Image - Knowlpedia

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

f(S)=\{f(s): s\in S\}\subseteq B.

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

Remarks

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 $f(S)=\{y\in\mathbb{R}:0\le y\le 4\}$ .