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12 x1 t05 01 inverse functions (2013)

N
Nigel Simmons

The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(

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Inverse Functions
Inverse Functions If y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.
Inverse Functions If y = f(x) is a function, then for each x in the domain, there is a maximum of one y value. The relation obtained by interchanging x and y is x = f(y)
Inverse Functions If y = f(x) is a function, then for each x in the domain, there is a maximum 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
Inverse Functions If y = f(x) is a function, then for each x in the domain, there is a maximum 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 If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised y  f 1  x 
Inverse Functions If y = f(x) is a function, then for each x in the domain, there is a maximum 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 If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised y  f 1  x  A function and its inverse function are reflections of each other in the line y = x.
Inverse Functions If y = f(x) is a function, then for each x in the domain, there is a maximum 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 If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised y  f 1  x  A function and its inverse function are reflections of each other in the line y = x. If a, b  is a point on y  f  x , then b, a  is a point on y  f 1  x 
Inverse Functions If y = f(x) is a function, then for each x in the domain, there is a maximum 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 If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised y  f 1  x  A function and its inverse function are reflections of each other in the 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 
Inverse Functions If y = f(x) is a function, then for each x in the domain, there is a maximum 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 If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised y  f 1  x  A function and its inverse function are reflections of each other in the 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 
Testing For Inverse Functions
Testing For Inverse Functions (1) Use a horizontal line test
Testing For Inverse Functions (1) Use a horizontal line test e.g. i  y  x 2 y x
Testing For Inverse Functions (1) Use a horizontal line test e.g. i  y  x 2 y x Only has an inverse relation
Testing For Inverse Functions (1) Use a horizontal line test e.g. i  y  x 2 y ii  y  x 3 y x x Only has an inverse relation
Testing For Inverse Functions (1) Use a horizontal line test e.g. i  y  x 2 y ii  y  x 3 y x x Only has an inverse relation Has 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. e.g. i  y  x 2 y ii  y  x 3 y x x Only has an inverse relation Has 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. 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
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
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
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
If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;
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
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   x
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   x e.g. 2x 1 f x  3  2x
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   x e.g. 2x 1 f x  3  2x 2x 1 2 y 1 y x 3  2x 3 2y
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   x e.g. 2x 1 f x  3  2x 2x 1 2 y 1 y 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
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   x e.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 2 y 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
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   x e.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 2 y 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
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   x e.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
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   x e.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
Restricting The Domain
Restricting The Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.
Restricting The Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range as possible.
Restricting The Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range as possible. e.g. i  y  x 3 y y  x3 x
Restricting The Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range as possible. e.g. i  y  x 3 y Domain: all real x y  x3 Range: all real y x
Restricting The Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range 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
Restricting The Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range 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
Restricting The Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range 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
Restricting The Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range 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
y  ex ii  y  e x y 1 x
y  ex ii  y  e x y Domain: all real x Range: y > 0 1
y  ex ii  y  e x y Domain: all real x Range: y > 0 1 1 x f :xe y  y  log x
y  ex ii  y  e x y Domain: all real x Range: y > 0 1 1 x f :xe y  y  log x Domain: x > 0 Range: all real y
y  ex ii  y  e x y Domain: all real x Range: y > 0 1 1 x f :xe y  y  log x Domain: x > 0 Range: all real y
y  ex ii  y  e x y Domain: all real x y  log x Range: y > 0 1 1 x 1 f :xe y  y  log x Domain: x > 0 Range: all real y
iii  y  x 2 y  x2 y x
iii  y  x 2 y  x2 y Domain: all real x Range: y  0 x
iii  y  x 2 y  x2 y Domain: all real x Range: y  0 NO INVERSE x
iii  y  x 2 y  x2 y Domain: all real x Range: y  0 NO INVERSE x Restricted Domain: x  0
iii  y  x 2 y  x2 y Domain: all real x Range: y  0 NO INVERSE x Restricted Domain: x  0 Range: y  0
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
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
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
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
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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12 x1 t05 01 inverse functions (2013)

  • 2. Inverse Functions If y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.
  • 3. Inverse Functions If y = f(x) is a function, then for each x in the domain, there is a maximum of one y value. The relation obtained by interchanging x and y is x = f(y)
  • 4. Inverse Functions If y = f(x) is a function, then for each x in the domain, there is a maximum 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 Functions If y = f(x) is a function, then for each x in the domain, there is a maximum 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 If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised y  f 1  x 
  • 6. Inverse Functions If y = f(x) is a function, then for each x in the domain, there is a maximum 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 If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised y  f 1  x  A function and its inverse function are reflections of each other in the line y = x.
  • 7. Inverse Functions If y = f(x) is a function, then for each x in the domain, there is a maximum 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 If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised y  f 1  x  A function and its inverse function are reflections of each other in the 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 Functions If y = f(x) is a function, then for each x in the domain, there is a maximum 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 If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised y  f 1  x  A function and its inverse function are reflections of each other in the 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 Functions If y = f(x) is a function, then for each x in the domain, there is a maximum 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 If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised y  f 1  x  A function and its inverse function are reflections of each other in the 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 test e.g. i  y  x 2 y x
  • 13. Testing For Inverse Functions (1) Use a horizontal line test e.g. i  y  x 2 y x Only has an inverse relation
  • 14. Testing For Inverse Functions (1) Use a horizontal line test e.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 test e.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 an inverse function), then;
  • 22. 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
  • 23. 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   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   x e.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   x e.g. 2x 1 f x  3  2x 2x 1 2 y 1 y 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   x e.g. 2x 1 f x  3  2x 2x 1 2 y 1 y 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   x e.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 2 y 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   x e.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 2 y 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   x e.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   x e.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
  • 32. Restricting The Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.
  • 33. Restricting The Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range as possible.
  • 34. Restricting The Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range as possible. e.g. i  y  x 3 y y  x3 x
  • 35. Restricting The Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range as possible. e.g. i  y  x 3 y Domain: all real x y  x3 Range: all real y x
  • 36. Restricting The Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range 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 Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range 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 Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range 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 Domain If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function. When restricting the domain you need to capture as much of the range 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 x f :xe y  y  log x
  • 43. y  ex ii  y  e x y Domain: all real x Range: y > 0 1 1 x f :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 x f :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 1 f :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