1 Functions MyModule

8/12/2019 1 Functions MyModule

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My

dditionalMathematicsModule

Form 4(Version 2007)

Topic: 1

FUNCTIONS

by

NgKL(M.Ed.,B.Sc.Hons.,Dip.Ed.,Dip.Edu.Mgt.,Cert.NPQH)


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1.1 CONCEPTS OF RELATIONS

1. The relation between two sets, P and Q, is the pairin o! e"e#en$sin set P with elements in set Q.

2. P Q

(a) Set P = %o#ain(b) Set Q = &o%o#ain(c) Each elements in set P = ob'e&$(d) Each elements in set Q = i#ae

(e) The images {4, 9, 16}in set Q = rane

. elations can be re!resented b"#

(a) Arro %iara#s$ (b) Or%ere% pairs$

% & {(', ), (, '*), (4, 64)} '

'*

4 64

(c) *rap+s,

64

'*

'

4

4

9

16

'+

I-PORTANT POINTS


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' 4

. Fo/rt"!es o relations$

(a) One$oone re"a$ion (b) One$o#any re"a$ion

% & % &

' 1 '

'* 1' 4

(c) -any$oone re"a$ion (d) -any$o#any re"a$ion

% & % &

'

9

4

1. -om!lete the table below o the ollowing relations.

% &

(a)

(b) (, +), (+, 9), (*, 1)/

'

4

6

4

10

14

Eer&ise 1.1 a ,


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(c)

"

2

' 14

Ansers$

o#ain Co%o#ain Ob'e&$ I#ae Rane

(a)

(b)

(&)

1. 3ase% on $+e re"a$ions i4en5 i%en$i!y $+e i#aes or ob'e&$s.

(a) % &

* 10

6

Eer&ise 1.1 b ,


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(b) (, +), (, 31), (4, 9)/

(c) 6

4

'

6 9

Ansers$

(a)

Ob'e&$ o! 6 Ob'e&$ o! I#ae o! I#ae o! 7

(b)

Ob'e&$ o! 8 Ob'e&$ o! 9 I#ae o! I#ae o!

(c)

Ob'e&$ o! 2 Ob'e&$ o! 6 I#ae o! 6 I#ae o! 9

2. S$a$e $+e $ype o! re"a$ions !or ea&+ o! $+e !o""oin.

(a) % &Anser,

a !

b

c r


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(b) (, s), (4, t), (+, s)/ Anser$

(c) 6

4 Anser$ '

! s

(d) % &

Anser$

(e) % &

m Anser$ 1

n

' !

1.2 CONCEPTS OF F:NCTIONS

1. 5!/n&$ion = s!ecial relation where e4ery ob'e&$in the domain hason"y one i#ae in the rane.

2. One$oonerelation and #any$oonerelation are nctions.

. 7ne3to3man" relation and man"3to3man" relation are no$nctions.

. 5 nction can be e2!ressed b" !/n&$ion no$a$ion, in which thenction is re!resented b" the s"mbol f and the ob8ect b" thes"mbolx.f : x y is read as ;function f maps x to y


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E2am!le$ ! , , ;. ction ma!s 2 to 2.

!() = ;. 2 is the image o 2 nder nction

.

1. State whether the ollowing relation are nctions and gi


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Anser,

m a

n b

!

2. >ri$e $+e !o""oin re"a$ions in !/n&$ion no$a$ion.

(a) 2 2' Anser$

9

' 4 1 1

(b) 2 42'3

Anser,

1. e$er#ine $+e i#ae o! $+e ob'e&$ !or ea&+ o! $+e !o""oin !/n&$ions.

(a) (2) = 42 +, 2 = 1, +

1

31 '

1

1+

Eer&ise 1.2 b


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(b) (2) = 2' ', 2 = 0, 3

(c) (2) = 4 > 2, 2 = 1, 6

(d) (2) =)

' "+

, 2 = ', *

1. A !/n&$ion is %e!ine% as ! , ? 65 !in%

(a) the ob8ect i the image is 1.

Eer&ise 1.2 &


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(b) the


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(b) the 4.

. *i4en a !/n&$ion + , )

+' +"

5 !in%

(a) the


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2. A !/n&$ion is %e!ine% as !, = 22 # ? n. I! !(1) = an% !(2) =75

%e$er#ine $+e 4a"/es o! # an% n.

. *i4en !, =n%" +

45

%

n5 !(2) = 2 an% !(8) = . Fin% # an% n.


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. T+e arro %iara# s+os $+e !/n&$ion ! ,&"

p"

+

5 @5

%e$er#ine $+e 4a"/es o! p an% @

2&"

p"

+

' 3

4 +

1. CO-POSITE F:NCTIONS

A 3 C

! 2 (2) g(2)


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!()

? ! is the nction which ma!s set 5 onto set @ and is the nction thatma!s set @ onto set -, then set 5 can be ma!!ed directl" to set - b" acom!osite nction, re!resented b" !().

1. Ai


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'. Ai 1. Cind

(a) '(2)

(b) 'g(2)

(c) g(')


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. Ai


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4. Ai . Cind

(a) g(2)

(b) g(2)

(c) g(1)


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1. Ai


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1. Ai 9, ind (2).

'. Ai


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. Ai


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a 6

(b) 2 '2'>

1

(c) 2 + > 42

c

4 3 9

(d) 2 4)

2 +

'b

+

1+

d

e

+

9


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(e) 2+2

)2

+

3

+

'. Ietermine the in


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(c) g $ 2 B42

)2'

+

, 2 34

. (a) Cind the in


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(a) Cind the in


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(d) Ai 1 (2) and state and gi


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(c) Ai 1 (2) and state and gi


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. Ai 16. Cind(a) the


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6. Ai1(), (b) hg(2). K4

marLsMSPM'*Paper

+

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

State(a) the range o the relation,

(b) the t"!e o the relation. K'marLsM

SPM'*Paper +

Pas$ ears SP- Pa ers

P = 1, ', /Q = ', 4, 6, , 10/

d

e

. w

2

"

.


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4. Ai1$ 2 B '-2 (

+, where m and

L are constant, ind the


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(a) h>1 (+),(b) gh('). K'

marLsMSPM'*Paper +

*. The nction w is deined as w(2) = 2'+

, 2 '. Cind(a) w>1(2),(b) w>1(4). K

marLsMSPM'*Paper +

. The ollowing inormation reers to the nctions h and g.

Cind gh>1 (2). KmarLsM

SPM'*Paper +

+ , 2

, 1

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1 Functions MyModule