1. Inverse Functions
Section 7.4

2. What You Will Learn: How to find the inverses of linear functions.
How to find inverses of nonlinear functions.

3. A Question If I ask you to do the following: Take the number 3
Multiply it by 3
Add 7
Subtract 2
Divide by 2
How would you get back to the original number?
These two “functions” are inverses of one another.

4. Inverse Relations X 4 2 -2 Y -1 1 X -2 -1 1 Y 4 2
An inverse relation maps the output values back to their original input values. This means that the domain of the inverse relation is the range of the original relation and that the range of inverse relation is the domain of the original relation.
Original relation: Inverse relation: X -2 -1 1 Y 4 2 X 4 2 -2 Y -1 1

5. Finding Inverse Equations
Find the inverse of the relation y = 2x – 4
Now try graphing them. What do you notice?
These functions are inverses of one another.

6. You Try!
Find the inverse of: y = -3x + 6

7. A “Definition”
Functions f and g are inverses of each other provided: f(g(x)) = x and g(f(x)) = x The function g is denoted by f-1, read as “f inverse”.

8. Verifying Inverse Functions
Verify that f(x) = 2x – 4 and f-1 = ½ x + 2 are inverses.

9. You Try
Verify that f(x) = -3x + 6 and f-1(x) = -1/3x + 2 are inverses.

10. Graphs of Inverse Functions
What is the inverse of y = x2?
What is the inverse of y = x3?
Graph each function and its inverse on the same set of axes. Are the inverse relations actually functions?

11. Finding Inverses with Restricted Domains
Find the inverse of f(x) = x2, for
Is the relation, with the restriction, a function?

12. Another Line Test
This will help you to determine if an inverse relation is a function:
If no horizontal line intersects the graph of a function f more than once, then the inverse of f is itself a function. Otherwise known as the “horizontal line test”.

13. Another Example
Consider the function Determine whether the inverse of f is a function. Then find the inverse.
Step 1: graph the function
Step 2: Switch x and y.
Step 3: Solve for y.

14. You Try
Consider the function Determine whether the inverse of f is a function and then find the inverse.

15. Homework
Homework: page 426, even, 26, 28, all, 36, 38, 42, 44, even, 58