Domain, codomain and image

Three sets are associated with every function, and it is important not to confuse them.

Definition — Domain, codomain, image

• The domain of a function $f:A\to B$ is the set $A$ of the elements on which $f$ is defined.
• The codomain is the set $B$, that is, the “destination” environment.
• The image (or image set) is $\mathrm{Im}(f) = \{f(x) : x\in A\}\subseteq B$, that is, the values that the function actually takes.

Careful — Codomain and image do not coincide

The codomain and the image do not coincide in general. The codomain is the container set, which we declare in advance; the image is the set of values that are reached. For example $f:\mathbb{R}\to\mathbb{R}$, $f(x)=x^2$, has codomain $\mathbb{R}$ but image $[0,+\infty)$, because squares are non-negative.

Links

Topics: Functions and properties
Concepts: Codomain · Domain · Image
Functions: Parabola