ASL STEM

Definition

If f:D->Y is a map (a.k.a. function, transformation, etc.) over a domain D, then the range of f, also called the image of D under f, is defined as the set of all values that f can take as its argument varies over D, i.e., Range(f)=f(D)={f(X):X in D}. Note that among mathematicians, the word "image" is used more commonly than "range." The range is a subset of Y and does not have to be all of Y.

Source: http://mathworld.wolfram.com/Range.html

Example

The function f:R->R (where R is the set of all real numbers) defined as f(x)=x^2 has a range that is a subset of the codomain. The image/range of f is only the set of non-negative reals, which is a subset of R. Had the codomain been the set of all non-negative reals, then the image/range of f would be equal to the codomain.