The document discusses inverse relations and inverse functions. It defines an inverse relation as interchanging the x and y variables of a relation. An inverse function exists when the inverse relation is also a function, as determined by the horizontal line test or uniqueness of solutions. Examples are provided to illustrate inverse relations that are functions and those that are not. Key properties of inverse functions are stated.

1. Inverse Relations

2. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y  x 3  x inverse relation is x  y 3  y

3. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y  x 3  x inverse relation is x  y 3  y
The domain of the relation is the range of its inverse relation

4. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y  x 3  x inverse relation is x  y 3  y
The domain of the relation is the range of its inverse relation
The range of the relation is the domain of its inverse relation

5. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y  x 3  x inverse relation is x  y 3  y
The domain of the relation is the range of its inverse relation
The range of the relation is the domain of its inverse relation
A relation and its inverse relation are reflections of each other in
the line y = x.

6. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y  x 3  x inverse relation is x  y 3  y
The domain of the relation is the range of its inverse relation
The range of the relation is the domain of its inverse relation
A relation and its inverse relation are reflections of each other in
the line y = x.
e.g. y  x 2

7. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y  x 3  x inverse relation is x  y 3  y
The domain of the relation is the range of its inverse relation
The range of the relation is the domain of its inverse relation
A relation and its inverse relation are reflections of each other in
the line y = x.
e.g. y  x 2
domain: all real x
range: y  0

8. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y  x 3  x inverse relation is x  y 3  y
The domain of the relation is the range of its inverse relation
The range of the relation is the domain of its inverse relation
A relation and its inverse relation are reflections of each other in
the line y = x.
e.g. y  x 2 inverse relation: x  y 2
domain: all real x
range: y  0

9. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y  x 3  x inverse relation is x  y 3  y
The domain of the relation is the range of its inverse relation
The range of the relation is the domain of its inverse relation
A relation and its inverse relation are reflections of each other in
the 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. Inverse Functions

11. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.

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. 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. 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. 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. 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
Only has an
inverse relation

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
x
Only has an OR
inverse relation x  y2
y x

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
x
Only has an OR
inverse relationx  y2
y x
NOT UNIQUE

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 x
Only has an OR
inverse relationx  y2
y x
NOT UNIQUE

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 x
Only has an OR
inverse relationx  y2
y x
NOT UNIQUE

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 x
Only has an OR
Has an
inverse relationx  y2
inverse function
y x
NOT UNIQUE

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 x
Only has an OR OR
Has an
inverse relationx  y2 x  y3
inverse function
y x y3 x
NOT UNIQUE

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 x
Only has an OR OR
Has an
inverse relationx  y2 x  y3
inverse function
y x y3 x
NOT UNIQUE UNIQUE

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

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. x2
f  x 
x2

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. x2
f  x 
x2
x2 y2
y x
x2 y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. x2
f  x 
x2
x2 y2
y 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. 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. 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. 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. 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. 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. 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. 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. 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. (ii) Draw the inverse relation
y
x

33. (ii) Draw the inverse relation
y
x

34. (ii) Draw the inverse relation
y
x

35. (ii) Draw the inverse relation
y
x
Exercise 2H; 1aceg, 2, 3bdf, 5ac, 6bd, 7ac, 9bde, 10adfhj