Inverse of Functions

1. Inverse of Functions Finding the inverse of functions Finding the inverse function

2. Inverse Functions Functions f and g are inverse functions if: - f(g(x)) = x - g(f(x)) = x For all values of x in the domain of f and g

3. Finding the Inverse of a Function by Reversing Operations The trick is to “undo” all the operations on x in reverse order. That means once we list the operation of a function and inverse each operation in reverse order, we will have the inverse

4. Example 1 • Find the inverse of the following function: f(x) = 2x – 4 The function has two steps: • Multiply by 2. • Subtract 4. Thus, f-1(x) must have two steps: • Add 4. • Divide by 2. Consequently, f-1(x) = [(x + 4)/2] .

5. Example 1 We can verify that this is the inverse of f(x) : f-1(f(x)) = f-1(2x - 4) = [((2x - 4) + 4)/2] = [2x/2] = x . f(f-1(x)) = f([(x + 4)/2]) = 2([(x + 4)/2]) - 4 = (x + 4) - 4 = x .

6. Example 2 • Find the inverse of f(x) = 3(x - 5) . Original function: • Subtract 5. • Multiply by 3. Finish problem on your own

7. Example 2 - answer • f-1(x) = [x/3] + 5 Check: f-1(f(x)) = f-1(3(x - 5)) = [(3(x - 5))/3] + 5 = (x - 5) + 5 = x . f(f-1(x)) = f([x/3] + 5) = 3(([x/3] + 5) - 5) = 3([x/3]) = x .

8. Example 3 • Find the inverse of f(x) = , x ³ 2 (we must restrict the domain because f(x) is undefined for x < 2 ). Original function: • Subtract 2. • Take the square root.

9. Example 3 - solution • f-1(x) = x2 + 2 • Check on your own Note: • When we take the inverse of a function, the domain and range switch. In example 3, the domain of f is x ³ 2 and the range of f is f(x) ³ 0 . Thus, the domain of f-1 is x ³ 0 and the range of f-1 is f-1(x) ³ 2 .

10. Finding the Inverse of a Function by Isolating f-1(x) • Taking the inverse "reverses" x and f(x) . • In the original function, substitute "x " for "f(x) " and substitute "f-1(x) " for "x ". • Then, solve for f-1(x) using inverse operations in the usual manner.

11. In other words: Switch the x and the y coordinate. Then, solve for the “new” y.

12. Example 1 • Find the inverse of: f(x) = [(2x - 1)/5] . 1. Substitute: x = [(2(f-1(x)) - 1)/5] 2. Solve for f-1(x) :

13. Example 1 - solution • f-1(x) = (5x + 1)/2 • Check on your own

14. Example 2 • [1/((x - 1)2)] , x ¹ 1 , f(x) > 0 . 1. Substitute: x = [1/((f-1(x) - 1)2)] 2. Solve for f-1(x) :

15. Example 2 - solution • f-1(x) =(1/Öx) + 1 • Domain of f-1 : x > 0 . Range of f-1 : f-1(x) ¹ 1 .

16. Finding the Inverse of a Function by Graphing Graphically: • The inverse of a function is a reflection of that function over the line y = x .

17. Graphically Inverse of a Graph • In other words, all points (x, y) = (a, b) become (x, y) = (b, a) • The x and y coordinates of each point switch

18. One approach • To find the inverse of a function, reflect the function over the line y = x . Or, find several points on the graph of y = f(x) , switch their x and y coordinates, and graph the resulting points. Connect these points with a line or curve that mirrors the line or curve of the original function.

19. Example 1 • Find the inverse y = 2x - 1 by graphing:

20. Question • Is any inverse of a function also a function? • Explain your answer.