(1) The functions f and g are given as: f(x) = x + 1 fg(x) = x^2 + 2x - 4 (2) The functions f and g are given as: f(x) = x^2 - 5 gf(x) = 2x^2 - 9 (3) The question asks to find the function g such that the composite functions equal the given functions.

1. 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

2. 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

3. 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}
B
A Celebrating Festivals
Chinese New Year
Malay
Chinese Hari Raya Aidilfitri
Indian Deepavali

4. 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 diagram
P Q
Favourite Sports

5. ( b ) Ordered pairs
{( Faouzi,Netball),( }
( c ) Graphs
Set Q
Netball
Set P
Fauzi

6. 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) an arrow diagram
( b) graphs
( c) ordered pairs
( a ) Arrow diagram
factor of Q
P
8 4
10 5
14 7

7. ( b ) Graphs
Set Q
7
5
4
Set P
8 10 14
( b ) Ordered pairs
{（8, 4 ), (10, 5 ) , (14, 7 ) }

8. 1.1.2 Identifying the Domain, Codomain,Object, Image and
Range of a Relation
Learning Outcomes:
At the end of this lesson, students should be able to
( a ) identify domain, codomain, object, image and range of
a relation.

9. The arrow diagram below shows a relation one quarter of
from set M to set N
N
M One quarter of
8 2
12 3
20 5
24 6
36 7
Domain= { 8, 12, 20, 24, 36 } Object = 8, 12, 20, 24
Codomain= { 2, 3, 5, 6, 7 }
Image = 2, 3, 5, 6
Range = { 2, 3, 5, 6 }

10. 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
M
square of
2 4
3 7
4 9
5 16
25
Domain= { 8, 12, 20, 24, 36 }
The image of 3 = 9
Codomain= { 2, 3, 5, 6, 7 }
Range = { 4, 9, 16, 25
The object which has 16 as its image = 4 }

11. 1.1.3 Classifying Relation
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.

12. State the type of relations for following arrow diagram
(a) Multiple of
(b) Examination
Kadir
3 9 Nabil PMR
4 12 Siu Lin
5 15 Muthu
SPM
6 18
many to one
One to one
(d) Factor of
(c) Factor of
4 2
4
64 3
6 6
4
8 8
24
One to many Many to many

13. 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)｝
(d) Set B
8
6
4
2
Set A
3 5 9 12

14. 1.2 Functions
Learning Outcomes:
At the end of this lesson, students should be able to
recognise function as a special relation.

15. 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.
p Q
A Multipler of B
Factor of
3 9 2
12 4
4
5 15 3
6 18 6
This relation is a function because This relation is not a function
every object has only one image because object 6 has two image

16. A B R S
One third of
Factor of
3 1 2
2 6
6
3 3
9 4 8
This relation is not a function This relation is not a function
because not every element in the because object 6 has two image
codomain has to be related.
Set D
(d)
This relation is not a function
8 because object 5 has two
6 image
4
2
3 5 9 Set C
12

17. 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)

18. 1.2.2 Expressing Function Using Function Notation
Learning Outcomes:
At the end of this lesson, students should be able to
express functions using function notations.

19. 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 x to 2 x 3 is written as :
f : x  2 x 3 or f ( x) 2 x 3
A function can be represented by lower-case alphabet
such as f , g h and others

20. f : x  2 x is read as “function f maps x to 2x”,
f ( x) 2x is read as “ 2x is the image of x under the
function f”,
or f of x is equal to 2x.

21. Write the functions below by using function notation.
Let the function be
g.
Square root The notation is:
4 2 g : x  ｘ or g ( x) x
9 3
16 4
x g
x
4 2
9 3
16 4

22. Write the functions below by using function notation.
(a) (b) Marks
Half of
Science 82
2 1
4 2 History 75
8 4 68
English
( c) Price of tickets
f
Child RM3 (d) 1
4 4
Adult RM7 1
RM4 8 15
Senior
1
citizen 15
8

23. 1.2.3 Determine Domain, codomain, object image and
range of a Function
The arrow diagram shows the function f : x  3x 1
x 3x 1
f
1
4 4
1
8 15
1
15
8
Domain codomain

24. 1. Given that f ( x) 3x 1， find the value of f(0), f(3)
and f(10)
f ( x) 3x 1
x 3x 1
f (0) 3(0) 1 f
1 0 1
3 29
f (3) 3(3) 1 10 8
8
f (10) 3(10) 1
29

25. 1. Given that f ( x) 7 cos x， find the value of x=0 and
0
x=60
f ( x) 7 cos x
x 3x 1
f
f (0) 7cos (0)
7(1) 0 7
7 3.5
600
f (600 ) 7cos(600 )
1
7
2
3.5

26. 2. Given that h( x) 1 3x ， find the value of h(-3) and
h(5)
h( x) 1 3x
x 1-3x
h( 3) 1 3( 3) h
10 -3 10
5 14
h(5) 1 3(5)
14
14

27. 3. Given that f ( x) 3x 4， find the value if
(b) f ( x) 10
(a) f ( x) 11
f ( x) 10
3 x x )4
f( 11
3x 10 4
3x 11 4
3x 6
3x 15 x 2
x 5 x 3x 4
f
5 11
-2 10

28. 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

29. Composite Functions
Given that f ( x) x 3 and g ( x) 2 x 1, find
(a) fg
fg f [ g ( x)]
f [ g ( x)] x
g ( x) 3
f(2 x( x1])
g ) 2x 1 3
2x 4

30. Composite Functions
Given that f ( x) x 3 and g ( x) 2 x 1, find
(a) gf
gf g[ f ( x)]
g [ f ( x)] 2 x 1
(x
[ f 3) 2x 3) 1
g ( x ( x)]
2x 7

31. Composite Functions
Given that f ( x) x 3 and g ( x) 2 x 1, find
(a) f 2 (b) f 4
f 2
f f f 4
f 2f 2
2 2
f [ f ( x)] f [ f ( x)]
2
f ( x 3) f ( x 6)
xx 3 3 xx 6 6
2 4
f x 6 f x 12

32. Given that f ( x) 2 x and g ( x) 3 2 x, find
(a) fg (2) (b) gf ( 2)
fg ( x) f [ g ( x)] gf ( x) g[ f ( x)]
f (3 2 x) g ( 2 x)
x
2 (3 2x)
3 2 x)
(2 x
fg ( x) 6 4x gf ( x) 3 4x
fg (2) 6 4(2) gf ( 2) 3 4( 2)
2 11

33. Determine one of the Functions when the Composite
Function and Other Function are Given
A function f is defined by . Find the function g
f :xx 1
in each of the following
(a) fg : x  2x 2
3 (b) gf : x2 3x 5
2
2
g[ f ( x)] x 3x 5
f [ g ( x)] 2x 3
Let y x 1
2
g (x ) 1
x 2x 3 x y 1
2 g ( y) ( y 1)2 3( y 1) 5
g ( x) 2x 2
g ( y) y2 2 y 1 3y 3 5
g ( y) y2 y 3
g ( x) x2 x 3

34. Exercise 1.3.2 ( Page14) 3-1-09 to 6-1-09
(1) Given that f ( x) x 3 and g ( x) 3x 1, find
(a) Find the composite functions fg and gf fg 3x 2
gf 3x 8
(b) What are the value of fg(2), gf(-3) and gf(-5) 8 1 7
(2) Given that f ( x) 4 x 5, find f 2 16 x 20
The composite function f 2, Hence, find
2 1
(a) f and f 2 ( 2) (b) value of x which f 2 ( x) 9
2
12 11
28
16

35. Exercise 3-1-09 to 6-1-09
(1) if f : x  x 1, find the function g such that
2
fg : x  x 2x 4 g ( x) x2 2x 5
2
(2) if f : x  x 5, find the function g such that
gf : x  2x2 9 g ( x) 2 x 1