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. Inverse functions satisfy the properties f^-1(f(x)) = x and f(f^-1(x)) = x. Examples are provided to demonstrate testing if an inverse function exists.

1. Inverse Functions

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

31. Restricting The Domain

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