Restriction of a function

Definition

Let $f \colon A \to B$ be a function and let $S$ be a subset of $A$. The restriction of $f$ to $S$ is the function $f|_S \colon S \to B$ with $f|_S(s) = f(s)$ for each $s \in S$.

Remarks

The graph of $f|_S$ is $\{(s, f(s)) \mid s \in S\}$, a subset of the graph of $f$. The function $f$ is an extension of $f|_S$. Example: $x \mapsto x^2$ on $\mathbb{R}$ is not injective, but its restriction to $[0, \infty)$ is injective.

Explanations

L1 human

To restrict a function, you keep its rule and allow fewer inputs. Restrictions often make a function injective, so that it has an inverse on a smaller domain.