#Function What is function? Domain, range, Co-domain, Pre-image, of the function

1. 6/28/2020 JANAK SINGH SAUD 1

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

3. -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
https://www.geogebra.org/m/mzxvxe44
B = {9, 4 ,1, 16 }
Click here
INPUT
OUTPUT
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4. In arrow diagram
In ordered pair form
f = { (- 3, 9), (- 2, 4), (- 1, 1), (1, 1), (2, 4), (3, 9), (4, 16) }
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5. 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
set A have a relation
with only one
member of set B
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Does each member of
set A have a relation
with only one
member of set B

6. 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.
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7. Let us consider a following example
Person
A
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8. 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.:
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A function from set A to set B is denoted as f : A → B, g : A → B, h : A → B, etc.

9. 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
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Click here
f(x) = x2
Every input as a
unique output

10. -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
https://www.geogebra.org/m/mzxvxe44
B = {9, 4 ,1, 16 }
Click here
Pre-image
Image
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11. -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
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f(- 3) = 9
f( 3) = 9
f(- 2) = 4
f (2) = 4
f( - 1) = 1
f( 1)= 1
f(x) = x2

12. 6/28/2020 JANAK SINGH SAUD 12
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
4 is image of – 2 & 2
1 is image of – 1 & 1
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

13. • Set of pre-image is domain
• Set of image of the function is Range
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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

14. 6/28/2020 JANAK SINGH SAUD 14
A f
f(x)
A
f
x
Functional value

15. 6/23/2020 JANAK SINGH SAUD 15
Range
Domain
Function (f)
Another
function
Domain
Range
https://tasyaluftwascher.blogspot.com/2017/01/newest-for-images-
drawing-cute-water.htmlSource

16. 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}
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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.

17. 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
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Representing in arrow diagram
1
3
-2
2
6
- 4
A

18. 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)
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19. 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
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• 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)}

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

21. 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
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d) In a graph :
The points with ordered pairs: (2, 4),
(3, 6), (4, 8)

22. 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
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Functional value means we
need to simply put the value of
x in the given function

23. 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
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24. 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
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25. 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
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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

26. PRACTICE QUESTIONS
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27. 1. Which of the following are the function of not?
Give reason.
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28. 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
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29. 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
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30. 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.
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31. 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.
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2
4
6
7
11
15
A B
f

32. 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?
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33. 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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