1. College Algebra Chapter 4 Exponential and Logarithmic Functions
Section 4.1
Inverse Functions
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2. Concepts Identify One-to-One Functions
Determine Whether Two Functions Are Inverses
Find the Inverse of a Function

3. Concept 1 Identify One-to-One Functions

4. Identify One-to-One Functions
One-to-One Function: A function f is a one-to-one function if for a and b in the domain of f, if a ≠ b, then f(a) ≠ f(b), or equivalently, if f(a) = f(b), then a = b. A function y = f (x) is a one-to-one function if no horizontal line intersects the graph in more than one place.

5. Example 1
Determine if the relation defines y as a one-to-one function of x. {(2,1), (4,2), (7,3), (-2,1)}

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

7. Skill Practice 1 Determine whether the function is one-to-one.

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.

11. Skill Practice 2
Use the horizontal line test to determine if the graph 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 f(a) = f(b) then a = b) f(x) = 3x + 2

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

14. Skill Practice 3 Determine whether the function is one-to-one.
f(x) = -4x + 1
f(x) = |x| - 3

15. Concept 2 Determine Whether Two Functions Are Inverses

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 f(x) 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.

19. Skill Practice 4
Determine whether the function are inverses.

20. Concept 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 1: Replace f(x) by y. Step 2: Interchange x and y. Step 3: Solve for y. Step 4: Replace y by

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

23. Skill Practice 5
Write an equation for the inverse function for f(x) = 4x + 3.

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

25. Skill Practice 6
Write an equation for the inverse function for the one-to-one function defined by

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

27. Skill Practice 7
write an equation of the inverse.

28. Skill Practice 8
find an equation of the inverse.

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