Codomain

If $f:X\to Y$ is a function, then the codomain of $f$ is the set $Y$.

The codomain is not determined solely by the rule $x\mapsto f(x)$; it is specified as part of the function’s type. Many notions (notably surjectivity) depend on the codomain, not just on the actual outputs attained.

Examples:

• If $f:\mathbb{R}\to\mathbb{R}$ is given by $f(x)=x^2$, then the codomain is $\mathbb{R}$ even though $f(x)\ge 0$ always.
• If the same rule is viewed as $f:\mathbb{R}\to[0,\infty)$, then the codomain is $[0,\infty)$.
• The codomain of $\sin:\mathbb{R}\to\mathbb{R}$ is $\mathbb{R}$ (even though the range is contained in $[-1,1]$).