1. Precalculus Essentials
Fifth Edition
Chapter 1
Functions and Graphs
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2. 1.8 Inverse Functions

3. Objectives Verify inverse functions. Find the inverse of a function.
Use the horizontal line test to determine if a function has an inverse function.
Use the graph of a one-to-one function to graph its inverse function.
Find the inverse of a function and graph both functions on the same axes.

4. Definition of the Inverse of a Function
Let f and g be two functions such that f(g(x)) = x for every x in the domain of g and g(f(x)) = x for every x in the domain of f.

5. Example: Verifying Inverse Functions
Show that each function is the inverse of the other:
Solution:
f(g(x)) = g(f(x)) = x verifies that f and g are inverse functions.

6. Finding the Inverse of a Function (1 of 2)
The equation for the inverse of a function f can be found as follows:
Replace f(x) with y in the equation for f(x).
Interchange x and y.
Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.

7. Finding the Inverse of a Function (2 of 2)

8. Example: Finding the Inverse of a Function
Find the inverse of f (x) = 2x + 7.
Solution:
Step 1 Replace f(x) with y: y = 2x + 7.
Step 2 Interchange x and y: x = 2y + 7.

9. The Horizontal Line Test for Inverse Functions

10. Example: Applying the Horizontal Line Test
Which of the following graphs represent functions that have inverse functions?
Solution: Graph b represents a function that has an inverse.

11. Graphs of f and f inverse

12. Example: Graphing the Inverse Function (1 of 2)
Solution:

13. Example: Graphing the Inverse Function (2 of 2)
We verify our solution by observing the reflection of the graph about the line y = x.