This document discusses inverses of relations. It defines an inverse as the relationship obtained by reversing the order of coordinates in each ordered pair. An example shows how to find the inverse of a relation and determine if the original relation and inverse are functions. The inverse relation theorem states that the inverse can be found by switching x and y, the graph of the inverse is a reflection over y=x, and the domain of the inverse is the range of the original and vice versa. Another example shows finding the inverse of a function and graphing both. The horizontal line test is introduced to determine if an inverse is a function. Homework problems are assigned.

1. Section 8-2
Inverses of Relations

2. What are inverses and how do
we use them?

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. 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. 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. Inverse of a Relation

7. Inverse of a Relation
The relationship obtained by reversing the order of
coordinates of each ordered pair in the relation

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. 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. 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. ***NOTICE***
Domain of f = range of g =
Range of f = domain of g =

12. ***NOTICE***
Domain of f = range of g =
{-1, 2, 3}
Range of f = domain of g =

13. ***NOTICE***
Domain of f = range of g =
{-1, 2, 3}
Range of f = domain of g =
{-8, 0, 4, 5}

14. Inverse Relation Theorem
Suppose f is a relation and g is the inverse of f.

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. 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. 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. Example 2
Consider the function y = 4x -1
a. Find its inverse

19. Example 2
Consider the function y = 4x -1
a. Find its inverse
x = 4y - 1

20. Example 2
Consider the function y = 4x -1
a. Find its inverse
x = 4y - 1
-1 -1

21. Example 2
Consider the function y = 4x -1
a. Find its inverse
x = 4y - 1
-1 -1
x - 1 = 4y

22. Example 2
Consider the function y = 4x -1
a. Find its inverse
x = 4y - 1
-1 -1
x - 1 = 4y
4 4

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. Example 2
b. Graph the two equations.

25. Example 2
b. Graph the two equations.

26. Example 2
b. Graph the two equations.

27. Example 2
b. Graph the two equations.

28. Horizontal-Line Theorem

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. Example 3
Is the inverse a function? How do you know?
2
y = x − 3x + 2

31. Example 3
Is the inverse a function? How do you know?
2
y = x − 3x + 2

32. Example 3
Is the inverse a function? How do you know?
2
y = x − 3x + 2

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. Homework

35. Homework
p. 487 #1-19
“Maybe it’s easier to like someone else’s life, and live
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change our lives into lives we might like.” - Tish Grier