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

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

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

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

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

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