Notes 3-8

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Inverse Functions

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Notes 3-8

  1. 1. Section 3-8 Inverse Functions
  2. 2. Warm-up Indicate how you would “undo” each operation or composite of operations. 1. Turn east and walk 50 meters, then turn north and walk 30 meters. 4 2. Multiply a number by . 5 3. Add -70 to a number, then multiply the result by 14. 4. Square a positive number, then cube it.
  3. 3. Inverse of a function:
  4. 4. Inverse of a function: A function that will “undo” what another function had previously done
  5. 5. Inverse of a function: A function that will “undo” what another function had previously done When the independent variable is switched with the dependent variable
  6. 6. Inverse of a function: A function that will “undo” what another function had previously done When the independent variable is switched with the dependent variable **Notation: The inverse of f is f-1
  7. 7. Example 1 Let S = {(1,1), (2, 4), (3, 9), (4, 16)}. a. Find the inverse S-1. b. Describe S and its inverse in words.
  8. 8. Example 1 Let S = {(1,1), (2, 4), (3, 9), (4, 16)}. a. Find the inverse S-1. S-1 = {(1,1), (4, 2), (9, 3), (16, 4)} b. Describe S and its inverse in words.
  9. 9. Example 1 Let S = {(1,1), (2, 4), (3, 9), (4, 16)}. a. Find the inverse S-1. S-1 = {(1,1), (4, 2), (9, 3), (16, 4)} b. Describe S and its inverse in words. S is a squaring function, where the independent variable is squared to obtain the dependent variable.
  10. 10. Example 1 Let S = {(1,1), (2, 4), (3, 9), (4, 16)}. a. Find the inverse S-1. S-1 = {(1,1), (4, 2), (9, 3), (16, 4)} b. Describe S and its inverse in words. S is a squaring function, where the independent variable is squared to obtain the dependent variable. Its inverse is a positive square root function, where you would square root the independent variable to get the dependent variable.
  11. 11. Just a little note:
  12. 12. Just a little note: As the independent variable switches with the dependent variable, the domain switches with the range.
  13. 13. Theorem (Horizontal-Line Test)
  14. 14. Theorem (Horizontal-Line Test) If you can draw a horizontal line on the graph of f and it touches the graph more than once, then the INVERSE of f is not a function.
  15. 15. Theorem (Horizontal-Line Test) If you can draw a horizontal line on the graph of f and it touches the graph more than once, then the INVERSE of f is not a function. The horizontal-line test tells us nothing about the original function...remember that!
  16. 16. Example 2 Give an equation for the inverse and tell whether it is a function. a. f ( x ) = 6x + 5
  17. 17. Example 2 Give an equation for the inverse and tell whether it is a function. a. f ( x ) = 6x + 5 y = 6x + 5
  18. 18. Example 2 Give an equation for the inverse and tell whether it is a function. a. f ( x ) = 6x + 5 y = 6x + 5 x = 6y + 5
  19. 19. Example 2 Give an equation for the inverse and tell whether it is a function. a. f ( x ) = 6x + 5 y = 6x + 5 x = 6y + 5 −5 −5
  20. 20. Example 2 Give an equation for the inverse and tell whether it is a function. a. f ( x ) = 6x + 5 y = 6x + 5 x = 6y + 5 −5 −5 x − 5 = 6y
  21. 21. Example 2 Give an equation for the inverse and tell whether it is a function. a. f ( x ) = 6x + 5 y = 6x + 5 x = 6y + 5 −5 −5 x − 5 = 6y x−5 y= 6
  22. 22. Example 2 Give an equation for the inverse and tell whether it is a function. a. f ( x ) = 6x + 5 y = 6x + 5 x = 6y + 5 −5 −5 x − 5 = 6y x−5 y= or 6
  23. 23. Example 2 Give an equation for the inverse and tell whether it is a function. a. f ( x ) = 6x + 5 y = 6x + 5 x = 6y + 5 −5 −5 x − 5 = 6y x−5 y= or 6 1 5 y= x− 6 6
  24. 24. Example 2 4 b. y = 3x − 1
  25. 25. Example 2 4 b. y = 3x − 1 4 x= 3y − 1
  26. 26. Example 2 4 b. y = 3x − 1 4 x= 3y − 1 4 3y − 1 = x
  27. 27. Example 2 4 b. y = 3x − 1 4 x= 3y − 1 4 3y − 1 = x 4 3y = + 1 x
  28. 28. Example 2 4 b. y = 3x − 1 4 x= 3y − 1 4 3y − 1 = x 4 3y = + 1 x
  29. 29. Example 2 4 b. y = 3x − 1 4 x= 3y − 1 4 3y − 1 = x 4 4x 3y = + 1 3y = + xx x
  30. 30. Example 2 4 b. y = 3x − 1 4 x= 3y − 1 4 3y − 1 = x 4 4x 3y = + 1 3y = + xx x
  31. 31. Example 2 4 b. y = 3x − 1 4 x= 3y − 1 4 3y − 1 = x 4+x 4 4x 3y = + 1 3y = 3y = + xx x x
  32. 32. Example 2 4 b. y = 3x − 1 4 x= 3y − 1 4 3y − 1 = x 4+x 4 4x 3y = + 1 3y = 3y = + xx x x 4+x y= 3x
  33. 33. Question: How do you verify that two functions are inverses of each other?
  34. 34. Question: How do you verify that two functions are inverses of each other? Use the Inverse Function Theorem!
  35. 35. Question: How do you verify that two functions are inverses of each other? Use the Inverse Function Theorem! The IFT says that two functions f and g are inverses of each other IFF f(g(x)) = x for all x in the domain of g AND g(f(x)) = x for all x in the domain of f.
  36. 36. Example 3 Verify that the functions in Example 2a are inverses of each other.
  37. 37. Example 3 Verify that the functions in Example 2a are inverses of each other. To do this, we have to show that f(g(x)) = x and g(f(x)) = x.
  38. 38. Example 3 Verify that the functions in Example 2a are inverses of each other. To do this, we have to show that f(g(x)) = x and g(f(x)) = x. Let’s calculate this together.
  39. 39. Example 4 Explain why the functions f and g, with f(m) = m2 and g(m) = m-2 are not inverses.
  40. 40. Example 4 Explain why the functions f and g, with f(m) = m2 and g(m) = m-2 are not inverses. Calculate f(g(m)). If this composite does not give us a value of m, then we know they are not inverses. If it does, then we have to check g(f(m)).
  41. 41. Homework
  42. 42. Homework p. 212 # 1 - 20

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