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