1. Functions and Their Inverses
Students will be able to find the inverse of a function and to determine if the inverse is a function.

2. Inverse Relations You have seen the word inverse used in various ways.
The additive inverse of 3 is –3.
The multiplicative inverse of 5 is .
You can also find and apply inverses to relations and functions. To graph the inverse relation, you can reflect each point across the line y = x. This is equivalent to switching the x- and y-values in each ordered pair of the relation.
FHS
Functions

3. Remember! A relation is a set of ordered pairs.
A function is a relation in which each x- value has, at most, one y-value paired with it.
To find the inverse of a function, switch places with the x and y variables and solve for y.
FHS
Functions

4. So the inverse function is written this way:
Example 2
Use inverse operations to write the inverse of
y = 3(x – 7).
Switch places with the x and the y. Now, solve for y.
Divide by 3.
Add 7.
So the inverse function is written this way:
To Check: Use a sample input. 
FHS
Functions

5. Inverse Functions
We have learned that the inverse of a function f(x) “undoes” f(x). However, the inverse may or may not be a function.
Recall that the vertical-line test can help you determine whether a relation is a function.
Similarly, the horizontal-line test can help you determine whether the inverse of a function is a function.
FHS
Functions

6. Using the Horizontal-Line Test
Example 5
Using the Horizontal-Line Test
Use the horizontal-line test to determine whether the inverse of the red relation is a function.
The inverse is a not a function because a horizontal line passes through more than one point on the graph.
FHS
Functions