Codomain (Function)

math

Definition

Codomain (Function)

The codomain of a function $f:D\to C$ is the set $C$ of permitted output values declared as part of the function:

$$\mathrm{cod}(f)=C,f(D)=\{f(x)\mid x\in D\}\subseteq C.$$

Unlike the image, the codomain may contain elements that the function never attains.

Example

Squaring

For $f:\mathbb{R}\to \mathbb{R}$ with $f(x)=x^2$, the codomain is $\mathbb{R}$, while the image is $[0,\infty )$. Changing the codomain to $[0,\infty )$ makes the same rule surjective.