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Add Math Form 4 Question

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Submitted By asyrafzaman
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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)
4
9

3
4

2
-3

Prime
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
2

b)

B

A

B

4

2

3

6

4

8

c)

B

4

1

5

3

9

2

7

4

Function notation f : x  …………….. or f ( x ) = ……………

A

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) 

3
4

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

8

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5.0 Assessment( 30 minutes)
1)

( a)many to many

2)

(i) 8x+11
(ii) -5
1
p = , q = 16
2

5)

p,q,r



k=3

4)



-2

3)

( b ) range =

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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 http://mathsmozac.blogspot.com
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iii) f ( x ) = px +

1 q http://sahatmozac.blogspot.com

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

x

|
1

_ -1

|
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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