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ADDITIONAL
MATHEMATICS
MODULE 1
FUNCTIONS
Organized by
Jabatan Pelajaran Pulau Pinang 2006
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CHAPTER 1 : FUNCTIONS
Contents
1.1 Concept map
Page
2
1.2 Determine domain , codomain , object,
image and range of relation
3
1.3 Classifying the types of relations
3
2.1 Recognize functions as a special relation.
2.2 Expressing functions using function
notation.
2.3 Determine domain , object , image and
range
4-5
3.0 Composite Functions
6 -9
4.0 SPM Questions
9 – 10
5.0 Assessment test
11 – 12
6.0 Answers
13 – 14
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CONCEPT MAP
FUNCTIONS
Relations
Object
images
………….,,
……………
……………
Functions
Function
Notation
Type
of
relation
y
Or
………………
Composite
Functions
Inverse
Functions
f: x
One to
one
Many to
one
………..
fg ( x ) = …………….
Object
f(x)=y
 ………………
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1.1 Functions
Express the relation between the following pairs of sets in the form of arrow diagram,
ordered pair and graph.
Arrow diagram
Ordered pair
Graph
a ) Set A =
 Kelantan, Perak ,
Selangor 
Set B =  Shah Alam
, Kota Bharu ,Ipoh 
Relation: ‘ City of the
state in Malaysia ‘
b )Set A
=  triangle,rectangle,
pentagon 
Set B =  3,4,5 
Relation : ‘ Number of
Sides’
1.2 Determine domain , codomain , object, image and range of relation.
List down the domain , codomain , objects , images and the range of the following relation
.
3
9
2
5
1
4
-2
3
-3
1
Set P
Diagram 1
Set Q
Domain =  ……………………………………… 
Codomain =  ……………………………………… 
Object
=……………………
Image
=……………………
Range
=…………………...
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1.3 Classifying the types of relations
State the type of the following relations
a)
x
2
x
3
4
2
9
16
4
x
X
4
2
-2
-3
36
6
……………………………………………..
c)
x
b)
x
X2
………………………………………….
Type of number
d)
3
4
2
-3
4
Prime
9
Even
-3
9
……………………………………………..
……………………………………………
2.0 Functions
2.1 Recognize functions as a special relation.
2.2 Expressing functions using function notation.
2.3 Determine domain , object , image and range
2.1 Identify each of the following relations is a function or not.
a)
A
B
A
B
b)
p
1
q
2
r
3
A
B
c)
p
a
a
p
q
b
b
q
r
c
c
r
d
……………………………
……………………………
……………………………
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2.2 Express each of the following functions using function notation.
a)
A
B
2
b)
A
B
4
2
3
6
4
8
Function notation
f : x  ……………..
or f ( x ) = ……………
c)
A
B
4
1
5
3
9
2
7
4
16
3
9
Function notation
g : x  ……………..
or g ( x ) = ……………
Function notation
h : x  ……………..
or h ( x ) = ……………
2.3 a)Find the image for each of the following functions.
( i ) f : x  2x + 9
( ii ) f : x 
f (5 ) =
5x  3
2
x
+6
5
find h ( -2 )
iii) h : x 
f (-3 ) =
…………………………
…………………………
…………………………..
b ) Find the object for each of the following functions.
i )f : x  2x – 3 , find the
object when the image is 5.
2x  8
, find the
3
object when the image is 3.
ii )f : x 
6
– 7 , find the
x
object when the image is -5.
i )f : x 
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c ) Find the value of x for each of the following function.
8
2x  1
for which g (x) = 4
ii )
i ) f ( x ) = 2x + 7
for which f ( x ) = 3
g(x)=
iii)
x 3
2
for which h ( x ) = x
h(x) =
3.0 Composite Functions
g
f
a
b
c
fg

f: a  b
g: b  c
gf : a  c
3.1 ( a ) Find the value for each of the following composite functions
i ) f( x ) = x + 2 and
g ( x) = 5x + 3
find fg ( 2 ) =
ii ) g ( x ) = 2 +5x
iii) f(x) = 3x+
find g2(4)
=
1
and g ( x )
2
1
.Find fg ( 1 )
x 1
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( b ) Find the following composite function.
( i ) f( x ) = 2x + 3
g( x ) = 1 – x
fg ( x ) =
( ii ) f( x ) = 2x + 3
g( x ) = 2 + 5 x2
gf ( x ) =
( iii ) f ( x ) = 1 g(x)=
x
2
4
x
fg ( x ) =
( c) Find the value of x for each of the following composite function.
3
x
g ( x ) = 2x + 1
i) f ( x ) =
fg ( x ) = 5
ii )f( x ) = 2x + 4
iii) f ( x ) = 1 -
g ( x ) = x -2
g (x ) =
fg ( x ) = 2
4
x
fg ( x ) = -1
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x
,
2
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( d ) Solve
i )Given the function
f: x  4x + k ,
g : x x – 2
fg : x  mx + 8
ii )Given the function
f: x  9 – 2x ,
g : x  ax + b and
fg: x  1 – 6x
iii)Given the function
f: x  2x – 1 ,
g : x  4x and
gf : x  ax + b
Find the value of k and m
Find the value of a and b.
Find the value of a and b
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4.0 SPM QUESTIONS
SPM 2004 ( paper 1, question no 1 )
1. Diagram 1 shows the relation between set P and set Q
 w
d
x
e
y
f
z
Set P
Set Q
Diagram 1
State
( a ) the range of the relation,
( b ) the type of relation.
Answer:
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[2 marks]
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SPM 2004 ( paper 1, question no 3)
1. Given the function h ( x ) =
6
, x  0 and the composite function hg ( x ) = 3x , find
x
(a)g(x)
( b ) the value of x when gh ( x ) = 5.
Answer:
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[ 4 marks]
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SPM 2005 Question
1. In Diagram 1 , the function h maps x to y and the function g maps y to z.
x
y
h
g
z
8
5
2
Diagram 1
Determine
( a ) gh ( 2 )
Answer:
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5.0 Assessment( 30 minutes)
1)
P
a
q
b
r
c
s
A
Diagram 1
B
The diagram above shows the relation between set A and set B. State
a) the type of relation
b) the range of relation
Answer:
2) Given that f : x  2x + 7 find the object when image is 3.
Answer:
3) Given that f ( x ) = 10 – kx and f ( 2 ) = 4 ( k constant ) . Find the value of k.
Answer:
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4) Given that f ( x ) = 4x -1 and g ( x ) = 2x + 3 .
Find
i ) fg ( x )
ii ) fg ( - 2 )
Answer:
5 ) Given the function f : x  px + 2 and g : x  qx + 3 . If the composite function fg is
such that fg ( x ) = 8x + 8 , find the values of p and q.
Answer:
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6.0 ANSWERS
1.1
Arrow diagram
Ordered pair
KB
Kel

SA
Sel
(kel, kb), (Sel,
S.Alam), (
Perak,Ipoh) 
Ipoh
Per
A
Graph
KB
SA
Ipoh
B
Kel

3
Tri
(Tri, 3), (rec,
4), ( Pen,5) 
43
rec
Per
5
4
5
Pen
Sel
3
A
B
Pen
rec
Tri
1.3
Domain =  -3,-2,1,2,3  , codomain =  1,3,4,5,9  , Object =1,2,3,-2,-3
Image = 1,3,4,5,9 , Range =  1,4,9 
( a ) one to one ( b ) One to many( c ) many to one( d ) many to many
2.1
( a ) function
2.2
(a) 2x (b) x2 (c) 2x + 3
2.3
a) ( I ) 19
(ii) -6 (iii ) 3
3.1
a) ( i) 15
(ii ) 102
b) (i) 5 – 2x
(ii) 20x2+60x+47
1.2
( b ) Not function
1
(ii) 1
5
d) (i) k=16,m=4
c) (i) -
4.0 SPM QUESTIONS
SPM 2004( P1,Q1) a)
SPM 2004( P1,Q3)
SPM 2005( P1,Q1)
( c ) function
b) (i) 2 (ii)
1
1
(iii) 3 c) (i) -2 (ii)
(iii) -3
2
2
( iii) 2
(iii) 1-
2
x
(iii) 1
(ii) a=3,b=4
(iii) a=8,b= - 4
range =  x,y

(a) g(x) =
2
, x 0
3
b )many to one
( b) x = 15
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5.0 Assessment( 30 minutes)
1)
( a)many to many
2)
-2
3)
k=3
4)
(i) 8x+11
(ii) -5
1
p = , q = 16
2
5)
( b ) range =

p,q,r

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ADDITIONAL
MATHEMATICS
MODULE 2
FUNCTIONS
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CHAPTER 1 : FUNCTIONS
Contents
Page
1.0 Inverse Functions ( concept map )
2–4
2.0 Absolute Function
4-6
3.0 SPM Questions
7–8
4.0 Assessment ( 30 minutes )
9 – 10
5.0 Answers
11 – 13
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CHAPTER 1 : FUNCTIONS
FUNCTIONS
Ordered pairs
Arrow diagram
Relations
Functions
graph
Object
images
Domain
Codomain ,
Range
Function
Notation
Type
of
relation
f: x
y
Or
f(x) = y
One to
one
Many to
one
image
Inverse
Functions
Composite
Functions
fg ( x ) = f [ g(x) ]
Object
One to
many
Many
to many
f(x)=y
 f-1( y ) = x
1.0 Inverse Functions
1.1 Determine the object by inverse mapping
1.2 Determine the inverse functions
a ) Find the inverse function of each of the following functions.
x
ii) f ( x ) =
i)f(x)=x+3
5
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iii ) f ( x ) =
3x  1
2
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iv ) f ( x ) = 7 – 5x
v)f(x)=
3 x
4
vi ) f( x ) =
5  4x
3
b ) Find the inverse function of each of the following functions in terms of p and q
i ) f ( x ) = px - q
ii ) f ( x ) =
x p
q
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iii) f ( x ) = px +
1
q
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c ) Given the function f : x 
x2
, x  1 and g(x) =2x -6 , find f-1 g .
x 1
Answer:
d ) Inverse function f is define by f-1 : x 
x 5
1
, x  . Find f ( 2 )
2x 1
2
Answer:
2.0 Absolute Function
1. Sketch the graph of each of the following functions
a)f(x)=x
b)f(x)=x+1
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c) f (x ) = | x |
d)f(x)=|x+1|
e ) f (x) = |x| + 1
f)f(x)=|x|-1
2.Sketch the graph of each the following functions and state the corresponding range.
a) f : x  2x – 3 for 0  x  4
b) f : x  |2x – 3| for 0  x  4
f(x)
f(x)
Range :……………………………….
Range : …………………………………………
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c) f : x  | 5 – 2x |for -1  x  4
d) f : x  |9 – 2x| for 0  x  6
f(x)
Range :……………………………….
Range :……………………………….
e) f : x  |2x| – 1 for -1  x  3
f ) f : x  | 3x | for -2  x  2
Range :……………………………….
Range :……………………………….
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3.0 SPM QUESTIONS
SPM 2004 Question.
1. Given the functions h : x  4x + m and h-1 : x  2kx +
5
, where m and k are
8
constants, find the value of m and of k.
[ 3 marks]
Answer:
SPM 2005 ( Paper 1, Question 1 )
2. In Diagram 1 , the function h maps x to y and the function g maps y to z.
x
y
h
g
z
8
5
2
Diagram 1
Determine
( a ) h-1 ( 5 )
Answer:
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SPM 2005 ( Paper 1, Question 2 )
1.The function w is defined as w ( x ) =
( a ) w-1 ( x ),
( b ) w-1 ( 4 ).
Answer:
5
, x  2.
2 x
[ 3 marks]
SPM 2005 ( Paper 1, Question 3 )
1.The following information refers to the functions h and g.
h: x
g: x
 2x – 3
 4x - 1
Find gh-1 ( x ).
[ 3 marks]
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4.0 Assessment ( 30 minutes )
1.A function f is defined by f: x  6 Find
a)f(x)
1
x
2
b ) f-1 ( 5 )
Answer:
2. Inverse function f is defined by f-1 : x 
5  4x
3
find f ( 2 ).
Answer:
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3.Given the function f : x  2x - m and inverse function f-1 : x  nx +
7
3
Find the value of m and n.
Answer:
4. Sketch the graph of the function f ( x ) = |2x – 5 | for 0  x  6. Hence , state the
corresponding range.
Answer:
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5.0 ANSWERS
1.1
2x  1
7x
5  3x
(iv)
(v) 3 – 4x (vi)
3
5
4
xq
x 1
(b)
(ii ) xq + p (iii) p
p pq
2x  4
7
( c)
, x
7  2x
2
( d ) -1
( a )( i) x- 3 ( ii) 5x (iii)
Absolute function
f(x)
a)
f(x)
b)
_1
|
-1
x
f(x)
c)
x
f(x)
d)
_1
|
-1
x
f(x)
e)
x
f(x)
f)
_1
|
-1
|
-1
x
_ -1
|
1
x
|
1
|
2 x
2.
a)
f(x)
b)
_8
_
.
f(x)
_5
|
1
|
2 x
-3
 3  f ( x)  8
0  f(x)  5
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c)
f(x)
_7
f(x)
d)
_9
|
2
|
4
|
3 x
0  f(x)  7
0  f(x)  9
f(x)
_5
e)
|
5 x
f(x)
_6
f)
_1
|
-1
|
_ -1 1
|
3x
-1  f(x)  5
|
-2
|
2
0  f(x)  6
SPM 2004 ( P1,Q2)
1
5
K= , m=8
2
SPM 2005 ( P1,Q2)
(a) 2
SPM 2005 ( P1,Q2)
2x  5
3
(a)
, x 0 ( b )
x
4
SPM 2005 ( P1,Q3)
2x+ 5
Assessment ( 30 minutes )
1)( a) 12 – 2x ( b ) 8
1
2) 4
1
3) n = , m=7
2
4)
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x
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a)
f(x)
_7
_5
|
2
|
3
|
6
x
The corresponding range of f(x ) = 0  f(x)  7
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