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1.1 CONCEPTS OF RELATIONS1.The relation between two sets, P and Q, is the pairing of elements in set P with elements in set Q.
2. P Q
(a) Set P = domain(b) Set Q = codomain(c) Each elements in set P = object(d) Each elements in set Q = image(e) The images (4, 9, 16( in set Q = range3. Relations can be represented by;(a) Arrow diagrams: (b) Ordered pairs: X Y ((2, 8), (3, 27), (4, 64)(
2 8
3 27
4 64(c) Graphs:
64
27
8 2 3 4
4. Four types of relations:
(a)One-to-one relation(b) One-to-many relation X Y X Y
2 8 18 2
3
3 27 12 4
(c) Many-to-one relation (d) Many-to-many relation
X Y X Y
2 8
3 9
4
1.Complete the table below of the following relations.
X
Y
(a)
(b){(3, 5), (5, 9), (7, 13)}
(c) z
y
x
2 8 14
Answers:DomainCodomainObjectImageRange
(a)
(b)
(c)
1.Based on the relations given, identify the images or objects.
(a) X Y
7 10
8
3 6
(b) {(3, 5), (3, -1), (4, 9)}
(c) 6
4
2
3 6 9
Answers:
(a)Object of 6Object of 8Image of 3Image of 7
(b)Object of 5Object of 9Image of 3Image of 4
(c)Object of 2Object of 6Image of 6Image of 9
2.State the type of relations for each of the following.
(a) X Y
Answer:
a p
b q
c r
(b){(3, s), (4, t), (5, s)}
Answer:
(c) 6
4Answer: 2
p q s
(d) X Y
Answer:
(e) X Y
mAnswer: 1
n
2 p
1.2CONCEPTS OF FUNCTIONS
1.A function = special relation where every object in the domain has only one image in the range.
2.One-to-one relation and many-to-one relation are functions.
3.One-to-many relation and many-to-many relation are not functions.
4.A function can be expressed by function notation, in which the function is represented by the symbol f and the object by the symbol x.
f : x ( y is read as function f maps x to y, or,
f(x) = y which is read as y is the image of x under function f
Example: f : :x ( 3x . fuction f maps x to 3x. f(x) = 3x . 3x is the image of x under function f.
1.State whether the following relation are functions and give reasons.
(a) X Y
Answer:
(b){(1, 3), (3, 5), (5, 7)}
Answer:
(c){(a, p), (a, q), (b, r)}
Answer:
(d) 7 Answer:
5
3
a b c
(e) X Y
Answer:
m
a
n
b
p
2.Write the following relations in function notation. f
(a) x x2 Answer:
3 9
2 4
1 1
f
(b) x 4x2 - 3
Answer:
1.Determine the image of the object for each of the following functions.(a) f(x) = 4x + 5, x = 1, 5
(b) f(x) = x2 + 2, x = 0, -3
(c) f(x) = (4 x(, x = 1, 6
(d) f(x) = , x = 2, 7
1. A function is defined as f : x (3x + 6, find
(a) the object if the image is 18.
(b)the value of x if f(x) = 2x.
2.A function is defined as f : x ( x2 + 6x, find
(a) the object if the image is 7.
(b) the object that maps to itself.
3.Given a function g : x ( , x 2, find
(a) the value of x if g(x) = 6
(b) the value of x if g(x) = x 4.
4.Given a function h : x ( , find
(a) the value of x if h(x) = 3.
(b)the object if the image is 5.
1.Given f(x) = px + q, f(0) = -5 and f(3) = 7. Find the values of p and q.
2.A function is defined as f: x = 2x2 mx + n. If f(1) = 4 and f(2) =7, determine the values of m and n.
3. Given f: x = , x
EMBED Equation.3 , f(2) =2 and f(5) = 4. Find m and n.4.The arrow diagram shows the function f :x ( , x q, determine the values of p and q x
2 -3
4 5
1.3 COMPOSITE FUNCTIONS
A B C
f g
gf(x)
If f is the function which maps set A onto set B and g is the function that maps set B onto set C, then set A can be mapped directly to set C by a composite function, represented by gf(x).
1.Given f:x 2x + 5 and g:x 3x 4. Find(a) fg(x)
(b) gf(x)
(c) fg(3)
2.Given f:x 5x + 2 and g:x x2 1. Find
(a) f2(x)
(b) f2g(x)
(c) fg(2)3.Given f:x , x ( 2 and g:x x + 4. Find
(a) fg(x)
(b) gf(x)
(c) f2(x)4.Given f:x 2x + 3 and g:x 2x 3. Find
(a) fg(x)
(b) gf(x)
(c) fg(1)
1.Given f(x) = 2x + 5 and fg(x) = 8x 5, find g(x).
2.Given f(x) = x + 3 and fg(x) = x2 + 2x + 1, find g(x).
2. Given f(x) = , x ( -2 and and fg(x) =, x ( , find g(x).
1.Given g(x) = 3x and fg(x) = 15x 9, find f(x).
2.Given f(x) = x - 3 and gf(x) = x2 - 4x + 5, find g(x).
3. Given f(x) = , x ( 0 and and fg(x) =, x ( 0, find g(x).
1.4 INVERSE FUNCTIONS
X f Y
f1 If f is the function which maps elements in set X onto the elements in set Y, then when the elements in set Y are mapped onto the elements in set X, the function is called an inverse function of f.
The notation for the inverse function of f is written by f1 When f(x) = y, then f1 (y) = x. When f1f = x, then ff1 = x.
Not all functions have inverse functions. The inverse function exists if and only if the function is a one-to-one relation.
1.Find the value of each object by inverse mapping. f
(a) x x2 + 2
3 11
a 6
f
(b) x 2x2 3
1
f
(c) x 5 4x
c 3
4 - 9
f
(d) x
f
(e) x
3 - 3
f 52. Determine the inverse function of the following functions.(a) f : x 2x + 4
(b) f : x
(c) g : x , x ( -4
3.(a)Find the inverse function, h-1(x) of the function h : x , x ( 0. What is the value for h-1(2)?
(b)Find the inverse function, f-1(x) of the function f : x , x ( - 2. What is the value for f -1(3)?
(a)Find the inverse function, g-1(x) of the function g : x 7x - 3. What is the value for g-1(5)?4.(a)Given the inverse function, f 1 : x 3x + 2, determine the function f(x).
(b)Given the inverse function, g 1 : x , x ( -5, determine the function g(x).
(c)Given the inverse function, f 1 : x , determine the function f(x).
(d)Given the inverse function, h 1 : x , x ( - , determine the function h(x).
5.(a)Given function, f(x) = 7x + 2, determine f 1 (x) and state and give reason whether the inverse function exists.
(b)Given function, g(x) = x2 + 9, determine g 1 (x) and state and give reason whether the inverse function exists.
(c)Given function, f(x) = x3 - 16, determine f 1 (x) and state and give reason whether the inverse function exists.
1.Given the function g : (x) , x ( m, find
(a)the value of m.
(b)g 1(x).
2.Given f : x 2x2 5 and g : x x + 3, find
(a)fg(x).
(b)the value of gf(-1).
3.Given the function h(x) = 3x + 5, find the value of x
(a)if h2(x) = h(-x)
(b)when x is mapped onto itself.
4.Given the function g : x 3x + 2, find the function f(x) if
(a)fg : x 2x2 + 5
(b)gf : x 2x 3
5.Given f(x) = px + q and f2(x) = 4x 16. Find
(a)the values of p an q.
(b)the value of ff1(2).
6.Given that f(x) = 5x - 8, find
(a)f(2).
(b)the values of the objects that have the image 7.
1.Based on the the above information, the relation between P and Q is defined by the set of ordered pairs {(1, 2), (1, 4), (2, 6), (2, 8)}. State
(a)the image of 1.
(b)the object of 2.
[2 marks]
SPM2003/Paper 1
2.Given that g : x 5x + 1 and h : x x2 2x + 3, find
(a)g1(3),
(b)hg(x).
[4 marks]
SPM2003/Paper 1
3.Diagram 1 shows the relation between set P and set Q.
State(a) the range of the relation,
(b) the type of the relation.
[2 marks]SPM2004/Paper 1
4.Given the functions h : x 4x + m and h1 : x 2kx +, where m and k are constant, find the value of m and of k .
[3 marks]SPM2004/Paper 1
5.Given the function h(x) = , x ( 0 and the composite function hg(x) = 3x, find
(a)g(x),
(b)the value of x when gh(x) = 5.
[4 marks]
SPM2004/Paper 1
6.In Diagram 1, the function h maps x to y and the function g maps to z.
x h y g z
58
2 DIAGRAM 1
Determine(a) h1 (5),(b) gh(2).
[2 marks]SPM2005/Paper 17.The function w is defined as w(x) = , x ( 2. Find
(a)w1(x),
(b) w1(4).
[3 marks]
SPM2005/Paper 1
8.The following information refers to the functions h and g. Find gh1 (x).
[3 marks]SPM2005/Paper 1My
Additional
Mathematics
Module
Form 4
(Version 2007)
Topic: 1
FUNCTIONS
by
NgKL
(M.Ed.,B.Sc.Hons.,Dip.Ed.,Dip.Edu.Mgt.,Cert.NPQH)
Kajang High School, Kajang, Selangor
IMPORTANT POINTS
4
9
16
25
2
3
4
Exercise 1.1(a):
4
10
14
2
4
6
Exercise 1.1(b):
Exercise 1.2(a):
1
-1
2
1
15
Exercise 1.2(b)
Exercise 1.2(c)
Exercise 1.2(d)
gf(x)
f(x)
x
Exercise 1.3(a)
Exercise 1.3(b)
Exercise 1.3(c)
y
x
Exercise 1.4(a)
2
b
5
15
d
e
5
9
Problem Solving
Past Years SPM Papers
P = {1, 2, 3}
Q = {2, 4, 6, 8, 10}
. w
x
y
. z
d
e
f
h : x 2x 3
g : x 4x - 1
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