Image (range) - Imported Knowlpedia Content

Let $f:X\to Y$ be a function.

The image (or range) of $f$ is

f(X):=\{f(x):x\in X\}\subseteq Y.

More generally, for a subset $A\subseteq X$ , the image of $A$ under $f$ is

f(A):=\{f(a):a\in A\}\subseteq Y.

Remarks

The image captures the "actual outputs" of $f$ and is the natural codomain for which $f$ becomes surjective (if one replaces $Y$ by $f(X)$ ).

Examples

If $f:\mathbb{R}\to\mathbb{R}$ , $f(x)=x^2$ , then $f(\mathbb{R})=[0,\infty)$ .
If $f(x)=x^2$ and $A=[-1,2]$ , then $f(A)=[0,4]$ .
If $f:\mathbb{R}\to\mathbb{R}$ , $f(x)=e^x$ , then $f(\mathbb{R})=(0,\infty)$ .