**Function** is a **special type of “Relation”**.

We have already studied about the concept 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.