# My Add Maths Module - Functions

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```					My 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 1.1 CONCEPTS OF RELATIONS 1. The relation between two sets, P and Q, is the pairing of elements in set P with elements in set Q. P
2 3 4

2.

Q
4 9 16 25

(a) (b) (c) (d) (e) 3.

Set P = domain Set Q = codomain Each elements in set P = object Each elements in set Q = image The images 4, 9, 16 in set Q = range

Relations can be represented by; (a) Arrow diagrams: X 2 Y 8 (b) Ordered pairs: (2, 8), (3, 27), (4, 64)

3 4

27 64

(c) Graphs: 64

27 8

2

3

4

4.

Four types of relations: (a) One-to-one relation X 2 Y 8 (b) One-to-many relation X 18 Y 2 3 3 27 12 4

(c) Many-to-one relation X 2 3 4 Y 8 9

(d) Many-to-many relation X Y

Exercise 1.1(a): 1. Complete the table below of the following relations. X (a)
2 4 6

Y
4 10 14

(b)

{(3, 5), (5, 9), (7, 13)}

(c)

z

y x

2 Answers: Domain (a)

8

14

Codomain

Object

Image

Range

(b)

(c)

Exercise 1.1(b): 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 Answers: Object of 6 (a) Object of 5 (b) Object of 2 (c)

6

9

Object of 8

Image of 3

Image of 7

Object of 9

Image of 3

Image of 4

Object of 6

Image of 6

Image of 9

2.

State the type of relations for each of the following. (a) X a b c Y Answer: p q r

(b)

{(3, s), (4, t), (5, s)}

(c)

6 4 2 Answer:

p

q

s

(d)

X

(e)

X

1 n 2 p

1.2

CONCEPTS OF FUNCTIONS 1. A function = special relation where every object in the domain has only one image in the range. One-to-one relation and many-to-one relation are functions. One-to-many relation and many-to-many relation are not functions. 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.

2. 3. 4.

Exercise 1.2(a): 1. State whether the following relation are functions and give reasons. (a) X Y Answer:

(b)

{(1, 3), (3, 5), (5, 7)}

(c)

{(a, p), (a, q), (b, r)}

(d)

7 5 3 a b c

(e)

X

a n b p

2.

Write the following relations in function notation. f (a) 3 2 1 x x2 9 4 1 Answer:

f (b) x
1 1 -1 2 15

4x2 - 3 Answer:

Exercise 1.2(b) 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) =

2 x 3

, x = 2, 7

Exercise 1.2(c) 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 

3 x2

, 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 

2x  5 3

, find

(a) the value of x if h(x) = 3.

(b) the object if the image is 5.

Exercise 1.2(d) 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 =

4 mx  n

,x 

n , f(2) = 2 and f(5) = 4. Find m and n. m

4.

The arrow diagram shows the function f :x  determine the values of p and q x
x p xq

x p xq

, x  q,

2

-3

4

5

1.3

COMPOSITE FUNCTIONS A B C

f x
f(x)

g gf(x)

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).

Exercise 1.3(a) 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 → (a) fg(x)

3 , x  2 and g:x → x + 4. Find x2

(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)

Exercise 1.3(b) 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) =

2 2 , x  -2 and and fg(x) = , x  ─ ½, find g(x). x2 2x  1

Exercise 1.3(c) 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) =

2 4 , x  0 and and fg(x) =  5 , x  0, find g(x). x x

1.4

INVERSE FUNCTIONS X f Y

x

y

f—1      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 f—1 When f(x) = y, then f—1 (y) = x. When f—1f = x, then ff—1 = x. Not all functions have inverse functions. The inverse function exists if and only if the function is a one-to-one relation.

Exercise 1.4(a) 1. Find the value of each object by inverse mapping. f (a) 3 a x x2 + 2 11 6

f (b) x 2x 2 – 3

1 2
b

5 15

f (c) c 4 x 5 – 4x 3 -9

f (d) x

x 4 3

d e

5 9

f (e) x

x3 x 5
-3 5

3 f

2.

Determine the inverse function of the following functions. (a) f : x → 2x + 4

(b) f : x →

2x  5 6

(c) g : x →

2x  3 , x  -4 x4

3.

(a) Find the inverse function, h-1(x) of the function h : x → What is the value for h-1(2)?

3  2 , x  0. x

(b) Find the inverse function, f-1(x) of the function f : x → What is the value for f -1(3)?

3 , x  - 2. x2

(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 f(x).

– 1

: x → 3x + 2, determine the function

(b) Given the inverse function, g function g(x).

– 1

: x →

2 , x  -5, determine the x 5

(c) Given the inverse function, f f(x).

–1

:x→

3  4x , determine the function 2

(d) Given the inverse function, h function h(x).

–1

:x→

5x , x  - ½, determine the 2x  3

5.

(a) Given function, f(x) = 7x + 2, determine f reason whether the inverse function exists.

– 1

(x) and state and give

(b) Given function, g(x) = x2 + 9, determine g reason whether the inverse function exists.

– 1

(x) and state and give

(c) Given function, f(x) = x3 - 16, determine f reason whether the inverse function exists.

– 1

(x) and state and give

Problem Solving 1. Given the function g : (x) → (a) the value of m. (b) g –1(x).
3 , x  m, find x4

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 ff–1(2).

6.

Given that f(x) = │5x - 8│, find (a) f(2). (b) the values of the objects that have the image 7.

Past Years SPM Papers P = {1, 2, 3} Q = {2, 4, 6, 8, 10} 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) g–1(3), (b) hg(x).

[4 marks]
SPM2003/Paper 1

3.

Diagram 1 shows the relation between set P and set Q.
. w d x e y f . z

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 h–1 : x → 2kx + are constant, find the value of m and of k .

5 , where m and k 8 [3 marks]
SPM2004/Paper 1

5.

Given the function h(x) =

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

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 5 2 Determine (a) h–1 (5), (b) gh(2). y g z 8 DIAGRAM 1

[2 marks]
SPM2005/Paper 1

7.

The function w is defined as w(x) = (a) w–1(x), (b) w–1(4).

5 , x  2. Find 2x

[3 marks]
SPM2005/Paper 1

8.

The following information refers to the functions h and g. h : x → 2x – 3 g : x →4x - 1 Find gh–1 (x). [3 marks]
SPM2005/Paper 1

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