Restriction of a function

A restriction of a function is a new function obtained by limiting the domain: if $f:A\to B$ is a function and $S\subseteq A$, then the restriction $f|_S:S\to B$ is defined by

A restriction changes the domain while keeping the same codomain . The graph of $f|_S$ is obtained by intersecting the graph of $f$ with the subset $S\times B$ of the Cartesian product $A\times B$.

Examples:

• If $f:\mathbb{R}\to\mathbb{R}$ is $f(x)=x^2$ and $S=\mathbb{Z}\subseteq\mathbb{R}$, then $f|_S:\mathbb{Z}\to\mathbb{R}$ is the squaring function on integers.
• If $A$ is a set and $S\subseteq A$, then $\mathrm{id}_A|_S:S\to A$ is the inclusion map $s\mapsto s$.