11 X1 T02 08 inverse functions (2010)

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

Inverse Relations

The domain of the relation is the range of its inverse relation

Inverse Relations

The range of the relation is the domain of its inverse relation

Inverse Relations

A relation and its inverse relation are reflections of each other in
the line y = x.

Inverse Relations

the line y = x.
e.g. y  x 2

Inverse Relations

the line y = x.
e.g. y  x 2
domain: all real x
range: y  0

Inverse Relations

the line y = x.
e.g. y  x 2 inverse relation: x  y 2
domain: all real x
range: y  0

Inverse Relations

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

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

Inverse Functions
Testing For Inverse Functions
(1) Use a horizontal line test

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.

Inverse Functions
i  y  x 2 y

x

Inverse Functions
i  y  x 2 y

x
Only has an
inverse relation

Inverse Functions
i  y  x 2 y

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

Inverse Functions
i  y  x 2 y

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

Inverse Functions
i  y  x 2 y ii  y  x 3 y

x x
Only has an OR
y x
NOT UNIQUE

Inverse Functions
i  y  x 2 y ii  y  x 3 y

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

Inverse Functions
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

Inverse Functions
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

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


f 1  f  x   x AND f  f 1  x   x

e.g. x2
f  x 
x2


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


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


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


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


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


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

(ii) Draw the inverse relation
y

x

(ii) Draw the inverse relation
y

x

Exercise 2H; 1aceg, 2, 3bdf, 5ac, 6bd, 7ac, 9bde, 10adfhj

11 X1 T02 08 inverse functions (2010)

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11 X1 T02 08 inverse functions (2010)