11 x1 t02 08 inverse functions (2013)

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11 x1 t02 08 inverse functions (2013)

  1. 1. Inverse Relations
  2. 2. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  y
  3. 3. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relation
  4. 4. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relation
  5. 5. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x.
  6. 6. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x. e.g. y  x 2
  7. 7. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x. e.g. y  x 2 domain: all real x range: y  0
  8. 8. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x. e.g. y  x 2 inverse relation: x  y 2 domain: all real x range: y  0
  9. 9. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x. e.g. y  x 2 inverse relation: x  y 2 domain: all real x domain: x  0 range: y  0 range: all real y
  10. 10. Inverse Functions
  11. 11. Inverse FunctionsIf an inverse relation of a function, is a function, then it is calledan inverse function.
  12. 12. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test
  13. 13. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.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.
  14. 14. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.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. i  y  x 2 y x
  15. 15. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.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. i  y  x 2 y x
  16. 16. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.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. i  y  x 2 y xOnly has aninverse relation
  17. 17. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.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. i  y  x 2 y xOnly has an ORinverse relation x  y2 y x
  18. 18. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.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. i  y  x 2 y xOnly has an ORinverse relationx  y2 y x NOT UNIQUE
  19. 19. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.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. i  y  x 2 y ii  y  x 3 y x xOnly has an ORinverse relationx  y2 y x NOT UNIQUE
  20. 20. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.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. i  y  x 2 y ii  y  x 3 y x xOnly has an ORinverse relationx  y2 y x NOT UNIQUE
  21. 21. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.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. i  y  x 2 y ii  y  x 3 y x xOnly has an OR Has aninverse relationx  y2 inverse function y x NOT UNIQUE
  22. 22. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.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. i  y  x 2 y ii  y  x 3 y x xOnly has an OR OR Has aninverse relationx  y2 x  y3 inverse function y x y3 x NOT UNIQUE
  23. 23. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.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. i  y  x 2 y ii  y  x 3 y x xOnly has an OR OR Has aninverse relationx  y2 x  y3 inverse function y x y3 x NOT UNIQUE UNIQUE
  24. 24. 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
  25. 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. x2 f  x  x2
  26. 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. x2 f  x  x2 x2 y2y x x2 y2
  27. 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. x2 f  x  x2 x2 y2y x x2 y2  y  2 x  y  2 xy  2 x  y  2  x  1 y  2 x  2 2x  2 y 1 x
  28. 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. x2  x22 f  x  2  x2 x 2 f 1  f  x     x2 x2 y2 1   y x  x 2 x2 y2  y  2 x  y  2 xy  2 x  y  2  x  1 y  2 x  2 2x  2 y 1 x
  29. 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. x2  x22 f  x  2  x2 x 2 f 1  f  x     x2 x2 y2 1  y x  x 2 x2 y2 2x  4  2x  4  y  2 x  y  2  x2 x2 xy  2 x  y  2 4x  x  1 y  2 x  2  4 2x  2 x y 1 x
  30. 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. x2  x22  2x  2   2 f  x  2    x2 x 2 1 x  f 1  f  x     f  f 1  x     x2  2x  2   2 x2 y2 1    y x  x 2  1 x  x2 y2 2x  4  2x  4  y  2 x  y  2  x2 x2 xy  2 x  y  2 4x  x  1 y  2 x  2  4 2x  2 x y 1 x
  31. 31. 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. x2  x22  2x  2   2 f  x  2    x2 x 2 1 x  f 1  f  x     f  f 1  x     x2  2x  2   2 x2 y2 1    y x  x 2  1 x  x2 y2 2x  4  2x  4 2x  2  2  2x  y  2 x  y  2   x2 x2 2x  2  2  2x xy  2 x  y  2 4x 4x  x  1 y  2 x  2   4 4 2x  2 x x y 1 x
  32. 32. (ii) Draw the inverse relation y x
  33. 33. (ii) Draw the inverse relation y x
  34. 34. (ii) Draw the inverse relation y x
  35. 35. (ii) Draw the inverse relation y x Exercise 2H; 1aceg, 2, 3bdf, 5ac, 6bd, 7ac, 9bde, 10adfhj

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