Inverse Functions: Horizontal Line Test & Algebraic Calculations

1. Key Concept 1

2. Apply the Horizontal Line Test Graph the function f(x)= 4x2 + 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) = x5+ x3– 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 graphingcalculator, 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 . 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.   Find Inverse Functions Algebraically Example 2

7. Find Inverse Functions Algebraically Example 2

8. 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. 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; Find Inverse Functions Algebraically Example 2

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

12. 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. 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) = x2 – 2, x 0 and are inverses of each other. Example 3

18. 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) = 96,000 + 80x. Find f–1(x) and explain what f–1(x) and x represent in the inversefunction. Example 5