Inverse Functions

1

Inverse Functions

If is a function consisting of ordered pairs , ,

then there is a relation called the ,

whose elements are the ord

inverse o

ered pair

f

s , .

f

f x y

y x

2

Inverse Functions

1

1

If is one-to-one, then the inverse of is

called the inverse function of denoted by

.

f f

y f x f y

f

f

x

3

Example 2.4.10

1

5 3Given ,

4

a. Find .

5 3

45 3

4

xf x

f x

xf x

xy

4

1

5 3

45 3

44 5 3

4 3 5

4 3

5

4 3

5

xy

yx

x y

x y

xy

xf x

5

1 1

1 1

b. Find and

4 3

5

4 35 3

5

44 3 3

44

4

f f x f f x

f f x f f x

xf

x

x

x

x

1

5 3

44 3

5

xf x

xf x

6

1 1

5 3

4

5 34 3

4

55 3 3

5

5

5

f f x f f x

xf

x

x

x

x

1

5 3

44 3

5

xf x

xf x

7

-2 -1 1 2

-2

-1

1

2

x

y

1

1

c. Show the graph of and in one

coordinate plane.

5 3

4

4 3

5

f f

xf x

xf x

x 0 -3/5

y 3/4 0

x 0 3/4

y -3/5 0

f

1f y x

8

Properties

1

1

1

1 1

1.

2.

3. The graphs of and are symmetric

with respect to the line .

4.

Dom f Rng f

Rng f Dom f

f f

y x

f f x f f x x

9

Example 2.4.1

1

Given 1 2,

a. Find .

1 2

1 2

1 2

g x x

g x

g x x

y x

x y

2

2

21

1 2

1 2

1 2

1 2

x y

x y

x y

g x x

10

1

21

1

1

b. Find the domain and range of

1 2

1 2

2 0 2,

2 1,

1,

2,

g

g x x

g x x

x Dom g

x Rng g

Dom g

Rng g

11

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

1

21

1

1

c. Show that the graph of and are

symmetric with respect to the line

1 2

2,

1 2

1,

2,

g g

y x

g x x

Dom g

g x x

Dom g

Rng g

x 2 3

y 1 2

y x

g

1g

12

1 1

1 1

2

2

2

d. Show that

1 2

1 1 2 2

1 1

1 1

g g x g g x x

g g x g g x

g x

x

x

x

x

21

1 2

1 2

g x x

g x x

13

1 1

2

2

1 2

1 2 1 2

2 2

2 2

g g x g g x

g x

x

x

x

x

21

1 2

1 2

g x x

g x x

14