2.01 add math module 01 functions papers add maths/function...3 1.1 functions express the relation...

FUNCT

Organi

Jabatan Pelajaran

ADDITMATHEM

MOD

http://mathsmoz

http://sahatmozac.blogspot.com

IONALATICS

IONS

zed by

Pulau Pinang 2006

ULE 1

ac.blogspot.com

1

CHAPTER 1 : FUNCTIONS

Contents Page1.1 Concept map 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

http://mathsmozac.blogspot.com


2

CONCEPT MAP

Functions

FunctionNotation

CompositeFunctions

InverseFunctions

Relations

Typeof

relation

f: x y

Or………………

Objectimages

………….,,……………

……………

FUNCTIONS

One toone

Many toone

fg ( x ) = …………….

……….. Object

f ( x ) = y ………………



3

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 Grapha ) Set A = Kelantan, Perak ,

Selangor

Set B = Shah Alam

, Kota Bharu ,Ipoh Relation: ‘ City of thestate in Malaysia ‘b )Set A= triangle,rectangle,

pentagon Set B = 3,4,5 Relation : ‘ Number ofSides’

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.

Domain = ……………………………………… Codomain = ……………………………………… Object =……………………Image =……………………Range =…………………...

3

2

-2

5

4

3

9

Set P Set QDiagram 1

-3 1

1



1.3 Classifying the types of relations

State the type of the following relations

a)

…………………………………

c)

…………………………………

2.0 Functions2.1 Recognize functions as a spe2.2 Expressing functions using f2.3 Determine domain , object , i

2.1 Identify each of the following

a)

……………………………

b)

…

x

2

4

6

AB

p

q

r

1

2

3

3

24

9-3

x X2

http://


…………..

b )

………………………………………….

…………..

d)

……………………………………………

cial relation.unction notation.mage and range

relations is a function or not.

4

16

2

x

36

4

9

Prime

-3

x

3

24

-29

-3

Even

xX

Type of number

c)A B

p

q

r

a

b

c

A B

a

b

c

p

q

r

……

ma
4
……………

thsmozac.b

……… ………

d

logspot.com

……………
………

2.2 Express each of the following functions using function notation.

a)

Functf : xor f ( x

b ) c)

2.3 a)

( i ) f

f

………

b ) Fin

i )f : xobject


ion notation……………) = …………

Find the imag

: x 2x + 9

(5 ) =

……………

d the object f

2x – 3 , fiwhen the im

A

2

3

4

B

4

6

A B

2

3

4

9

A B

1

2

5

7

..…

Functig : xor g (

e for each of the f

……

( ii ) f

f

………

or each of the fol

nd theage is 5. ii )f : x

object

8

http://ma
5
on notation……………

x ) = ………

ollowing fun

: x 2

35 x

(-3 ) =

……………

lowing functi

3

82 x,

when the im

4

thsmozac.b

…
..…
Functih : xor h (

ctions.

……

iii) h :

fin

………

ons.

find the

age is 3.

i )f : x

object

16

logspot.com

on notation……………

x ) = …………

x5

x+6

d h ( -2 )

……………

x

6– 7 , fi

when the im

3

n

a

..…

……..

d the

ge is -5.

9

6

c ) Find the value of x for each of the following function.

i ) f ( x ) = 2x + 7for which f ( x ) = 3

ii ) g ( x ) =12

8

xfor which g (x) = 4

iii) h(x) =2

3x

for which h ( x ) = x

3.0 Composite Functions

3.1 ( a ) Find the value for each of the following composite functions

i ) f( x ) = x + 2 andg ( x) = 5x + 3find fg ( 2 ) =

ii ) g ( x ) = 2 +5x

find g2(4)

iii) f(x) = 3x+2

1and g ( x )

=1

1

x.Find fg ( 1 )

acb

fg

f: a bg: b c

gf : a c

f g



7

( b ) Find the following composite function.

( i ) f( x ) = 2x + 3g( x ) = 1 – x

fg ( x ) =

( ii ) f( x ) = 2x + 3g( x ) = 2 + 5 x2

gf ( x ) =

( iii ) f ( x ) = 1 -2

x

g ( x ) =x

4

fg ( x ) =

( c) Find the value of x for each of the following composite function.

i) f ( x ) =x

3

g ( x ) = 2x + 1

fg ( x ) = 5

ii )f( x ) = 2x + 4

g ( x ) = x -2

fg ( x ) = 2

iii) f ( x ) = 1 -2

x,

g (x ) =x

4

fg ( x ) = -1



8

( d ) Solve

i )Given the functionf: x 4x + k ,g : x x – 2fg : x mx + 8

Find the value of k and m

ii )Given the functionf: x 9 – 2x ,g : x ax + b andfg: x 1 – 6x

Find the value of a and b.

iii)Given the functionf: x 2x – 1 ,g : x 4x andgf : x ax + b

Find the value of a and b



9

4.0 SPM QUESTIONS

SPM 2004 ( paper 1, question no 1 )

1. Diagram 1 shows the relation between set P and set Q

State( a ) the range of the relation,( b ) the type of relation. [2 marks]

Answer:

d

e

f

x

y

z

w

Set P Set Q

Diagram 1



10

SPM 2004 ( paper 1, question no 3)

1. Given the function h ( x ) =x

6, x 0 and the composite function hg ( x ) = 3x , find

( a ) g ( x )( b ) the value of x when gh ( x ) = 5. [ 4 marks]

Answer:



11

SPM 2005 Question

1. In Diagram 1 , the function h maps x to y and the function g maps y to z.

Determine( a ) gh ( 2 )

Answer:

y zx

2

5

8

h g

Diagram 1



12

5.0 Assessment( 30 minutes)

1)

The diagram above shows the relation between set A and set B. Statea) the type of relationb) the range of relation

Answer:

2) Given that f : x2x + 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:

a

b

c

q

r

s

P

ABDiagram 1



13

4) Given that f ( x ) = 4x -1 and g ( x ) = 2x + 3 .Findi ) fg ( x )ii ) fg ( - 2 )

Answer:

5 ) Given the function f : x px + 2 and g : x qx + 3 . If the composite function fg issuch that fg ( x ) = 8x + 8 , find the values of p and q.Answer:



14

6.0 ANSWERS

1.1

Arrow diagram Ordered pair Graph

(kel, kb), (Sel,

S.Alam), (Perak,Ipoh)

(Tri, 3), (rec,

4), ( Pen,5)

1.2 Domain = -3,-2,1,2,3 , codomain = 1,3,4,5,9 ,

Image = 1,3,4,5,9 , Range = 1,4,9 1.3 ( a ) one to one ( b ) One to many( c ) many to one( d ) many

2.1 ( a ) function ( b ) Not function ( c ) function

2.2 (a) 2x (b) x2 (c) 2x + 3

2.3 a) ( I ) 19 (ii) -6 (iii ) 3 b) (i) 2 (ii)2

1(iii) 3 c

3.1 a) ( i) 15 (ii ) 102 ( iii) 2

b) (i) 5 – 2x (ii) 20x2+60x+47 (iii) 1-x

2

c) (i) -5

1(ii) 1 (iii) 1

d) (i) k=16,m=4 (ii) a=3,b=4 (iii) a=8,b= - 4

4.0 SPM QUESTIONSSPM 2004( P1,Q1) a) range = x,y b )many to one

SPM 2004( P1,Q3) (a) g(x) =3

2, x 0 ( b) x = 15

SPM 2005( P1,Q1) 8

Tri

rec

Pen

4

5

3

AB

Kel

Sel

PerIpoh

SA

KB

AB

KB

SA

Ipoh

3

3

4

5



Object =1,2,

to many

) (i) -2 (ii)

Kel Sel Per

recPen

3,-2,-3

2

1(iii) -3

Tri

15

5.0 Assessment( 30 minutes)

1) ( a)many to many ( b ) range = p,q,r

2) -2

3) k = 3

4) (i) 8x+11 (ii) -5

5) p =2

1, q = 16



1

FUNCT

ADDITMATHEM

MOD

http://mathsmoz


IONALATICS

6

IONS

ULE 2

ac.blogspot.com

17


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



18


1.0 Inverse Functions

1.1 Determine the object by inverse mapping1.2 Determine the inverse functions

a ) Find the inverse function of each of the following functions.

i ) f ( x ) = x + 3 ii) f ( x ) =5

xiii ) f ( x ) =

2

13 x

Functions

FunctionNotation

CompositeFunctions

InverseFunctions

Relations

Typeof

relation

f: x y

Orf(x) = y

f ( x ) = y f-1( y ) = x

ObjectimagesDomain

Codomain ,Range

FUNCTIONS

Arrow diagram

Ordered pairs

graph

One toone

One tomany

Many toone

Manyto many

fg ( x ) = f [ g(x) ]

image Object



19

iv ) f ( x ) = 7 – 5x v ) f ( x ) =4

3 xvi ) f( x ) =

3

45 x

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 ) =q

px iii) f ( x ) = px +

q

1



20

c ) Given the function f : x1

2

x

x, x 1 and g(x) =2x -6 , find f-1 g .

Answer:

d ) Inverse function f is define by f-1 : x 12

5

x

x, x

2

1. Find f ( 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



21

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

Range :……………………………….

b) f : x |2x – 3| for 0 x 4

Range : …………………………………………

f(x) f(x)



22

c) f : x | 5 – 2x |for -1 x 4

Range :……………………………….

d) f : x |9 – 2x| for 0 x 6

Range :……………………………….

e) f : x |2x| – 1 for -1 x 3

Range :……………………………….

f ) f : x | 3x | for -2 x 2

Range :……………………………….

f(x)



23

3.0 SPM QUESTIONS

SPM 2004 Question.

1. Given the functions h : x 4x + m and h-1 : x 2kx +8

5, where m and k are

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.

Determine( a ) h-1 ( 5 )

Answer:

y zx

2

5

8

h g

Diagram 1



24

SPM 2005 ( Paper 1, Question 2 )

1.The function w is defined as w ( x ) =x2

5, x 2.

( a ) w-1 ( x ),( b ) w-1 ( 4 ). [ 3 marks]

Answer:

SPM 2005 ( Paper 1, Question 3 )1.The following information refers to the functions h and g.

Find gh-1 ( x ). [ 3 marks]

h: x 2x – 3

g: x 4x - 1



25

4.0 Assessment ( 30 minutes )

1.A function f is defined by f: x 6 -2

1x

Finda ) f ( x ) b ) f-1 ( 5 )

Answer:

2. Inverse function f is defined by f-1 : x 3

45 x

find f ( 2 ).

Answer:



26

3.Given the function f : x 2x - m and inverse function f-1 : x nx +3

7

Find the value of m and n.

Answer:

4. Sketch the graph of the function f ( x ) = |2x – 5 | for 0x6. Hence , state thecorresponding range.

Answer:



28

0 f(x) 7 0 f(x) 9

-1 f(x) 5 0 f(x) 6

SPM 2004 ( P1,Q2)

K=8

1, m=-

2

5

SPM 2005 ( P1,Q2)(a) 2SPM 2005 ( P1,Q2)

( a )x

x 52 , x 0 ( b )

4

3

SPM 2005 ( P1,Q3)2x+ 5Assessment ( 30 minutes )1)( a) 12 – 2x ( b ) 8

2) -4

1

3) n =2

1, m=7

4)

f(x)

x

f )_ 6

|2

|-2

f(x)

x

d )

_ 9

|4

|5

f(x)

x

e )_ 5

|1

|-1 _ -1

_ 1

|3

f(x)

x

c )_ 7

|2

|3



The corresponding range of f(x

f(x)a )

_ 5

|2

|3

_ 7

|6

http:


29

) = 0 f(x) 7

x

//mathsmozac.blogspot.com

2.01 add math module 01 functions papers add maths/function...3 1.1 functions express the relation...

Documents