Image

An image is the set of values a function takes on a subset of its domain: if $f:A\to B$ is a function and $S\subseteq A$, then

The image (often called the range) of $f$ is $f(A)$.

Images are built from subsets of the domain and are paired conceptually with preimages of subsets of the codomain. A function is surjective precisely when $f(A)$ equals its codomain .

Examples:

• For $f:\mathbb{Z}\to\mathbb{Z}$ given by $f(n)=2n$, the image $f(\mathbb{Z})$ is the set $\{2k: k\in\mathbb{Z}\}$ of even integers.
• For $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=x^2$ and $S=\{x\in\mathbb{R}:-1\le x\le 2\}$, the image is $f(S)=\{y\in\mathbb{R}:0\le y\le 4\}$.