1. Inverse Functions Algebra III, Sec. 1.9 Objective
You will learn how to find inverse functions graphically and algebraically.

2. Important Vocabulary
Inverse Function - a function that reverses another function
Horizontal Line Test – a method to determine if a function is one-to-one

3. Inverse Functions
For a function f that is defined by a set of ordered pairs, to form the inverse function of f, interchange the first and second coordinates of each of the ordered pairs. For a function f and its inverse f-1, the domain of f is equal to the range of f-1, and the range of f is equal to the domain of f-1.

4. Inverse Functions
To verify that two functions, f and g, are inverse functions of each other, perform the composition of each function.

5. Example
Find the inverse of f(x) = 8x. Then verify that both f(f- 1(x)) and f-1(f(x)) are equal to the identity function. The function f multiplies each input (x) by 8. To “undo” this function, you need to divide each input by 8. So…

6. Example 1
Find the inverse of f(x) = 8x. Then verify that both f(f- 1(x)) and f-1(f(x)) are equal to the identity function.

7. Example 2
Which of these functions is the inverse of f(x) = 5x + 8? or ✔

8. Example 2 (cont.)
Which of these functions is the inverse of f(x) = 5x + 8? or ✖

9. Example (on your handout)
Verify that the functions and are inverse functions of each other. ✔

10. The Graph of an Inverse Function
If the point (a, b) lies on the graph of f, then the point (b, a) must lie on the graph of f-1 and vice versa. The graph of f-1 is a reflection of the graph of f in the line y = x.

11. Example 3
Sketch the graphs of the inverse functions on the same rectangular coordinate system, and show that the graphs are reflections of each other in the line y = x. x
f-1(x) x
f-1(x) -7 -2 -4 -1 2 1 5 x
f(x) -2 -1 1 2 x
f(x) x
f(x) -2 -7 -1 -4 1 2 5

12. Example 4
Sketch the graphs of the inverse functions on the same rectangular coordinate system, and show that the graphs are reflections of each other in the line y = x.

13. One-to-One Functions
To tell whether a function has an inverse function from its graph, use the horizontal line test. A function f is one-to-one if each value of the dependent variable corresponds to exactly one value of the independent variable. A function f has inverse function if and only if f is one-to-one.
* No horizontal line intersects the graph at more than one point
* It passes the horizontal line test

14. Example 5
Use the graph of the function and the Horizontal Line Test to determine whether the function has an inverse.
Does it pass the horizontal line test?
Yes
So f has an inverse

15. Example (on your handout)
Does the graph of the function have an inverse function? Explain.
Does it pass the horizontal line test? No
So f does not have an inverse

16. Finding Inverse Functions Algebraically
To find the inverse of a function f algebraically,
Use the horizontal line test to decide whether f has an inverse function.
In the equation for f(x), replace f(x) with y.
Interchange the roles of x and y, and solve for y.
Replace y with f-1(x) in the new equation.
Verify that f and f-1 are inverse functions of each other by showing that the domain of f is the range of f-1 and vice versa, and the compositions give you the identity x.

17. Example 6
Find the inverse of f(x) = -4x – 9.

18. Example 7
Find the inverse of

19. Example (on your handout)
Find the inverse of f(x) = 4x – 5.

20. Practice
Sec 1.9, pg 90 – 91 # 23, 27, 35 (also sketch f and f-1), 37-40, 57, 69, 71