Section 8-2
Inverses of Relations
What are inverses and how do
       we use them?
Warm-up
Determine if each set of ordered pairs is a function.
           1. {(3, 5), (5, 5), (7, 5), (9, 5)}




       2....
Warm-up
Determine if each set of ordered pairs is a function.
           1. {(3, 5), (5, 5), (7, 5), (9, 5)}


           ...
Warm-up
Determine if each set of ordered pairs is a function.
           1. {(3, 5), (5, 5), (7, 5), (9, 5)}


           ...
Inverse of a Relation
Inverse of a Relation


The relationship obtained by reversing the order of
 coordinates of each ordered pair in the relat...
Example 1
     g = {(4, 3), (0, -1), (5, 2), (-8, -1)}

    a. Identify the inverse of g. Call it f.




    b. Is g a fun...
Example 1
     g = {(4, 3), (0, -1), (5, 2), (-8, -1)}

    a. Identify the inverse of g. Call it f.

     f = {(3, 4), (-...
Example 1
     g = {(4, 3), (0, -1), (5, 2), (-8, -1)}

    a. Identify the inverse of g. Call it f.

     f = {(3, 4), (-...
***NOTICE***
      Domain of f = range of g =




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


              {-1, 2, 3}


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


              {-1, 2, 3}


      Range of f = domain of g =


            ...
Inverse Relation Theorem
 Suppose f is a relation and g is the inverse of f.
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 ...
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 ...
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 ...
Example 2
     Consider the function y = 4x -1
           a. Find its inverse
Example 2
     Consider the function y = 4x -1
           a. Find its inverse
               x = 4y - 1
Example 2
     Consider the function y = 4x -1
           a. Find its inverse
                x = 4y - 1
               -1...
Example 2
     Consider the function y = 4x -1
           a. Find its inverse
                x = 4y - 1
               -1...
Example 2
     Consider the function y = 4x -1
           a. Find its inverse
                x = 4y - 1
               -1...
Example 2
     Consider the function y = 4x -1
           a. Find its inverse
                x = 4y - 1
               -1...
Example 2
     b. Graph the two equations.
Example 2
     b. Graph the two equations.
Example 2
     b. Graph the two equations.
Example 2
     b. Graph the two equations.
Horizontal-Line Theorem
Horizontal-Line Theorem

    If you can draw a horizontal line on a graph and it
intersects the graph more than once, then...
Example 3
 Is the inverse a function? How do you know?
                    2
             y = x − 3x + 2
Example 3
 Is the inverse a function? How do you know?
                    2
             y = x − 3x + 2
Example 3
 Is the inverse a function? How do you know?
                    2
             y = x − 3x + 2
Example 3
     Is the inverse a function? How do you know?
                          2
                   y = x − 3x + 2

...
Homework
Homework


                     p. 487 #1-19




“Maybe it’s easier to like someone else’s life, and live
vicariously thro...
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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

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