1.8 Inverse Functions

1. 1.8 Inverse Functions

2. Any function can be represented by a set of ordered pairs. For example: f(x) = x + 5 → goes from the set A = {1, 2, 3, 4} to the set B {5, 6, 7, 8} This can also be represented by: f(x) = x + 5: {(1, 5), (2, 6), (3, 7), (4, 8)}

3. Inverse Functions • An inverse function, denoted by , is found by interchanging the first and second coordinates of each ordered pair. f(x) = x + 4: {(1, 5), (2, 6), (3, 7), (4, 8)} = : {(5, 1), (6, 2), (7, 3), (8, 4)}

4. Domain of f(x) Range of f(x) x f(x)

5. If functions are inverses: • If f(x) and g(x) are two functions that are inverses of each other, then: • (f ○ g) (x) = x • (g ○ f) (x) = x

6. f(x) = x + 4 g(x) = x - 4 • To verify that these 2 functions are inverses, you must show that f(g(x)) = x and g(f(x)) = x f(g(x)) g(f(x)) = f(x - 4) = g(x + 4) = x - 4 + 4 = x + 4 - 4 = x = x

7. Verify that the functions are inverses: • f(x) = 3x – 2 g(x) = x + 2 • f(x) = x – 3 g(x) = 12 + 4x • f(x) = 2x + 4 g(x) = x - 2

8. Finding inverses informally: What does this function do? How can this be “undone”?

9. Find the inverses of the following: • f(x) = x – 3 • f(x) = 7x • f(x) = • f(x) =

10. Graphs of Inverse Functions: • If the point (a, b) lies on the graph of f(x), then the point (b, a) must lie on the graph of the inverse function. • This means that the graph of is a reflection of the graph of f(x) over the line y = x

11. Graph the function and its inverse. f(x) = x + 2

12. Graph the function and its inverse. f(x) = 2x - 3

13. Do all functions have an inverse? Think about what an inverse function actually does

14. Horizontal Line Test • A function f has an inverse if and only if no horizontal line intersects the graph of f at more than one point.

15. Does the function have an inverse?

16. Does the function have an inverse?

17. Does the function have an inverse?

18. One-to-One Function • A function is said to be one-to-one if for every input, there is exactly one output and for every output, there is exactly one input. • If a function is one-to-one, the function has an inverse

19. f(x) = x² Does every input have exactly 1 output? -2 4 Does every output have exactly 1 input? -1 1 0 0 1 1 The function is not 1-1 2 4 3 9

20. f(x) = x³ Does every input have exactly 1 output? Does every output have exactly 1 input? The function is 1-1

21. 1.8 Inverse Functions Finding Inverse Functions Algebraically

22. Find the inverse of the function: For complicated functions, it is best to find the inverse function algebraically.

23. Finding Inverse Functions Algebraically • Use the horizontal line test to determine whether f has an inverse • In the equation f(x), replace f(x) with y • Interchange the roles of x and y, then solve for y • Replace y with in the new equation • Verify your answer

24. 1) Does it pass the horizontal line test?

25. 2) Replace f(x) with y 3) Interchange x and y and solve for y

26. 2) Replace f(x) with y 3) Interchange x and y and solve for y 4) Replace y with 5) Verify your answer

27. Find the inverse of the following functions.

31. Review 1.7 & 1.8 • Basic operations on functions • Composition of functions • Domain of functions (interval notation) • Finding Inverses • Verifying Inverses • Graphing functions vs. inverses • Domains of inverses

32. One-to-One Function • A function is said to be one-to-one if for every input, there is exactly one output and for every output, there is exactly one input. • If a function is one-to-one, the function has an inverse

33. f(x) = x³ Does every input have exactly 1 output? Does every output have exactly 1 input? The function is 1-1

34. f(x) = x² Does every input have exactly 1 output? -2 4 Does every output have exactly 1 input? -1 1 0 0 1 1 The function is not 1-1 2 4 3 9

35. Interval Notation • Identify the domain of the function using interval notation:

36. Graph the function and determine whether or not it has an inverse

37. Find the inverse of the function

38. Verify the inverse of the function