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12 x1 t05 01 inverse functions (2012)

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  • 1. Inverse Functions
  • 2. Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is amaximum of one y value.
  • 3. Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is amaximum of one y value.The relation obtained by interchanging x and y is x = f(y)
  • 4. Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is amaximum of one y value.The relation obtained by interchanging x and y is x = f(y)e.g. y  x 3  x  x  y 3  y
  • 5. Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is amaximum of one y value.The relation obtained by interchanging x and y is x = f(y)e.g. y  x 3  x  x  y 3  yIf in this new relation, for each x value in the domain there is amaximum of one y value, (i.e. it is a function), then it is called theinverse function to y = f(x) and is symbolised y  f 1  x 
  • 6. Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is amaximum of one y value.The relation obtained by interchanging x and y is x = f(y)e.g. y  x 3  x  x  y 3  yIf in this new relation, for each x value in the domain there is amaximum of one y value, (i.e. it is a function), then it is called theinverse function to y = f(x) and is symbolised y  f 1  x A function and its inverse function are reflections of each other inthe line y = x.
  • 7. Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is amaximum of one y value.The relation obtained by interchanging x and y is x = f(y)e.g. y  x 3  x  x  y 3  yIf in this new relation, for each x value in the domain there is amaximum of one y value, (i.e. it is a function), then it is called theinverse function to y = f(x) and is symbolised y  f 1  x A function and its inverse function are reflections of each other inthe line y = x.If a, b  is a point on y  f  x , then b, a  is a point on y  f 1  x 
  • 8. Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is amaximum of one y value.The relation obtained by interchanging x and y is x = f(y)e.g. y  x 3  x  x  y 3  yIf in this new relation, for each x value in the domain there is amaximum of one y value, (i.e. it is a function), then it is called theinverse function to y = f(x) and is symbolised y  f 1  x A function and its inverse function are reflections of each other inthe line y = x.If a, b  is a point on y  f  x , then b, a  is a point on y  f 1  x The domain of y  f  x  is the range of y  f 1  x 
  • 9. Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is amaximum of one y value.The relation obtained by interchanging x and y is x = f(y)e.g. y  x 3  x  x  y 3  yIf in this new relation, for each x value in the domain there is amaximum of one y value, (i.e. it is a function), then it is called theinverse function to y = f(x) and is symbolised y  f 1  x A function and its inverse function are reflections of each other inthe line y = x.If a, b  is a point on y  f  x , then b, a  is a point on y  f 1  x The domain of y  f  x  is the range of y  f 1  x The range of y  f  x  is the domain of y  f 1  x 
  • 10. Testing For Inverse Functions
  • 11. Testing For Inverse Functions(1) Use a horizontal line test
  • 12. Testing For Inverse Functions(1) Use a horizontal line teste.g. i  y  x 2 y x
  • 13. Testing For Inverse Functions(1) Use a horizontal line teste.g. i  y  x 2 y x Only has an inverse relation
  • 14. Testing For Inverse Functions(1) Use a horizontal line teste.g. i  y  x 2 y ii  y  x 3 y x x Only has an inverse relation
  • 15. Testing For Inverse Functions(1) Use a horizontal line teste.g. i  y  x 2 y ii  y  x 3 y x x Only has an inverse relation Has an inverse function
  • 16. Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique.e.g. i  y  x 2 y ii  y  x 3 y x x Only has an inverse relation Has an inverse function
  • 17. Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique.e.g. i  y  x 2 y ii  y  x 3 y x x Only has an inverse relation Has an inverse function OR x  y2
  • 18. Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique.e.g. i  y  x 2 y ii  y  x 3 y x x Only has an inverse relation Has an inverse function OR x  y2 y x NOT UNIQUE
  • 19. Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique.e.g. i  y  x 2 y ii  y  x 3 y x x Only has an inverse relation Has an inverse function OR OR x  y2 x  y3 y x NOT UNIQUE
  • 20. Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique.e.g. i  y  x 2 y ii  y  x 3 y x x Only has an inverse relation Has an inverse function OR OR x  y2 x  y3 y x y3 x NOT UNIQUE UNIQUE
  • 21. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has aninverse function), then;
  • 22. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has aninverse function), then; f 1  f  x   x
  • 23. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has aninverse function), then; f 1  f  x   x AND f  f 1  x   x
  • 24. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. 2x 1 f x  3  2x
  • 25. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. 2x 1 f x  3  2x 2x 1 2 y 1y x 3  2x 3 2y
  • 26. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. 2x 1 f x  3  2x 2x 1 2 y 1y x 3  2x 3 2y 3  2 y x  2 y  1 3 x  2 xy  2 y  1 2 x  2  y  3 x  1 3x  1 y 2x  2
  • 27. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. 2x 1  2x 1   f x  3  1 3  2x 3  2x  f 1  f  x     2x 1  2x 1 2 y 1 2 2y x  3  2x  3  2x 3 2y 3  2 y x  2 y  1 3 x  2 xy  2 y  1 2 x  2  y  3 x  1 3x  1 y 2x  2
  • 28. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. 2x 1  2x 1   f x  3  1 3  2x 3  2x  f 1  f  x     2x 1  2x 1 2 y 1 2 2y x  3  2x  3  2x 3 2y 3  2 y x  2 y  1 6x  3  3  2x  3 x  2 xy  2 y  1 4x  2  6  4x 2 x  2  y  3 x  1  8x 3x  1 8 y x 2x  2
  • 29. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. 2x 1  2x 1    3x  1   f x  3  1 2  1 3  2x f  f 1  x    3  2x  2x  2  f 1  f  x     2x 1   3x  1  2x 1 2 y 1 2  2 3  2 y x  3  2x   2x  2  3  2x 3 2y 3  2 y x  2 y  1 6x  3  3  2x  3 x  2 xy  2 y  1 4x  2  6  4x 2 x  2  y  3 x  1  8x 3x  1 8 y x 2x  2
  • 30. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. 2x 1  2x 1    3x  1   f x  3  1 2  1 3  2x f  f 1  x    3  2x  2x  2  f 1  f  x     2x 1   3x  1  2x 1 2 y 1 2  2 3  2 y x  3  2x   2x  2  3  2x 3 2y 3  2 y x  2 y  1 6x  3  3  2x 6x  2  2x  2   3 x  2 xy  2 y  1 4x  2  6  4x 6x  6  6x  2 2 x  2  y  3 x  1  8x  8x 3x  1 8 8 y x x 2x  2
  • 31. Restricting The Domain
  • 32. Restricting The DomainIf a function does not have an inverse, we can obtain an inversefunction by restricting the domain of the original function.
  • 33. Restricting The DomainIf a function does not have an inverse, we can obtain an inversefunction by restricting the domain of the original function.When restricting the domain you need to capture as much of therange as possible.
  • 34. Restricting The DomainIf a function does not have an inverse, we can obtain an inversefunction by restricting the domain of the original function.When restricting the domain you need to capture as much of therange as possible. e.g. i  y  x 3 y y  x3 x
  • 35. Restricting The DomainIf a function does not have an inverse, we can obtain an inversefunction by restricting the domain of the original function.When restricting the domain you need to capture as much of therange as possible. e.g. i  y  x 3 y Domain: all real x y  x3 Range: all real y x
  • 36. Restricting The DomainIf a function does not have an inverse, we can obtain an inversefunction by restricting the domain of the original function.When restricting the domain you need to capture as much of therange as possible. e.g. i  y  x 3 y Domain: all real x y  x3 Range: all real y f 1 : x  y 3 x 1 y  x 3
  • 37. Restricting The DomainIf a function does not have an inverse, we can obtain an inversefunction by restricting the domain of the original function.When restricting the domain you need to capture as much of therange as possible. e.g. i  y  x 3 y Domain: all real x y  x3 Range: all real y f 1 : x  y 3 x 1 y  x 3 Domain: all real x Range: all real y
  • 38. Restricting The DomainIf a function does not have an inverse, we can obtain an inversefunction by restricting the domain of the original function.When restricting the domain you need to capture as much of therange as possible. e.g. i  y  x 3 y Domain: all real x y  x3 Range: all real y f 1 : x  y 3 x 1 y  x 3 Domain: all real x Range: all real y
  • 39. Restricting The DomainIf a function does not have an inverse, we can obtain an inversefunction by restricting the domain of the original function.When restricting the domain you need to capture as much of therange as possible. e.g. i  y  x 3 y Domain: all real x y  x3 Range: all real y f 1 : x  y 3 x 1 1 y  x 3 yx 3 Domain: all real x Range: all real y
  • 40. y  exii  y  e x y 1 x
  • 41. y  exii  y  e x y Domain: all real x Range: y > 0 1
  • 42. y  ex ii  y  e x y Domain: all real x Range: y > 0 1 1 xf :xe y  y  log x
  • 43. y  ex ii  y  e x y Domain: all real x Range: y > 0 1 1 xf :xe y  y  log x Domain: x > 0 Range: all real y
  • 44. y  ex ii  y  e x y Domain: all real x Range: y > 0 1 1 xf :xe y  y  log x Domain: x > 0 Range: all real y
  • 45. y  ex ii  y  e x y Domain: all real x y  log x Range: y > 0 1 1 x 1f :xe y  y  log x Domain: x > 0 Range: all real y
  • 46. iii  y  x 2 y  x2 y x
  • 47. iii  y  x 2 y  x2 y Domain: all real x Range: y  0 x
  • 48. iii  y  x 2 y  x2 y Domain: all real x Range: y  0 NO INVERSE x
  • 49. iii  y  x 2 y  x2 y Domain: all real x Range: y  0 NO INVERSE x Restricted Domain: x  0
  • 50. iii  y  x 2 y  x2 y Domain: all real x Range: y  0 NO INVERSE x Restricted Domain: x  0 Range: y  0
  • 51. iii  y  x 2 y  x2 y Domain: all real x Range: y  0 NO INVERSE x Restricted Domain: x  0 Range: y  0 f 1 : x  y 2 1 y  x 2
  • 52. iii  y  x 2 y  x2 y Domain: all real x Range: y  0 NO INVERSE x Restricted Domain: x  0 Range: y  0 f 1 : x  y 2 1 y  x 2 Domain: x  0 Range: y  0
  • 53. iii  y  x 2 y  x2 y Domain: all real x Range: y  0 NO INVERSE x Restricted Domain: x  0 Range: y  0 f 1 : x  y 2 1 y  x 2 Domain: x  0 Range: y  0
  • 54. iii  y  x 2 y  x2 1 y Domain: all real x yx 2 Range: y  0 NO INVERSE x Restricted Domain: x  0 Range: y  0 f 1 : x  y 2 1 y  x 2 Domain: x  0 Range: y  0
  • 55. iii  y  x 2 y  x2 1 y Domain: all real x yx 2 Range: y  0 NO INVERSE x Restricted Domain: x  0 Range: y  0 f 1 : x  y 2 1 y  x 2 Domain: x  0 Book 2 Exercise 1A; 2, 4bdf, 7, 9, 13, 14, 16, 19 Range: y  0

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