hi Cartis,

I've made a drawing below which may make this clearer.

When a function is defined the definition should include saying what set the function may be applied to.  This set is called the domain.

It should also include a statement of what set the function maps to.  This second set is called the codomain.

But some elements of the codomain may not be mapped onto.  So the word range is used for the subset of the codomain that actually gets mapped to.

I'll make up an example.

Domain = {counting numbers} Codomain = {counting numbers}  Fuinction f(x) : x ---> 5x

For this function the Range = {5,10,15,20,25,.......}

Other members of the codomain such as 3, 17, 29 etc are not mapped to by f.

The definition of the domain is important. 

If I change the domain and codomain in the above example both to {real numbers} then every member of the codomain is mapped to (since we can always take an element, y,  of the codomaiin, divide it by 5, and that answer (y/5) is the member of the domain that maps onto y.

Thus {range} = {codomain}

[I have seen a number of questions posted where the student is asked to say what the domain is for a particular function.  I consider this unreasonable since the domain is part of the definition and many answers are possible and equally valid.] 

Bob

I am sorry,I meant dom(f(x)),domain of f(x),and ran(f(x)),range of f(x)

Can you define codomain of a function like dom(f),ran(f)?