Chapter 2.3 Properties of

Functions

1

One-to-One Functions

A function is if and only

if whenever and are two numbers in the

domain of and th

one-to-one 1

e .

1

n

f

a b

f a b f a f b

2

One-to-One Functions

, then

Equivalently, is one-to-one if and only

if whenever .

One-to-one functions are also called

functioninjective s.

f a f b a b

f

3

Example 2.3.1

Determine if the following functions are

one-to-one or not by using the definition.

1. 3 7

3 7 3 7

3 3

Therefore, the function is 1-1.

y f x x

f a f b

a b

a b

a b

4

2 12.

2 1 2 1

2 1 2 1

2 2

Therefore, the function is 1-1.

xy g x

x

g a g b

a b

a b

a b b a

ab b ab a

b a

b a

5

23.

1 1 but 1 1.

Therefore, the function is not 1-1.

y h x x

h h

6

Horizontal Line Test

every horizontal li

A function is one-to-one if and only if

intersects the graph

of the func

ne

at most ontio e pn o in .int

f

7

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

Example 2.3.2

Determine if the following functions are

one-to-one or not by using horizontal

line test.

1. 3 7y f x x

1-1

8

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

2 12.

xy g x

x

1-1

9

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

23. y h x x

not 1-1

10

Onto Functions

A function from to , : is

called if for all there is an

onto

such that .

f A B f A B

b B

a A f a b

11

Onto Functions

Each element of has a in

under .

An onto function is also called .

pre-image

surjective

B

A f

12

Onto Functions

If is a function from to then is onto

if and only if .Rng

f B f

f

A

B

13

Example 2.3.3

Determine if the following functions are onto

given the indicated sets.

1. :

where 3 7

. Therefore is onto.

f R R

f x x

Rng f R f

14

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

2. : 0 2

2 1where

2

Therefore is onto.

g R R

xg x

x

Rng g R

g

15

3. : 0

2 1where

2

Therefore is not onto.

h R R

xh x

x

Rng h R R

h

16

2

4. :

where

0,

Therefore is not onto.

m R R

m x x

Rng m R

h

17

Bijective Functions

A function is if it is both

one-to-one a

bijec

nd o

tive

nto.

18

Example 2.3.4

Determine if the following functions are

bijective given the indicated sets.

1. :

where 3 7

Since is 1-1 and onto, is bijective.

f R R

f x x

f f

19

2. : 0 2

2 1where

Since is 1-1 and onto, is bijective.

g R R

xg x

x

g g

20

2

3. : 0,

where

Is 1-1?

Is onto?

Is bijective?

h R

h x x

h

h

h

21

Increasing Functions

1 2 1 2

1 2

A function is on an interval

if and only if

whenever

where and are any numbers in the

interva

increas

l.

ingf

f x f x x x

x x

22

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

2Consider 4 on ,0f x x

1 2

1 2

If then x x

f x f x

24 is increasing on ,0f x x 23

Decreasing Functions

1 2 1 2

1 2

A function is on an interval

if and only if

whenever

where and are any numbers in the

interva

decreas

l.

ingf

f x f x x x

x x

24

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

2Consider 4 on 0,f x x

1 2

1 2

If then x x

f x f x

24 is decreasing on 0,f x x 25

Monotonic Functions

If a function is either increasing or decreasing

on an interval, then it is said to be

on the i

monotoni

nter

c

val.

26

Example 2.3.5

1 2

1 2

1 2

1 2

Show by definition that 3 7

is monotonic increasing in R.

3 3

3 7 3 7

Therefore, is increasing on R.

f x x

x x

x x

x x

f x f x

f27

Example 2.3.6

1 2

1 2

1 2

1 2

Show by definition that 5

is monotonic decreasing in R.

5 5

Therefore, is decreasing on R.

g x x

x x

x x

x x

g x g x

g28

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

Example 2.3.7 Consider the graph of . Determine if the function is monotonic increasingor decreasing on the indicated intervals.

f

a. 0,

b. , 1

c. 2,0

d. R29

Bounded Functions

If the range of a function is bounded

above, then the function is sai

bounded

d to

ab

be

ove.

30

Example 2.3.7

2

2

Determine if the following functions

are bounded above or not.

1. 1

0,1

is bounded above by 1.

2. 1

1,

is not bounded above.

f x x

Rng f

f

g x x

Rng g

g

31

Bounded Functions

If the range of a function is bounded

below, then the function is sai

bounded

d to

be

be

low.

32

Example 2.3.8

2

2

Determine if the following functions

are bounded below or not.

1. 1

0,1

is bounded below by 0.

2. 1

1,

is bounded below by 1.

f x x

Rng f

f

g x x

Rng g

g

33

Bounded Functions

If the range of the function is both

bounded above and below, the

function is said to boube nded.

34

Example 2.3.9

2

Determine if the following functions

are bounded or not.

1. 1

0,1

is bounded above by 1.

is bounded below by 0.

Therefore, is bounded.

f x x

Rng f

f

f

f

35

22. 1

1,

is not bounded above.

is bounded below by 1.

Therefore, is not bounded.

g x x

Rng g

g

g

g

36