Functions

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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}

Chinese

Indian

Hari Raya Aidilfitri

Chinese New Year

Deepavali

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

Netball

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

( a) an arrow diagram

( b) graphs

( c) ordered pairs

factor of

( b ) Ordered pairs

( b ) Graphs

{ （ 8, 4 ), (10, 5 ) , (14, 7 ) }

8 10 14

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

One quarter of

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

{ 8, 12, 20, 24, 36 }

{ 2, 3, 5, 6, 7 }

{ 2, 3, 5, 6 }

8, 12, 20, 24

2, 3, 5, 6

Domain=

Codomain=

Range =

Object =

Image =

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

{ 8, 12, 20, 24, 36 }

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

Domain=

Codomain=Range =

The image of 3 =

The object which has 16 as its image =

Learning Outcomes:

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

1.1.3 Classifying Relation

Nabil PMR

SPMSiu Lin

(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) ｝

Set B(d)

Learning Outcomes:

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.

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

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

One third ofFactor of

Set D(d)

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:

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.

Square root

Let the function be g.The notation is:

:g x ｘ or ( )g x x

Half of

Write the functions below by using function notation.

(a) (b)

Senior

citizen

RM3Price of tickets

Science

English

History

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

( ) 33 ( ) 13f 8

( ) 310 1( ) 10f 29

fx 3 1x

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

( ) 7cosf x x

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

0 0( ) 7cos(60 60 )f

0603.5

fx 3 1x

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

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

( ) 15 53( )h

hx 1-3x

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

3 10 4x ( )f x 10

3 6x 2x

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

Composite Functions

( )a gf

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

[ ( )]g f x

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

Composite Functions

2( )a f

2f f f

f [ ( )]f x

( 3)x f

4( )b f

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

( 6)x 2f

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

(3 )2x

( )gf x [ ( )]g f x

( )2g x

3 2 x(2 )x

( )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

2 16 20f x

28 12 11

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

Functions

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