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FUNCTIONS

Name

........................................................................................

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CHAPTER 1 : FUNCTIONS

SET NOTATION

The notation of set is { } .

Examples.

1. A is the set of even numbers less than 12 .

A = { 2 , 4 , 6 , 8 , 10 }

2. B = { violet, indigo, blue, green, yellow , orange , red }

B is ……………………………………………………..

3. We can also use Venn Diagram to represent a set .

RELATION

A relation from set A to set B is the linking (or pairing) of the elements of set A to the

elements of set B .

1.1 Represent a relation

A relation between two sets can be represented by :-

(a) arrow diagram

(b) ordered pairs

(c) graphs

Example.

A relation from set A = { - 3 , 1 , 2 , 3 ) to set B = { 1 , 4 , 9 } is given by “power of 2

for “ . Represent the relation using

(a) arrow diagram

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(b) ordered pairs

(c) graph

Activity 1

1. Given that set C = { 2 , 4 , 5 } and set D = { 4 , 16, 18 , 25 } . Represent the relation

“multiple of “ between set C to set D using

(a) arrow diagram

(b) ordered pairs

2. Represent the given relation using ordered pairs and graph.

(a)

Chandra

Aisya

Razif

Chong

Hasnul

Lee

Siva

A B “child of”

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(b)

1.2 Identify domain, codomain , object , image and range of a relation .

Definition

Domain - First set

Codomain – Second set

Object – Elements in the domain

Image – Elements in the codomain that are linked to the objects.

Range – set of images

Example 1

Draw an arrow diagram to show a relation “ power of 2 for “ from set A = { - 4 , - 3, - 2 ,

2 , 3 } to set B = { - 9 , - 4 , 4 , 9 } . Then , identify the

(a) image for number 3

(b) image for number – 2

(c) object for number 4

(d) object for number 9

(e) domain

(f) codomain

(g) range

of the relation .

M N

5 M

10

15

2

3

4

4

“has factor

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Solution

Example 2

A relation is given in the form of ordered pairs , H = { (2 , 6) , (2 , 8) , (2,10) , (4,8),

(5,10) , (5,15) } . Identify the

(a) image of 5

(b) object of 8

(c) domain

(d) range

(e) codomain

Example 3

The relation below is shown in the graph .

Identify the

(a) image of 4

(b) object of 15

(c) domain

(d) range

(e) codomain

5

10

15

20

25

5

Set B

2

4

6

8

Set A

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Activity 2

1. Draw an arrow diagram to show a relation “ add 3 to “ from set P = { 1 , 2, 3 , 4 , 5 }

to set Q = { 1 , 2, 3 , 4 , 5 , 6 , 7 , 8 } . From the diagram , state the

(a) image of 3

(b) object which its image is 6

(c) domain

(d) codomain

(e) range

for the relation.

2. The relation that mapped set A = { 2 , 3 , 4 , 5} onto set B = { 6 , 8 , 9 } is “factor of” .

(a) Represent the relation using

(i) arrow diagram

(ii) ordered pairs

(iii) graph

(c) State the

(i) domain (iii) range (v) object of 3

(ii) codomain (iv) image of 8

1.3 Classifying the types of relation

Type of relation Example of Arrow diagram

1. One – to – one relation

Each object in the domain has only one

image in the codomain

2. One – to – many relation

Each object in the domain has more

than one image in the range

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3. Many – to – one relation

Each image in the range has more than

one object in the domain but each object

has only one image.

4. Many – to – many relation

Each object in the domain has more

than one image in the range and each

image in the range has more than one

object in the domain.

State the type of relation shown by each of the following relation.

Question Type of relation

1.

2. R = { (1,1) , (2,8) , (3,27) , (4,64) , (5,25) }

A B

“element of”

Carbon

Hydrogen

Oxygen Ozone

Metane

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3.

4. B = { (3,7) , (3,9) , (3,11) , (5,9) , (6,11) }

5.

Set B

Set A

5 4

1

1 4 4

1

1 3 4

1

1 2 4

1

1

1 4

1

1

4

1

3 2 1

“power of 2” P

0

4

9

- 3

- 2

2

3

Q

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2 . FUNCTIONS

2.1 Definition of a functions

A functions is a special relation whereby for EVERY object in the domain ,

there is ONE AND ONLY ONE image in the codomain.

Hence , the type of relations that are considered as a functions are

(i) one to one relation

(ii) many to one relation

Examples

Identify which of the following relations are functions. Give reason for your answer.

Question Answer / Reason

(a) R = { (1,2) , (3,4) , (5,6) , (7,8) }

(b)

(c)

(d) K = { (May,31 days),(June,30 days),

(July, 31 days),(Feb,28 days)}

“multiple of” A B

3

9

8

6

4

3

10 5

3

“three times” M N

1

2

3

4

3

6

9

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Activity 4

Determine whether each of the following relation are considered as a function or not .

Question

Function / not a function

1. H = { (1,4) , (3,4) , (5,7) , (8,9) }

2.

3.

4.

A B

March

April

May

28 days

30 days

31 days

B

A

C

D

X

Y

Z

Set A

Set B

2

4

6

8

1 2 3 4

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2.2 Express functions using function notation

(1) A function can be represented using small letter such as f ,g , h and etc .

(2) (a) f : x y ( is read as “function f maps x to y” ) can also be written as

f(x) = y .

(b) g : x 5x - 3 , hence g(x) = 5x - 3 ( is read as “5x – 3 is the image of x

under the function g” )

(c) h : m 4 – 2m , hence h(m) = 4 – 2m .

(d)

f : x 2x or f(x) = 2x

2.3 Domain , object , image and range of a function

Example 1

Domain = { 0 , 1 , 2 }

Objects = …., …., …..

Images = …. , …. , …..

Range = { ….,…..,…..}

Example 2

Find the image of f : x x + 4 given that x = { - 1 , 1 , 3 } . State the domain , object

and range .

“Square of”

domain codomain

2

4

6

4

16

36

g x x

2 - 3

2

1

0 0

-3

-2 4

1 4

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2.4 Determine the image of a function given the object or vice versa .

Example 1

(a) Given that f(x) = 4x – 1 . Find the

image of x = 3 .

(b) Given that 2( ) 4f x x . Evaluate

the value of f ( 4 ) .

(c) A function h is defined as

h : x 2 – 5x. Find the image for

x = -2 .

(d) Given the function m(x) = 3x . Find

the image for x = - 8 .

Example 2

A function g is defined as g : x 3

2

x , find the value of

(a) g(0) , g(- 2) , g(5) , g(8) (b) State the value of x such that

the function is not defined .

Example 3

Given that f : x 2x + 5 . Find the object for which the image is 9 .

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Example 4

Given the function g(x) = 2x – 9 . Find the value of the object that maps onto itself .

Example 5

Given that function f(x) = 4

3

x . Find the value of x such that f(x) = 2x .

Activity 5

1. A function f is defined as f : x 3x + 2 . Find

a. f(2)

b. f(3)

c. f(x + 1)

.

2. A function f is defined by f : x 4x – 3 . Find

(a) the object that has 9 as its image.

(b) the object that is mapped onto itself .

3. A function f is defined as f : 22 1x x .

(a) Find the image of - 3 , -1 , 0 , 1 , 3 .

(b) Find the objects that has 49 as it image .

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3 . COMPOSITE FUNCTION

3.1 Composition of two functions

(1) The figure above shows a function f that maps elements in set A to those in set

B and the function g that maps set B to set C . The combined effect of the two

functions can be represented by the function gf which is called the composite

function of g and f.

(2) fg = fg(x) = f[g(x)]

(3) f 2 = ff , f

3 = fff or ff

2 or f

2f and so on .

Example

For each pair of the following given function, find the composite function fg and gf .

(a) f(x) = 5x , g(x) = x + 3

(b) f(x) = 5x – 1 , g(x) = 2x + 3

A B C g f

x f(x) g(x)

gf

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(c) f : x 23x , g :x x – 4

(d) f : x 5

x , g : x 1 – 2x

3.2 Determine the image or object of a composite function

Examples;

1. Given, f : x 5x and g : x 2x – 4 . Find

(a) fg(x) (c) gf(x)

(b) fg(4) (d) gf(- 1)

2. Given that f : x 0,2

xx

and g :x 3x + 1 . Find

(a) fg(x) (b) gf(x)

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(c) fg(4) (d) gf(4)

3. Given that f : x 3 + 2x and g : x 8

x , x 0 . Find the value of fg(2) .

4. Functions f and g are defined as f :x x + 2 and g : x 3x – 4 respectively . Find

(a) f 2 (b) g

2

(c) f 2(2) (d) g

2 ( 3 )

5. Functions f and g are defined as f(x) = 3x and g(x) = 2x – 1 respectively. Find the

value of x if gf(x) = 11 .

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Activity 6

1. Given f : x 3x + 2 and g : x 4x – 6. Find

(a) fg ( x )

(b) gf ( x )

(c) gf ( 2 )

2. Functions f and g are defined as f : x 4x - 5 , g : x 3

, 0xx

respectively .

(a) Find the composite function gf and the value of gf(4) .

(b) Determine f 2(x)

3. Given f : x px + q and f 2 : x 25 8x . Find the value of p and q ( p > 0) .

4. INVERSE FUNCTION

4.1 Find the object, given its image and function by inverse mapping .

If f represent a function, the inverse function of f is denoted by f – 1

.

( take note that 1f 1

f )

Therefore , if f : x y then f(x) = y

if f – 1

: y x then f – 1

(y ) = x

Example

Function f is defined by f : x 2x – 1 .Find the value of

a. f – 1

(5)

b. f – 1

(- 3)

c. f – 1

(3) ?

f

x f(x)

f - 1

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4.2 Determining the inverse function

Examples

1. Given f(x) = 2x + 1 . Determine the inverse function f – 1

.

2. A function g is defined as g ( x ) = 5x - 8, determine g – 1

(x) .

3. Given that f : x 3 1

5

x , find the inverse function f

– 1(x) .

4. A function g is defined as g : x 3 – 2x . Determine

(a) g – 1

(x) (b) the value of g – 1

(2)

(c) the value of k if g – 1

(k)= k (d) g

– 1h(x) if given h : x

4

x .

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5. Given that f : x 3

2x , x 2 and g : x 4x – 3 . Find

(a) f – 1

(x) and state the value of x such (b) the value of f – 1

g(3)

that f – 1

is not defined .

6. Given f : x 2x + k and its inverse function f – 1

: x nx + 3

2 . Find

(a) the value of k and n (b) the value of f – 1

f(4)

7. Given f(x) = 4x + 7 and g(x) = 2x + 1 . Find

(a) fg – 1

(x) (b) the value of x such that gf(- x) = - 9

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8. Given f : x p – qx . determine

(a) f – 1

(x) in terms of p and q (b) the value of p and q if

f – 1

(8) = -1 and f(2) = - 7 .

Activity 8

1. Find the inverse function for each of the following

(a) f : x 3x + 4

(b) g : x 2x – 7

(c) 0,2

: xx

xf

(d) 2

32:

xxg

(e) 4,4

2:

x

xxf

2. Given the function f : x 3x+ 2 , find the value of f – 1

(5) .

3. A function f is defined as f : x 2 5

, 55

xx

x

. Find the value of f

– 1(1)

4. Given that f : x , 44

x ax

x

and f(7) = 4 . Find

(a) the value of a

(b) the value of f – 1

(- 2)

5. Given the function f : x mx + n and its inverse function 3

8:1 x

xf . Find the

value of m and n.

- 1

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4.3 The condition for the existence of an inverse function

There are two types of relation that are considered as a functions that is

(i) one to one

(ii) many to one

The reverse mapping for one to one relation is still one to one but the reverse

mapping for many to one relation is not a function because it is one to many relation.

Example

Determine whether each of the following functions has inverse functions. Explain your

answer .

Question Answer

(a)

(b)

f

g

3

5

9

1

3

1

5

1

9

2

3

4

5.2

4.6

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Topical Test

1. Express the relation “factor for” between Set A = {4, 6, 10, 15} and Set B = {2, 3, 5}

in the form of

(a) graphs

(b) ordered pairs

2. The arrow diagram shows the relation between Set A and Set B.

State the

(a) domain

(b) codomain

(c) range

(d) image for s

(e) object for image 1

3.

The figure above shows a function bxaxxf 2: . Find

(a) the value of a and b

(b) the image for x = 3.

4. Given the function 14)( xxf and 32)( xxg . Find the composite functions

(a) fg

(b) gf

(c) )(2 xf

(d) )1(2 g

r .

s .

t .

. 2

3

. 5

7

. 11

4

- 1

1

- 1

f

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5. Given 2,2

3)(

x

xxf and 14)( xxg . Find the value of

(a) )1(fg

(b) )2(2 g

6. Sketch the graph for the absolute function xxf 25)( in the domain 31 x .

Hence, state the range of the function.

7. Given the function 4: xxf and the function 0,2

: xx

xg . Find

(a) the inverse function )(1 xf

(b) the value of )3(1gf

8. Given the function khx

xf

10

: , where 5)1( f and 10)2( f .

Find

(a) the value of h and k

(b) the values of x such that f(x) = x

10. Functions f and g are defined as hxx

xf

,32

3: and 14: xxg

(a) State the value of h.

(b) Express the functions gf and )(1 xf

(c) Find the value of )3(1 gf

11. Given f(x) = px + q, f(0) = -5 and f(3) = 7. Find the value of p and q