# Codomain

A codomain in mathematics is the set of "output" values associated with (or mapped to) the domain of "input" arguments in a function. For any given function $\displaystyle f\colon A\rightarrow B$ , the set A, on which f is defined (the arguments), is called the domain, and B (the set of possible values) is called the codomain of f. The set of all actual values $\displaystyle \lbrace f(x) | x \in A \rbrace$ or f(A) is called the range of f.

Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values. The codomain is not to be confused with the range, which is in general only a subset of B; in lower-level mathematics education, however, range is often taught as being equivalent to codomain.

## Example

Let the function f be a function on the real numbers:

$\displaystyle f\colon \mathbb{R}\rightarrow\mathbb{R}$

defined by

$\displaystyle f\colon\,x\mapsto x^2.$

The codomain of f is R, but clearly f(x) never takes negative values, and thus the range is in fact the set R+—non-negative reals, i.e. the interval [0,∞):

$\displaystyle 0\leq f(x)<\infty.$

One could have defined the function g thus:

$\displaystyle g\colon\mathbb{R}\rightarrow\mathbb{R}^+_0$
$\displaystyle g\colon\,x\mapsto x^2.$

While f and g have the same effect on a given number, they are not, in the modern view, the same function since they have different codomains.

The codomain can affect whether or not the function is a surjection; in our example, g is a surjection while f is not. The codomain does not affect whether or not the function is an injection.