## Objectives

– Verify inverse functions

– Use the horizontal line test to determine if a

function is a one-to-one function.

– Find the inverse of a function.

– Given a graph, graph the inverse .

– Find the inverse of a function & graph both

functions simultaneously .

## What is an inverse function?

• A function that “undoes” the original function.

• A function “wraps an x” and the inverse would

“unwrap the x” resulting in x when the 2

functions are composed on each other.

**Example**

Given that f(x) = 7x − 2, use composition of

functions to show that f^{−1}(x) = (x + 2)/7.

## Do all functions have inverses?

• Yes, and no. Yes, they all will have inverses,

BUT we are only interested in the inverses if

they ARE A FUNCTION.

**• DO ALL FUNCTIONS HAVE INVERSES THAT**

ARE FUNCTIONS? NO.

• Recall, functions must pass the vertical line test

when graphed. If the inverse is to pass the

vertical line test , the original function must pass

the HORIZONTAL line test (be one-to-one)!

## One-to-One Functions

A function f(x) is a one-to-one function if

x- values do not share the same y-values.

Remember that a function will have

different x -values.

A one-to-one function will have different

x- values and different y -values.

## Why are one-to-one functions important?

**One-to-One Functions**

have

Inverse functions

## Horizontal Line Test

• Use to determine whether a function is

one-to-one.

• A function is **one-to-one** if and only if **no**

horizontal line intersects its graph more

than once

## Horizontal-Line Test

Graph f(x) = −3x + 4.

**Example:** From

the graph at the

left, determine

whether the

function is one-to-

one and thus

has an inverse

that is a function.

**Example:**

From the

graph at the

left, determine

whether the

function is

one-to-one

and thus has

an inverse that

is a function.

## How do you find an inverse?

• “Undo” the function.

OR

• Replace the x with y and solve for y .

## How to find the Inverse of a

One-to-One Function

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

2. Interchange x and y in the equation.

3. Solve this equation for y .

4. Replace y with f^{-1}(x).

**Any restrictions on x or y should be considered and**

included with the equation.

Remember: **Domain and Range are interchanged**

for inverses.

**Example**

Determine whether the function f(x) = 3x − 2

is one-to-one, and if it is, find a formula for

f ^{-1}(x).

• The graph of a function and

its inverse always mirror

each other through the line

y=x.

• Example:y = (1/3)x + 2 and

its inverse = 3(x-2)

• Every point on the graph

(x,y) exists on the inverse

as (y,x) (i.e. if (-6,0) is on

the graph, (0,-6) is on its

inverse.

## Graph of Inverse f^{-1} function

• The graph of f^{-1} is obtained by reflecting the

graph of f across the line y = x.

• To graph the inverse f^{-1} function:

Interchange the points on the graph of f to

obtain the points on the graph of f^{-1}.

**Example**

Graph f(x) = 3x − 2 and

using the same set of axes.

Then compare the two graphs .

Determine the domain and range of the function

and its inverse.

• If a function is one-to-one, then its

inverse is a function.

• The domain of a one-to-one function f is

the range of the inverse f^{-1}.

• The range of a one-to-one function f is

the domain of the inverse f^{-1}.

• A function that is increasing over its

domain or is decreasing over its domain

is a one-to-one function.

## Restricting a Domain

• When the inverse of a function is not a

function, the domain of the function can

be restricted to allow the inverse to be a

function.

• In such cases, it is convenient to consider

“part” of the function by restricting the

domain of f(x). If the domain is restricted,

then its inverse is a function.

## Restricting the Domain

**Recall that if a function is not one-to-one,**

then its inverse will not be a function.

**If we restrict the domain values of f(x) to those
greater**

than or equal to zero , we see that f(x) is now one-to-one

and its inverse is now a function.

**Example**

• For f(x) = x² - 1, x < 0:

a.) Find the equation for the inverse , f^{-1}

b.) Find the domain and range for the function and its
inverse.