Functions

A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.

A function which has either R or one of its subsets as its range is called a real valued function. Further, if its domain is also either R or a subset of R, it is called a real function.

In other words, a function f is a relation from a non-empty set A to a non-empty set B such that the domain of f is A and no two distinct ordered pairs in f  have the same first element. 
If f is a function from A to B and (a, b) ∈ f, then f (a) = b, where b is called the image of a under f and a is called the preimage of b under f.
The function f from A to B is denoted by f: A → B.
A funcion is a connection between 2 sets A and B f: A→B such that
1) All elements in A are associated to some element in B
2) This association is unique, that means one and only one.
Let's try to understand this with a simple anology,

Here, let's say `"X"_1` is a set of all children and `"X"_2` is a set of all womens. And `"X"_1` and `"X"_2` have connection as mother and children. 
So as per the definition there is a connention between 2 sets `"X"_1` and `"X"_2` such that all the elements of `"X"_1` are associated to some element in set `"X"_2` i.e all the childrens are related to a particualr mother, and this association is unique because no one child can have two or more mothers, but a mother can have more than one child. 

Consider the sets D and Y related to each ther as shown, clearly every element in the set D is related to exactly one element in the set Y. So the given relation is a function. f: D → Y.
Here, D is the domain of the function and Y is the co domain of the function.
f(1)= 5
Here, 5 is called the image of 1 under f and 1 is called preimage of 5 under f.
The range of this function is, Range= {2,3,5,7}
The range is a set of real numbers so the function is Real valued function.