What is Function in Mathematics?

A function in mathematics is just like relation between input and output. So a function takes input and returns an output. It is denoted as f.  Most of the time, you see it in below form.

f(x) = x² + 1 

For a given value of x, you will get the result of  x² + 1.

Value of x	Output
0 1 2 3 4	1 2 5 10 17

We see that for a particular value of variable x, we get a corresponding output. This phenomena is called function in mathematics.

We shall denote function of x as f(x), F(x), φ(x) etc.

The number x that we use for the input of the function is called the 'Argument' of the function. It may seems to us that we can pick any number as argument  and for this reason we can call it 'Independent Variable' and the output of the function, e.g. f(x), f(3), etc. depends upon the argument, can be called the 'Dependent  variable'.
We can draw a picture of a function on graph by taking argument-value pairs of the function and describing each by a point in the plane, with x coordinate given by the argument and y coordinate given by the value for that pair.