Codomain

The term range is sometimes ambiguously used to refer to either the codomain or the image of a function.

[1] The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. The image of a function is a subset of its codomain so it might not coincide with it.

Namely, a function that is not surjective has elements y in its codomain for which the equation f(x) = y does not have a solution.

A codomain is not part of a function f if f is defined as just a graph.

[2][3] For example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G).

For this reason, it is possible that h, when composed with f, might receive an argument for which no output is defined – negative numbers are not elements of the domain of h, which is the square root function.

Some transformations may have image equal to the whole codomain (in this case the matrices with rank 2) but many do not, instead mapping into some smaller subspace (the matrices with rank 1 or 0).

Take for example the matrix T given by which represents a linear transformation that maps the point (x, y) to (x, x).

The point (2, 3) is not in the image of T, but is still in the codomain since linear transformations from

Examining the differences between the image and codomain can often be useful for discovering properties of the function in question.

For example, it can be concluded that T does not have full rank since its image is smaller than the whole codomain.