1. College Algebra Chapter 4 Exponential and Logarithmic Functions
Section 4.1
Inverse Functions

2. 1. Identify One-to-One Functions
2. Determine Whether Two Functions Are Inverses
3. Find the Inverse of a Function

3. Identify One-to-One Functions
A function f is a one-to-one function if for a and b in the domain of f,
or equivalently,

4. Example 1:
Determine if the relation defines y as a one-to-one function of x.

5. Example 2:
Determine if the relation defines y as a one-to-one function of x.

7. Identify One-to-One Functions
A function y = f (x) is a one-to-one function if no horizontal line intersects the graph in more than one place.

8. Example 3:
Determine if the relation defines y as a one-to-one function of x.

9. Example 4:
Determine if the relation defines y as a one-to-one function of x.

10. Example 5:
Determine if the relation defines y as a one-to-one function of x.

12. Example 6:
Use the definition of a one-to-one function to determine whether the function is one-to-one.
(Show that if )

13. Example 7:
Use the definition of a one-to-one function to determine whether the function is one-to-one.
(Show that if )

15. 1. Identify One-to-One Functions
2. Determine Whether Two Functions Are Inverses
3. Find the Inverse of a Function

16. Determine Whether Two Functions Are Inverses
Inverse Functions:
Let f be a one-to-one function. Then g is the inverse of f if the following conditions are both true.
Given a function and its inverse , then the definition implies that

17. Example 8:
Determine whether the two functions are inverses.

18. Example 9:
Determine whether the two functions are inverses.

20. 1. Identify One-to-One Functions
2. Determine Whether Two Functions Are Inverses
3. Find the Inverse of a Function

21. Find the Inverse of a Function
Procedure to Find an Equation of an Inverse of a Function
For a one-to-one function defined by y = f (x), the equation of the inverse can be found as follows:
Step Replace f (x) by y.
Step Interchange x and y.
Step Solve for y.
Step Replace y by

22. Example 10:
A one-to-one function is given. Write an equation for the inverse function.

23. Example 10 continued:

25. Example 11:
A one-to-one function is given. Write an equation for the inverse function.

26. Example 11 continued:

27. Example 11 continued:

28. Example 11 continued:

29. Example 11 continued:

31. Example 12:
A one-to-one function is given. Write an equation for the inverse function.

32. Example 12 continued:

35. Example 13:
The graph of a function is given. Graph the inverse function.