Restricted Functions

What is a restricted function?

It is sometimes useful to look at a function that focuses on a subset of the domain. Let’s consider $f: A\to B$, where the domain set is $A$ and $f$ is a function from $A$ to $B$. If we take $C$—a subset of $A$—then the restricted function $f{\mid}_C : C\to B$ provides a function from $C$ to $B$. We can recognize it as restricting $f$ to $C$.

Examples

Let’s assume the two sets, $A=\{1,2,3,4,5,6,7,8,9\}$ and $B=\{a,e,i,o,u\}$, with a function $f: A\to B$ as follows: