Chapter 1 Functions

Learning Outcomes:

At the end of this lesson, students should be able to represent a relation using

( a ) arrow diagrams,

( b ) ordered pairs

( c ) graphs

1.1 Relations

A represents the set of races in Malaysia

B represents the set of festivals in Malaysia

A = { Malay, Chinese, Indian }

B = { Hari Raya Aidilfitri, Chinese New Year, Deepavali}

A B The elements of set A are associated with the elements of set B as depicted by diagram shown on the left.

This association between A and B is called a relation from A to B

1.1 RelationsA represents the set of races in Malaysia

B represents the set of festivals in Malaysia

A = { Malay, Chinese, Indian }

B = { Hari Raya Aidilfitri, Chinese New Year, Deepavali}

Malay

Chinese

Indian

Hari Raya Aidilfitri

Chinese New Year

Deepavali

AB

Celebrating Festivals

1.1.1 Representing a Relation

If set P is the set of students in Form 4 Newton and set Q is their favourite sports.

P = { } Q = { }

The relation between set P and set Q is favourite sports. This relation can also be represent by using

( a ) Arrow diagramP Q

Favourite Sports

( b ) Ordered pairs

{( Faouzi,Netball),( }

( c ) Graphs

Fauzi

Netball

Set Q

Set P

Given that set P ={8, 10, 14} and set Q={4, 5, 7}. The relation from set P to set Q is a factor of. Represent the above relation by using

( a ) Arrow diagram

PQ

( a) an arrow diagram

( b) graphs

( c) ordered pairs

8

10

14

4

5

7

factor of

( b ) Ordered pairs

( b ) Graphs

{ （ 8, 4 ), (10, 5 ) , (14, 7 ) }

8 10 14

4

5

7

Set P

Set Q

Learning Outcomes:

At the end of this lesson, students should be able to

( a ) identify domain, codomain, object, image and range of

a relation.

1.1.2 Identifying the Domain, Codomain,Object, Image and Range of a Relation

M

8

12

24

2

3

6

One quarter of

The arrow diagram below shows a relation one quarter of from set M to set N

N

20 5

7

{ 8, 12, 20, 24, 36 }

{ 2, 3, 5, 6, 7 }

{ 2, 3, 5, 6 }

8, 12, 20, 24

2, 3, 5, 6

36

Domain=

Codomain=

Range =

Object =

Image =

M

2

3

5

4

7

16

square of

Given that set R={ 2, 3, 4, 5} and set S={4, 7,9, 16, 25}. The relation is square of from set R to set S

N

4 9

25

{ 8, 12, 20, 24, 36 }

{ 2, 3, 5, 6, 7 }{ 4, 9, 16, 25 }

9

4

Domain=

Codomain=Range =

The image of 3 =

The object which has 16 as its image =

Learning Outcomes:

At the end of this lesson, students should be able to

( a ) classify relations into one to one, many to one, one to many and many to many relation.

1.1.3 Classifying Relation

3

654

15

129

18

Kadir

Muthu

Nabil PMR

SPMSiu Lin

24

644

8

6

8

4 2

4

36

(a) (b)

(c)(d)

One to onemany to one

One to many Many to many

Multiple of Examination

Factor ofFactor of

State the type of relations for following arrow diagram

State the type of relations for following ordered pairs

(a) { (3, 6), (3,9), (4,8),(5, 10)}

(b) {(Ahmad,Science), (Brian, Science),(Chandran, Mathematics)}

(c ) { ( a, 3), (b, 5),(b, 6), (c, 8) ｝

3 5 9

2

4

6

Set A

Set B(d)

12

8

Learning Outcomes:

At the end of this lesson, students should be able to

recognise function as a special relation.

1.2 Functions

1.2 Functions As a Special Relation

Function is a relation in which every element in the domain has a unique image in the codomain.

3

654

15

129

18 6

42

3

This relation is a function because every object has only one image

This relation is not a function because object 6 has two image

Multipler of Factor ofA Bp Q

3

9

63

21

4

This relation is not a function because not every element in the codomain has to be related.

8

62

3

One third ofFactor of

This relation is not a function because object 6 has two image

3 5

246

Set C

Set D(d)

8

9 12

This relation is not a function because object 5 has two image

A B R S

Exercise 1.1.3 Page 5

1. ( a), ( b ) , ( c ) , ( d )

2. ( a ) , ( b )

Skill Practice 1.1 Page 6

1 ( a) (b) ( c )

2 ( a) (b)

3 ( a) (b) (c)

4 ( a) (b) (c) (d) (e)

Learning Outcomes:

At the end of this lesson, students should be able to

express functions using function notations.

1.2.2 Expressing Function Using Function Notation

A function f from set A to set B is denoted by :f A B

This mean that all the elements in set A are mapped into set B by function f.

The function f which maps is written as :2 3x to x

: 2 ) 33 ( 2f x x or f x x

A function can be represented by lower-case alphabet such as ,f g h and others

: 2f x x is read as “function f maps x to 2x”,

( ) 2f x x is read as “ 2x is the image of x under the

function f”,

or f of x is equal to 2x.

Write the functions below by using function notation.

49

16

234

Square root

Let the function be g.The notation is:

:g x ｘ or ( )g x x

49

16

234

gx x

248

124

Half of

Write the functions below by using function notation.

(a) (b)

Child

Senior

citizen

Adult

RM4

RM7

RM3Price of tickets

Science

English

History

68

7582

Marks

(d)4

15

8

1

41

151

8

f( c)

4

15

8

1

41

151

8

f

1.2.3 Determine Domain, codomain, object image and range of a Function

: 3 1f x x The arrow diagram shows the function

x 3 1x

Domain codomain

( ) 3 1f x x 1. Given that ， find the value of f(0), f(3) and f(10)

( ) 3 1f x x ( ) 30 ( ) 10f

1

( ) 33 ( ) 13f 8

( ) 310 1( ) 10f 29

0

10

3

1

8

29

fx 3 1x

( ) 7cosf x x1. Given that ， find the value of x=0 and x=60

0

( ) 7cosf x x

( ) 7c )0 os (0f 7(1)

0 0( ) 7cos(60 60 )f

3.5

0 7

0603.5

fx 3 1x

7

172

( ) 1 3h x x 2. Given that ， find the value of h(-3) and h(5)

( ) 1 3h x x ( ) 1 33 ( 3)h

10

( ) 15 53( )h

14

-3

5

10

14

hx 1-3x

14

3. Given that ， find the value if( ) 3 4f x x

(a) (b)

( )f x

( ) 11f x ( ) 10f x

3 4x 11

3x 11 4

3x 15

x 5

3 10 4x ( )f x 10

3 6x 2x

5

-2

11

10

fx 3 4x

Exercise 1.2.4 ( page 10)

1. 2. 3. 4. 5. 6

Skill practice 1.2 (page 10)

1. 2. 4. 5. 7 8

19-1-2009

3 2 1( ) ( ) ,Given that f x and g x finxx d

Composite Functions

( )a fg

[ ( )]fg f g x

[ ( )]g xf x 3( )g x

f ( )g x (2 ])1x 2 1x 3

2 4x

3 2 1( ) ( ) ,Given that f x and g x finxx d

Composite Functions

( )a gf

gf

[ ( )]f xg 2 1( 3)x 2

[ ( )]g f x

xx 1[ ( )]f xg ( 3)x

2 7x

3 2 1( ) ( ) ,Given that f x and g x finxx d

Composite Functions

2( )a f

2f f f

f [ ( )]f x

( 3)x f

x 33x

6x 2f

4( )b f

4f 2 2f f2f 2[ ( )]f x

( 6)x 2f

x 66x

12x 4f

3( 22( ) ) ,Given that f x ax xnd g x find

( ) (2)a fg ( ) ( 2)b gf

( )fg x [ ( )]f g x

2(3 )f x

2 x

6 4x

(3 )2x

( )gf x [ ( )]g f x

( )2g x

3 2 x(2 )x

3 4x

( )fg x

(2)fg 6 4(2) 2

3 4( 2) ( 2)gf 11

( )gf x

Determine one of the Functions when the Composite Function and Other Function are Given

A function f is defined by . Find the function g in each of the following

: 1f x x

2( ) : 2 3a fg x x

[ ( )]f g x 22 3x

x 1 22 3x ( )g x

( )g x 22 2x

2( ) : 3 5b gf x x

[ ( )]g f x 2 3 5x x

1Let y x 1x y 21 1( ) ( ) 3( ) 5g y y y

2( ) 2 1 3 3 5g y y y y 2( ) 3g y y y 2( ) 3g x x x

1.3.2 ( 14)Exercise Page

(1) ( ) ( ) ,3 3 1Given that f x and g x findxx

(a) Find the composite functions fg and gf

(b) What are the value of fg(2), gf(-3) and gf(-5)

(2) ( ) 4 5,Gi xven that f x find

The composite function , Hence, find 2f2 21

( ) ( 2)2

a f and f

2( ) ( ) 9b value of x which f x

3 2fg x

3 8gf x

8 1 7

2 16 20f x

28 12 11

16

3-1-09 to 6-1-09

Exercise

(1) : 1,if f x x find the function g such that

2: 2 4fg x x x

2(2) : 5,if f x x find the function g such that

2: 2 9gf x x

3-1-09 to 6-1-09

2( ) 2 5g x x x

( ) 2 1g x x