MODUL PROGRAM IBNU SINA ADDITIONAL MATHEMATICS
MODULE 1 FUNCTIONS
Terbitan :YAYASAN PELAJARAN JOHOR JABATAN PELAJARAN NEGERI JOHOR
FU NCTIONS
FORM 4
MODULE 1 IBNU SINA TOPIC : FUNCTIONS Express Note :
1.
2.
Representing a relation -
representing a relation using arrow diagrams, ordered pairs and graphs
-
using the concept of domain, codomain, object, image and range
Classifying a relation -
one to one one to many many to one many to many
3.
Expressing functions
4.
Determining composite functions
5.
-
fg, gf, f 2 , f 3, ........
-
determining g from the composite function fg or gf
Determining the inverse function -
Let f -1 (x) = y, then x = f (y)
-
the condition of an inverse function
PAPER 1 1.
The arrow diagram represent the relation between the elements of two sets, P and Q. P -3 -2 2 3
Q 4
9
Based on the above diagram, state (a) the set of ordered pairs, (b) the objects of 4
PROGRAM IBNU SINA TAHUN 2010(ALL As)
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FU NCTIONS
2.
FORM 4
In the diagram below, the function f maps x to y and the function h maps y to z .
f
h
x
y
z
5
3 -1
Determine (a) f 1(3) (b) hf 1 3.
R = { 2, 3, 5} S = { 4, 5, 9, 10, 15} Based on the above information, the relation between R and S is defined by the set of ordered pairs {(2, 4),(5,10),(3,9),(2,5)} State (a) the image of 2 (b) the object of 5.
4.
Given the function f ( x ) ! 3 x 2
2 x , find
(a) the image of -1. (b) the objects that have the image 1.
5.
Given that f : x p 3 x 2 and g : x p x
x , find
(a) f 1(3), (b) g f ( x ) 6.
Given the functions
:xpt
x and h :
x , where t and r are constants, find the r
value of t and r .
PROGRAM IBNU SINA TAHUN 2010(ALL As)
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FU NCTIONS
7.
The function f is defined by fg : x p
f :x
p
x . Another function
FORM 4
g is such that
4 x , x { 2 . x 2
(a) Find the function g . (b) State the value of x such that the function g is undefined.
8.
The arrow diagram below shows the function f : x
x
p
px
qx 2 2
f
px+qx
-8
-
-16 Find (a) the value of p and of q. (b) the values of x that are mapped onto themselves.
9.
The functions f and g are defined by f ( x ) ! fg
10.
x 2
and g ( x ) ! 1 4 x respectively. Find
1
( x ).
Given that f ( x )
!
x
and f g ( x )
!
x , find g f ( x ) .
PAPER 2 1.
The function f and g are defined by f : x p 2 x 3 and g : x If the composite function fg is given by fg : x
p
2x2
p
x2
bx c respectively.
4 x 3 , find
(a) the value of b and of c . (b) the value of g 2 ( 1) .
2.
(a) The function f is defined by f : x p
x , x { . Find x
(i) f (ii) f 4
PROGRAM IBNU SINA TAHUN 2010(ALL As)
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FU NCTIONS
(b) Given the functions f : x p
x , g : x p x and g f : x
p
FORM 4
x , find the
value of p and of q.
3.
x
f
h x k
7
5 -5
-1
The above arrow diagram represents a function f : x p
h x k
, Find
(a) the value of h and of k , (b) the value of x such that the function f is undefined. (c) the image of 10 (d) the object that has the image -10.
4.
Given the functions f : x p 4 5 x and g : x p x 2 2 , find (a) (a) f , (b) the value of f 1g (4 )
5.
Given the function f ( x ) ! 5 2 x 2 and g ( x )
!
kx
, find the value of k for each of the
following cases: 1
(a) (a) g
9
!
3
(b) (b) f g ( x ) ! 2 x 2
12 x 13
END OF MODULE 2
PROGRAM IBNU SINA TAHUN 2010(ALL As)
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FU NCTIONS
FORM 4
MODULE 1 - ANSWERS TOPIC : FUNCTIONS PAPER 1 1.
(a) (-3,9), (3,9), (-2,4), (2,4) (b) -2,2
2.
(a) -1 (b) 5
3.
(a) 4 or 5 (b) 2 (a) 1
4.
(b) 5.
(a)
1
1
,
3
5 3
(b) 6.
7.
t
!
(a)
,r
!
:
p
(b) x =2 8.
(a) p=2, q=-3 (b) 0,
9.
10.
1 3
fg1(x) !
gf(x) !
x 1 8
2 6x
3
PROGRAM IBNU SINA TAHUN 2010(ALL As)
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FU NCTIONS
FORM 4
PAPER 2 1.
2.
(a) b=2 , c = -3 (b) -3 (a) (i)
1 ,
x {
0
(ii) x
x
(b) p= -6 , q=10 3.
4.
5.
(a) h=10 , k = 5 (b) x = 5 (c) 2 (d) 4 4x (a) f 1 : x p 5 (b) -2 (a) 2 (b) -1
END OF MODULE 1
PROGRAM IBNU SINA TAHUN 2010(ALL As)
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