Codomain

If $f\colon A\to B$ is a function , then the codomain of $f$ is the target set $B$ appearing in the declaration $f\colon A\to B$.

The codomain is part of the data of a function and should be distinguished from the image $f(A)\subseteq B$, which is determined by $f$.

Examples:

• If $f\colon\mathbb{R}\to\mathbb{R}$ is $f(x)=x^2$, then the codomain is $\mathbb{R}$ but the image is $[0,\infty)$.
• The same rule $x\mapsto x^2$ can be viewed as a function $g\colon\mathbb{R}\to[0,\infty)$ with codomain $[0,\infty)$, in which case $g$ is surjective .