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# Inverse Functions Help (page 2)

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By McGraw-Hill Professional
Updated on Oct 4, 2011

## One-To-One Functions

A function has an inverse if its graph passes the Horizontal Line Test—if any horizontal line touches the graph in more than one place, then the function will not have an inverse. Functions whose graphs pass the Horizontal Line Test are called one-to-one functions. For a one-to-one function, every x will be paired with exactly one y and every y will be paired with exactly one x .

#### Example

• The graph of f ( x ) is given in Figure 4.7. Is f ( x ) one to one?

Fig. 4.7

This graph fails the Horizontal Line Test, so f ( x ) is not one to one.

Fig. 4.8

For functions that are not one to one, we can restrict the domain to force the function to be one to one. The function whose graph is in Figure 4.9, f ( x ) = x 2 −3, is not one to one. If we restrict the domain to x ≥ 0, then the new function is one to one.

Finding the inverse function is not hard, but it can be a little tedious. The steps below show the process of algebraically switching x and y.

1. Replace f ( x ) with x , and replace x with y .
2. Solve this equation for y .
3. Replace y with f −1 ( x ).

Fig. 4.9

Fig. 4.10

#### Examples

Find f −1 ( x ).

• f ( x ) = 6 x + 14

• f ( x ) = 9( x - 4) 5

## Inverse Functions Practice Problems

#### Practice

1. Show that and g ( x ) = 2 x − 14 are inverses.
2. Show that and g ( x ) = x 3 + 8 are inverses.
3. Show that and g are inverses.
4. Use the graph of f ( x ) in Figure 4.11 to sketch the graph of f −1 ( x ).
5. Find f −1 ( x ) for f ( x ) = 5 x + 12.
6. Find g −1 ( x ) for .
7. Find f −1 ( x ) for

Fig. 4.11

#### Solutions

1.

2.

3.

4. The solid graph in Figure 4.12 is the graph of f ( x ), and the dashed graph is the graph of f −1 ( x ).

Fig. 4.12.

5.

6.

7.

Find practice problems and solutions for these concepts at Combinations of Functions and Inverse Functions Practice Test.

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