Image of a function

Let $f\colon A\to B$ be a function and let $S\subseteq A$ be a subset . The image of $S$ under $f$ is

In particular, the image (range) of $f$ is $f(A)\subseteq B$, a subset of the codomain .

Images interact well with unions and containments, and they are paired with preimages via inverse image operations.

Examples:

• If $f\colon\mathbb{Z}\to\mathbb{Z}$ is $f(n)=2n$, then $f(\mathbb{Z})$ is the set of even integers.
• If $f(x)=x^2$ on $\mathbb{R}$ and $S=[-2,1]$, then $f(S)=[0,4]$.