# What are the Functions?

After doing Relations in maths, the next topic under algebra is a function and its types. The function is a rule that connects every element of a non-empty set ‘A’ to a unique element of non-empty set ‘B’. Let’s understand this in more detail.

## The difference between relation and functions

I will explain this with an example. Set A and set B are shown below in an arrow diagram form. As you will observe, in the figure below. Set A elements; “a,b,c,d” are connected with Set B elements; “f,g,i,h” respectively. But element ‘e’ under set A is not connected with any element in set B. This arrow diagram is an example of relations.

Now, let’s take the same example, but this time element ‘e’ of set A is connected with element ‘j’ of set B. This is an example of functions. (Thinking why? look at the rules of functions below)

To become a function there are two rules to be followed.

## Conditions or the rule of functions

- Every element of set A must be connected with a unique element of set B. Look at, figure 2 above.
- Every element in Set A must be connected with 1 element only of set B. Lets take another example of functions as shown below:

## Domain, Range and Co-domain

Let’s assume A->B.

Function is defined as:

f(x)=*x*^{2}

Set A = {1, 2, 3, 4}

Set B = { 1, 2, 3, 4, 5, 9, 12, 16, 18}

Here x is 1, 2, 3 ,4 = Set A, this is domain.

*x*^{2} is what you map, 1, 4, 9, 16, this is range.

Set B is called co-domain.

So, range is always a subset of co-domain.

16 is an image of 4, 9 is an image of 3, 4 is an image of 2, and 1 is an image of 1. As shown below,

## How do you read a function?

If we have a function, ‘f’ that connects A to B. Then, the following is an example of a function with its symbols.