1. Key Concept 1

2. Example 1 Apply the Horizontal Line Test A. 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. The graph of f (x) = 4x 2 + 4x + 1 shows that it is possible to find a horizontal line that intersects the graph of f (x) more than once. Therefore, you can conclude that f –1 does not exist. Answer: no

3. Example 1 Apply the Horizontal Line Test B. 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. The graph of f (x) = x 5 + x 3 – 1 shows that it is not possible to find a horizontal line that intersects the graph of f (x) more than one point. Therefore, you can conclude that f –1 exists. Answer: yes

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

5. Key Concept 2

6. Example 2 Find Inverse Functions Algebraically A. 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. 

7. Example 2 Find Inverse Functions Algebraically

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

9. Example 2 Find Inverse Functions Algebraically Divide.

10. Example 2 Find Inverse Functions Algebraically Answer: f –1 exists; 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. 

11. Example 2 Find Inverse Functions Algebraically B. 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.

12. Example 2 Find Inverse Functions Algebraically Original function Replace f(x) with y. Interchange x and y. Divide each side by 2. Square each side.

13. Example 2 Find Inverse Functions Algebraically Add 1 to each side. Replace y with f – 1 (x). From the graph, you can see that f – 1 has domain and range. By restricting the domain of f – 1 to, the range remains. Only then are the domain and range of f equal to the range and domain of f –1, respectively. So,.

14. Example 2 Find Inverse Functions Algebraically Answer: f –1 exists with domain ;

15. Example 2 Determine whether f has an inverse function for. If it does, find the inverse function and state any restrictions on its domain. A. B. C. D.f –1 (x) does not exist.

16. Key Concept 3

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

18. Example 3 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.

19. Example 3 Verify Inverse Functions Answer:

20. Example 3 Show that f (x) = x 2 – 2, x  0 and are inverses of each other. A. B. C.D.

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

22. Example 4 Answer: 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.

23. Example 4 Use the graph of the function to graph its inverse function. A. B. C. D.