# What is a function in Math?

Function is a special type of “Relation”.

Let’s revise the basics of relation.

Relation is a form of association between two or more sets.

All the input data of relation is called Domain.
All the output data of relation is called Range.

For Example;
R = { (2, 5), (3, 7), (5, 7), (4, 2) }

The above set R is a form of relation.

## Function Definition

Function is a special kind of relation where it contains ordered pairs of x and y.
However, the unique point is that, in function, for every value of domain, there is only one y value.

Let us understand the concept of functions with examples;

Example 01
Consider the below relation.
Check if it is a function or not.

The above relation is a function because;

⟹ All the values of domain are paired with some output.
⟹ Each value of domain has one output.

Example 02
Check if the below relation is function or not.

Solution
The above relation is not a function because;

⟹ All the values of domain is not covered.
Here number 5 in the domain do not have any output.

Conclusion
In a function, all the domain must be paired with a output number.

Example 03
Check if the below relation is function or not.

The above relation is not a function because number 7 of domain is paired with multiple numbers like -4 & 2.

In a function, each domain value has one output only.

Conclusion
In a domain if an input has more than one output then the relation is not a function

Example 04
Check if the below relation is function or not.

Solution
The above relation is not a function.

Here the domain 2 is paired with multiple numbers.

Hence, number 2 results in multiple output of 7, 8 and 9

Conclusion
In a function every input has only one output.
It means the domain number can be linked to only one number in range.

Example 05
Check if the below relation is function or not.

The above relation is a function because;

⟹ All the numbers in domain is linked to single number.

⟹ In function, multiple domain can link to single output.
Hence, the linking of number 6 & 7 in domain to number 4 is not a problem for a function.

### How to represent a function?

Function are represented in similar way as Relation.

Let A & B are the two set and f is a function from A to B, we can technically represent function as;

It means that in a function when you input values from set A, you will get numbers from set B as an output.

#### Image of Function

The set of all possible output the function can produce is called image of function.

Suppose A & B are the given sets and x and f(x) are the input and output of the function respectively.

⟹ x is Input value of function.
Where x belongs to set A (i.e. x ϵ A )

⟹ f(x) is output value of function.
Where f(x) belongs to set B (i.e. f(x) ϵ B )

All the possible values of output f(x) is called image of function f.

## Solved Problems – Relations

(01) Check if the given relation is a function or not.

Solution
The above relation is not a function because all the domain inputs are not linked to the output.

Here the input value ” c ” is not linked to any other output.

(02) Check if the below relation is a function?

Solution
The above relation is a function because;

⟹ All the domain values are linked to output value.
⟹ Each domain value is linked to only one output.

(03) If x, y ϵ { 1, 2, 3, 4 }, check if the below expression is function or not.

f = { (x, y) : y = x + 1 }

Solution
Its given that x & y are elements which belong to set { 1, 2, 3, 4 }.
i.e. x, y ϵ { 1, 2, 3, 4 }

Putting the value of x in function f = { (x, y) : y = x + 1 }

Putting x = 1
Calculate y = x + 1
y = 1 + 1
y = 2
Hence we get ordered pair (1, 2)

Put x = 2
y = 2 + 1 = 3
Ordered Pair (2, 3)

Put x = 3
y = 3 +1 = 4
Ordered pair (3, 4)

Put x = 4
y = 4 +1 = 5
value of y = 5 is not possible since y ϵ { 1, 2, 3, 4 }

Combining all the ordered pair we get;
f = { (1, 2), (2, 3), (3, 4) }

The f is not a function because it do not link all the value of domain x = {1, 2, 3, 4 } to the output.

Below is the arrow diagram for function f.

(04) Check of the below relation is function or not.

f = { (x, y) : y = 3x, x ϵ {1, 2, 3}, y ϵ {3, 6, 9, 12} }

Solution
Let’s find the ordered pairs

Put x = 1
calculate y = 3x
y = 3. 1
y = 3
Ordered pair is (1, 3)

Put x =2
we get y = 3 . 2
y = 6
Ordered Pairs is (2, 6)

Put x = 3
y = 3 . 3
y = 9
Ordered pair is (3, 9)

Combining all the ordered pair we get;
f = { (1, 3), (2, 6), (3, 9) }

The above relation is a function because;
⟹ All the domain value (x) has been linked
⟹ All the domain value have only one output value.

Below is the arrow diagram of above function.