What is function?

A relation f between two non-empty sets A and B is called a function from A to B if, for each a ∈ A there exists only one b ∈B such that (a,b) ∈ f .That is, f ={(a,b)| for all a ∈ B, b∈B }. A function is also called as a mapping or transformation

• A function is also called as a mapping or transformation
• The set A is called the domain of the function f and the set B is called its co-domain.
• If f (a) = b, then b is called ‘image’ of a under f and a is called a ‘pre-image’ of b.
• The set of all images of the elements of A under f is called the ‘range’ of f.
• Every function is a relation. Thus, functions are subsets of relationsand relations are subsets of cartesian product
• The range of a function is a subset of its co-domain.

Example 

 Let X = {1,2, 3, 4} and Y = {2, 4,6, 8,10} and R = {(1,2),(2,4),(3,6),(4,8)}.

Show that R is a function and find its domain, co-domain and range?

Solution: From the diagram, we see that for each x∈ X , there exists only one y ∈Y . 

Thus all elements in X have only one image in Y. Therefore R is a function.

Domain X = {1,2,3,4}; Co-domain Y = {2,3,6,8,10}; Range of f = {2,4,6,8}.