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# AA Section 8-2

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Inverses of Relations

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• ### AA Section 8-2

1. 1. Section 8-2 Inverses of Relations
2. 2. What are inverses and how do we use them?
3. 3. Warm-up Determine if each set of ordered pairs is a function. 1. {(3, 5), (5, 5), (7, 5), (9, 5)} 2. {(1, 1), (2, 4), (3, 9), (1, -1), (2, -4)}
4. 4. Warm-up Determine if each set of ordered pairs is a function. 1. {(3, 5), (5, 5), (7, 5), (9, 5)} Yes 2. {(1, 1), (2, 4), (3, 9), (1, -1), (2, -4)}
5. 5. Warm-up Determine if each set of ordered pairs is a function. 1. {(3, 5), (5, 5), (7, 5), (9, 5)} Yes 2. {(1, 1), (2, 4), (3, 9), (1, -1), (2, -4)} No
6. 6. Inverse of a Relation
7. 7. Inverse of a Relation The relationship obtained by reversing the order of coordinates of each ordered pair in the relation
8. 8. Example 1 g = {(4, 3), (0, -1), (5, 2), (-8, -1)} a. Identify the inverse of g. Call it f. b. Is g a function? Is f a function?
9. 9. Example 1 g = {(4, 3), (0, -1), (5, 2), (-8, -1)} a. Identify the inverse of g. Call it f. f = {(3, 4), (-1, 0), (2, 5), (-1, -8)} b. Is g a function? Is f a function?
10. 10. Example 1 g = {(4, 3), (0, -1), (5, 2), (-8, -1)} a. Identify the inverse of g. Call it f. f = {(3, 4), (-1, 0), (2, 5), (-1, -8)} b. Is g a function? Is f a function? g is a function; f is not
11. 11. ***NOTICE*** Domain of f = range of g = Range of f = domain of g =
12. 12. ***NOTICE*** Domain of f = range of g = {-1, 2, 3} Range of f = domain of g =
13. 13. ***NOTICE*** Domain of f = range of g = {-1, 2, 3} Range of f = domain of g = {-8, 0, 4, 5}
14. 14. Inverse Relation Theorem Suppose f is a relation and g is the inverse of f.
15. 15. Inverse Relation Theorem Suppose f is a relation and g is the inverse of f. 1.A rule for g can be found by switching x and y
16. 16. Inverse Relation Theorem Suppose f is a relation and g is the inverse of f. 1.A rule for g can be found by switching x and y 2.The graph of g is the reﬂection of f over the line y = x
17. 17. Inverse Relation Theorem Suppose f is a relation and g is the inverse of f. 1.A rule for g can be found by switching x and y 2.The graph of g is the reﬂection of f over the line y = x 3.The domain of g is the range of f; the range of g is the domain of f
18. 18. Example 2 Consider the function y = 4x -1 a. Find its inverse
19. 19. Example 2 Consider the function y = 4x -1 a. Find its inverse x = 4y - 1
20. 20. Example 2 Consider the function y = 4x -1 a. Find its inverse x = 4y - 1 -1 -1
21. 21. Example 2 Consider the function y = 4x -1 a. Find its inverse x = 4y - 1 -1 -1 x - 1 = 4y
22. 22. Example 2 Consider the function y = 4x -1 a. Find its inverse x = 4y - 1 -1 -1 x - 1 = 4y 4 4
23. 23. Example 2 Consider the function y = 4x -1 a. Find its inverse x = 4y - 1 -1 -1 x - 1 = 4y 4 4 1 1 y= x+4 4
24. 24. Example 2 b. Graph the two equations.
25. 25. Example 2 b. Graph the two equations.
26. 26. Example 2 b. Graph the two equations.
27. 27. Example 2 b. Graph the two equations.
28. 28. Horizontal-Line Theorem
29. 29. Horizontal-Line Theorem If you can draw a horizontal line on a graph and it intersects the graph more than once, then the INVERSE is not a function; if it only touches once, then the INVERSE is a function
30. 30. Example 3 Is the inverse a function? How do you know? 2 y = x − 3x + 2
31. 31. Example 3 Is the inverse a function? How do you know? 2 y = x − 3x + 2
32. 32. Example 3 Is the inverse a function? How do you know? 2 y = x − 3x + 2
33. 33. Example 3 Is the inverse a function? How do you know? 2 y = x − 3x + 2 The inverse is not a function. There is at least one spot where a horizontal line can be drawn and it touches the graph more than once.
34. 34. Homework
35. 35. Homework p. 487 #1-19 “Maybe it’s easier to like someone else’s life, and live vicariously through it, than take some responsibility to change our lives into lives we might like.” - Tish Grier