Function : Introduction

What is function?
https://www.geogebra.org/m/mzxvxe44
INPUT Num.
OUTPUT Num.
(INPUT, OUTPUT)
(3, 9)
6/28/2020 JANAK SINGH SAUD 2
Click here
f(x) = x2
Every input as a
unique output

-3
3
-2
2
-1
1
4
9
4
1
16
A B
f
A = {-3, - 2, - 1 ,1, 2, 3,4 }
f(x) = x2
B = {9, 4 ,1, 16 }
Click here
INPUT
OUTPUT

In arrow diagram
In ordered pair form
f = { (- 3, 9), (- 2, 4), (- 1, 1), (1, 1), (2, 4), (3, 9), (4, 16) }

Find out again by looking at the matching diagram below and discussing
the answers to the questions asked in the group.
2
3
4
5
6
7
A B
R1
1
2
3
4
2
3
4
5
A B
R2
Does each member of set
A have a relation with
only one member of set
B? Is there no reflection
of any member of A to B?
1
2
3
4
2
3
4
5
6
A B
Does each member of
set A have a relation
with only one member
of set B? Are there any
members in set B
without relationship ?
R3
-1
1
-2
2
1
4
5
A B
R4
Does each member of
with only one
member of set B
Does each member of
with only one
member of set B

2
3
4
5
6
7
A B
R1
1
2
3
4
2
3
4
5
A B
R2
1
2
3
4
2
3
4
5
6
A B
R3
-1
1
-2
2
1
4
5
A B
R4
In R1 , R3and R4 of the above relations, each member of group A has relation
with only one member of group B and there is no any member in group B
without relation. These relations are functions.

Let us consider a following example
Person
A

Let A and B are two no-empty sets.
A relation from set A to a set B is said to
be a function if every elements of set A
has one and only one image in set B
Conclusion.:
A function from set A to set B is denoted as f : A → B, g : A → B, h : A → B, etc.

Domain, Co-domain and range of a function
• https://www.geogebra.org/m/mzxvxe44
INPUT Num.
OUTPUT Num.
(INPUT, OUTPUT)
(3, 9), (4, 16)
Pre- image
image
Click here
f(x) = x2
Every input as a
unique output

-3
3
-2
2
-1
1
4
9
4
1
16
A B
f In arrow diagramA = {-3, - 2, - 1 ,1, 2, 3,4 }
f(x) = x2
B = {9, 4 ,1, 16 }
Click here
Pre-image
Image

-3
3
-2
2
-1
1
4
9
4
1
16
2
A
B
fIn arrow diagram
In ordered pair form
f = { (- 3, 9), (- 2, 4), (- 1, 1), (1, 1), (2, 4), (3, 9), (4, 16) }
Domain
9
4
1
16
Range
Co-domain
f(- 3) = 9
f( 3) = 9
f(- 2) = 4
f (2) = 4
f( - 1) = 1
f( 1)= 1
f(x) = x2

A f
Domain
Range
Co-domain
f(- 3) = 9
f( 3) = 9
f(- 2) = 4
f (2) = 4
f( - 1) = 1
f( 1)= 1
f( 4)= 16
f(x) = x2
9 is image of – 3 & 3
16 is image of & 4
- 3 & 3 are pre-images of 9
- 2 & 2 are pre images of 4
- 1 & 1 are pre-images of 1
4 is pre-image of 16
Range is subset of co-domain

• Set of pre-image is domain
• Set of image of the function is Range
A f
Domain
Range
Co-domain
• If function f :A→B , then the set
A is DOMAIN of f and
• set B is CO-DOMAIN of f
• Set of elements of B which are
the images of elements of A is
RANGE

A f
f(x)
A
f
x
Functional value

Range
Domain
Function (f)
Another
function
Domain
Range
https://tasyaluftwascher.blogspot.com/2017/01/newest-for-images-
drawing-cute-water.htmlSource

EXAMPLE 1: In the give arrow diagram check whether f and g are function
or not. If function, find the domain range and co-domain of the function
• In first arrow diagram, f is a function because every element of set A has unique image in set B
Thus, domain = {1, 2, 3}, range = {1, 2, 3} and co-domain = {1, 2, 3, 4, 5}
1
2
3
1
2
3
4
5
A f
1
2
3
1
2
3
4
6
A g
In second arrow diagram, g is not a function because element 1 and 2 of set A have more than one
images in set B.

EXAMPLE 2 :Identify whether {(1, 2), (3, 6), (- 2, - 4)} is a function or not ?
• Since, every element of set A has only
one image in set B.
• Hence it is a function
Representing in arrow diagram
1
3
-2
2
6
- 4
A

Representation of the function
a) A table
b) A set of ordered pair form
c) An arrow diagram
d) A graph
e) A formula ( or an equation)

EXAMPLE 3: Let A = {2, 3, 4} and B= {4, 6, 8} such that f(2 ) = 4, f(3) = 6
and f(4) = 8. Represent the function f by all methods
• Here, A ={2, 3, 4} and B= {4, 6, 8}
a) In a tabular form
f(2)= 4 i.e. when x = 2, then y = 4
f(3)= 6 i.e. when x = 3 then y = 6
f(4) = 8 i.e. when x = 4 then y = 8
X
Y
3
6
2
4
4
8
(b) A set of ordered pair form:
∴ f = {(2, 4), (3, 6), (4, 8)}

(c) In an arrow diagram: f(2 ) = 4, f(3) = 6 and f(4) = 8.
2 corresponds with f(2) = 4
2
3
4
4
6
8
A B

e) In an equation:
When x= 2, then y = f(2) = 4 = 2 x 2
when x = 3, then y = f(3) = 6 = 2 x 3
when x = 4, then y = f(4) = 8 = 2 x 4
Hence, the function f can be expressed as y = 2x
or, f(x) = 2x
d) In a graph :
The points with ordered pairs: (2, 4),
(3, 6), (4, 8)

EXAMPLE 4: What will be the functional value
of 4 in the function f(x) = x2 ?
Here, f(x) = x2
f(4) = ?
f(4) = (4)2 = 16
Functional value means we
need to simply put the value of
x in the given function

EXAMPLE 5:If f(x) = {(x, y): y = 2x + 3} and x 𝜖 {2, 4, 6}, find the
range and represent the function in arrow diagram.
Here, f(x) = 3x – 2 and image of f(x) = 10
So, f(x) = 10
or, 3x – 2 = 10
or, 3x = 10 + 2
or, 3x = 12
or,
3𝑥
3
=
12
3
or, x = 4
Hence, the element in domain is 4 has image 10

EXAMPLE 6:What element in the domain has the
image 10 under the function f(x) = 3x – 2 ?
Here, f(x) = 3x – 2 and image of f(x) = 10
So, f(x) = 10
or, 3x – 2 = 10
or, 3x = 10 + 2
or, 3x = 12
or,
3𝑥
3
=
12
3
or, x = 4
Hence, the element in domain is 4 has image 10

EXAMPLE 7:If the range of the function f(x) = 3x – 4 is {- 1, 5}, find its
domain. Also represent the function in arrow diagram.
Here, f(x) = 3x – 4 and Range = {- 1,
5}
So, y = f(x) = 3x - 4
When, y = - 1
or, 3x – 4 = - 1
or, 3x = - 1 + 4
or, 3x = 3
or x = 1
Again when , y = 5
or, 3x – 4 = 5
or, 3x = 5 + 4
or, 3x = 9
or, x = 3
Hence, Domain = {1, 3}
Representing the function f in arrow
diagram
- 1
- 5
1
- 3
A B
f

PRACTICE QUESTIONS

1. Which of the following are the function of not?
Give reason.

2. Represent the following relations in arrow diagram. Identify whether they are
function of not .Find the domain, range and co-domain of the functions

3. Find the image of the functions
a) Find the image of 2 in the function f(x) = 5x
b) Find the image of - 2 in the function f(x) = - 6x
c) Find the image of 10 in the function f(x) =
𝑥+2
3
d) Find the image of 3 in the function g(x) = 4x - 5

4. Find range:
a) f(x) = 2x2- 3; Domain = {- 1, 0 , 1, 2}. Find its range and represent the
function in graph.
b) f(x) = 2x – 3; Domain= {4, 5, 6}. Find its range and represent the
function
c) If f(x) = 4x + 5 and domain D = {1, 2, 3}, find the range and list the
set of ordered pairs of f.
d) If f(x) = {(x, y): y = 2x + 3} and x 𝜖 {2, 4, 6}, find the range and
represent the function in arrow diagram.

4 (d). If f(x) = {(x, y): y = 2x + 3} and x 𝜖 {2, 4, 6}, find the range and represent the
function in arrow diagram.
• f(x) = {(x, y) y = 2x + 3, domain = {2, 4, 6}
When x = 2, y = 2x2 + 3 = 7
When x = 4, then y = 2x 4 + 3 = 11
When, x = 6, y = 2. 6 + 3 = 15
Hence, range = {7, 11, 15}
or, y ∈ {7, 11, 15}
Now, representing the function f in arrow diagram.
2
4
6
7
11
15
A B
f

5. Find the pre-image:
a) Find the pre-image of 16 of the function f(x) = 4x
b) Find the pre image of 5 in the function g(x) = 5x .
c) Find the pre- image of 8 in the function f(x) = 2x + 6.
d) What will be the pre- image of 4 in the function f(x) = 𝑥 ?
e) What element in the domain has image 7 under the function f(x) =
2𝑥 − 3
5
?
f) What will be the pre image of 8 in f(x) = 3
𝑥?

6. Find the domain:
a) If the range of the function f(x) = 4x – 5 is {- 1, 7}, find its domain.
Also represent the function in arrow diagram.
b) If range of a function f:x→ 3x + 5 is {2, 8}, find its domain . Also
represent in the graph .
c) If range R = {1, 7, 11}, find the domain in the function f :x → 4x – 5
d) Find the domain of the function f(x) = 4x + 2, if range R = {2, 6}. Also
represent the function in table.

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Function : Introduction

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Function : Introduction