MT F4 Chapter 1 Functions_1

Chapter 1 Functions

1.1 Relations – linking (pairing) of the elements of two sets.
A relation between two sets can be represented by
(a) ordered pairs
(b) arrow diagram
(c) graph

Example:

1 Set X = {4, 9, 16} Set Y = {2, 3, 4}

Set X is related to set Y by ‘square root of ’.

ordered pairs:

arrow diagram:

graph:

2 Set X = {4, 6} Set Y = {16, 18, 24}

Set X is related to set Y by ‘multiples of ’.

ordered pairs:

arrow diagram:

graph:

1.2 Domain, Codomain, Objects, Images and Range
Example:

1

Domain Codomain
Objects Images
Range
2
Set Q
Codomain
Images Set P
Range

Domain
Objects

1 Domain = { 3, 5 }
Codomain = { 6, 10, 15 }
Objects = 3, 5
Images = 10, 15
Range = { 10, 15 }

2 Draw the 2• •5
Set Q arrow diagram 4• •9
6• • 13

Set P Set P Set Q

Domain = { 2, 4, 6 } Important
Codamain = { 5, 9, 13 }
Objects = 2, 4, 6 ** Write the Domain, codomain, range
Images = 5, 9 in the { }
Range = { 5, 9 }
** But Objects, image without { }

1.3 Types of Relations
(i) One-to-one
(ii) Many-to-one
(iii) One-to-many
(iv) Many-to-many

(i) One-to-one

(ii) Many-to-one
(iii) One-to-many
(iv) Many-to-many

1. Graph below shows the relation between set E and set F.

Set F State
Set E (a) the images of 10,
(b) the objects of 6,
(c) the range of the relation.

2. Diagram 2 shows the relation between set A and set B.

State
(a)the type of the relation,
(b)the range of the relation.

Set A Set B

1.4 Functions
(i) A function is a special type of relation, each and every object in the
domain has only one image.

(a) one-to-one (b) many-to-one

(ii) Function notation function notation
arrow diagram

1

2

f :x

f (x) 

3

f :x

f (x) 

1.5 Object and Image of A Function

Example: arrow diagram

1 f 2  5

object image 2• •5
4• •9
f (4)  5 6 • •13

f (6) 13 ( object ) ( image )

2 gx  x 1

object image

3 hx  x2  2x 1

object image

 4 Given the function f x  x  4 ,

(a) find the image of 3,

object

x  3, f x  x  4

f 3  3  4 object image

7

the image of 3 = 7

(b) find the object that has the image 10.

f x  10 image 10

x  4  10
x  10  4

6

the object that has the image 10 is 6.
the image of 3 = 7

5 Given the function f : x  x2  4x 1 , find

(a) the image of 2,
(b) the objects that have the image 4.

(a) f x  x2  4x 1

f 2  22  4(2) 1

 5

(b) f x  4

x2  4x 1 4
x2  4x 1 4  0

x2  4x  5  0

x 1x  5  0

x  1 or 5

6 A function f is defined by f : x  5x  2.

Find
(a) the object that has the image 3,
(b) the object that is mapped onto itself.

7 Diagram below shows an arrow diagram representing the function

f : x  x2  2.

State
(a)the domain,
(b)the range,
(c)the image of 0,
(d)the objects of –1.

1. Given the function f : x  x2  2x  5 , find the objects that have

the image 8.

2. Given the function h : x  2x2 1, find the image of 2,