1. Key Concept 1

2. Apply the Horizontal Line Test
Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no.
Example 1

3. Apply the Horizontal Line Test
Graph the function f (x) = x 5 + x 3 – 1 using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no.
Example 1

4. Graph the function using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no.
Example 1

5. Key Concept 2

6. Determine whether f has an inverse function for
Find Inverse Functions Algebraically
Determine whether f has an inverse function for
. If it does, find the inverse function and state any restrictions on its domain.
The graph of f passes the horizontal line test. Therefore, f is a one-one function and has an inverse function. From the graph, you can see that f has domain
and range Now find f –1. 
Example 2

7. Find Inverse Functions Algebraically
Example 2

8. 2xy – y = x Isolate the y-terms. y(2x –1) = x Factor.
Find Inverse Functions Algebraically
Original function
Replace f(x) with y.
Interchange x and y.
2xy – x = y Multiply each side by 2y – 1. Then apply the Distributive Property.
2xy – y = x Isolate the y-terms.
y(2x –1) = x Factor.
Example 2

9. Find Inverse Functions Algebraically
Divide.
Example 2

10. Find Inverse Functions Algebraically
From the graph, you can see that f –1 has domain and range The domain and range of f is equal to the range and domain of f –1, respectively. Therefore, it is not necessary to restrict the domain of f –1. 
Answer: f –1 exists;
Example 2

11. Find Inverse Functions Algebraically
Determine whether f has an inverse function for If it does, find the inverse function and state any restrictions on its domain.
Example 2

12. Determine whether f has an inverse function for
Determine whether f has an inverse function for If it does, find the inverse function and state any restrictions on its domain.
Example 2

13. Key Concept 3

14. Show that f [g (x)] = x and g [f (x)] = x.
Verify Inverse Functions
Show that f [g (x)] = x and g [f (x)] = x.
Example 3

15. Verify Inverse Functions
Because f [g (x)] = x and g [f (x)] = x, f (x) and g (x) are inverse functions. This is supported graphically because f (x) and g (x) appear to be reflections of each other in the line y = x.
Example 3

16. Verify Inverse Functions
Answer:
Example 3

17. Show that f (x) = x 2 – 2, x  0 and are inverses of each other.
Example 3

18. Use the graph of relation A to sketch the graph of its inverse.
Find Inverse Functions Graphically
Use the graph of relation A to sketch the graph of its inverse.
Example 4

19. Find Inverse Functions Graphically
Graph the line y = x. Locate a few points on the graph of f (x). Reflect these points in y = x. Then connect them with a smooth curve that mirrors the curvature of f (x) in line y = x.
Answer:
Example 4

20. Use the graph of the function to graph its inverse function.
Example 4

21. Use an Inverse Function
MANUFACTURING The fixed costs for manufacturing one type of stereo system are $96,000 with variable cost of $80 per unit. The total cost f (x) of making x stereos is given by f (x) = 96, x. Find f –1(x) and explain what f –1(x) and x represent in the inverse function.
Example 5