to do the following:

Take the number 3

Multiply it by 3

Add 7

Subtract 2

Divide by 2

How would you get back to the original number?

These two “functions” are inverses of one another.

Inverse Relations

X -2 -1 0 1

Y 4 2 0 -2

X 4 2 0 -2

Y -2 -1 0 1

An inverse relation maps the output values back to their original input values. This means that the domain of the inverse relation is the range of the original relation and that the range of inverse relation is the domain of the original relation.

Original relation: Inverse relation:

Finding Inverse Equations

• Find the inverse of the relation y = 2x – 4

• Now try graphing them. What do you notice?

• These functions are inverses of one another.

You Try!• Find the inverse of: y = -3x + 6

A “Definition”

Functions f and g are inverses of each other provided:

f(g(x)) = x and g(f(x)) = x

The function g is denoted by f-1, read as “f inverse”.

Verifying Inverse Functions• Verify that f(x) = 2x – 4 and f-1 = ½ x + 2 are

inverses.

You Try• Verify that f(x) = -3x + 6 and f-1(x) = -1/3x + 2 are

inverses.

Graphs of Inverse Functions• What is the inverse of y = x2?

• What is the inverse of y = x3?

• Graph each function and its inverse on the same set of axes. Are the inverse relations actually functions?

Finding Inverses with Restricted Domains

0x• Find the inverse of f(x) = x2, for

• Is the relation, with the restriction, a function?

Another Line Test• This will help you to determine if an inverse

relation is a function:

• If no horizontal line intersects the graph of a function f more than once, then the inverse of f is itself a function. Otherwise known as the “horizontal line test”.

Another Example

22

1)( 3 xxf• Consider the function . Determine

whether the inverse of f is a function. Then find the inverse.

• Step 1: graph the function

• Step 2: Switch x and y.

• Step 3: Solve for y.

You Try

42)( 2 xxf• Consider the function . Determine whether the inverse of f is a function and then find the inverse.

Homework

Homework: page 426, 16-20 even, 26, 28, 33-35 all, 36, 38, 42, 44, 48-52 even, 58