Function:

Let A and B be any two non-empty sets. If to each element a (pre-image) belongs to A (domain), there is assigned , in some manner or the other, a unique element b (image) belongs to  B (codmain), then such an assignment f is called a mapping or function for A into B and we write f : A-> B.

Relation

Let A and B be two non-empty sets. Then the relation R from a set A to the set B is a Subset of A x B.
If (a,b) E R b .
functions-Relation-Domain-Range-Image-Preimage


Domain:

When R is a relation from A to B, then the set of first element in R is called the domain of the function.

Symbolically,   Domain of R = {a : (a,b) E R}.

Codomain:

The set B in the given figure is called Co-domain of the function.

Range

When R is a relation from A to B, then the set of second element in R is called the range of the function.
Symbolically,   Range of R = {b : (a,b) E R}.

Image:

 The element b = f(a) E B is called the image of a under f.

Pre-image:

The element a of A is called pre-image (or inverse image) under f.