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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