If $f : A \to B$ and if $A_0$ is a subset of $A$, we define the restriction of $f$ to $A_0$ to be the function mapping $A_0$ into $B$ whose rule is $$\{(a,f(a)) | a \in A_0\}$$ It is denoted by $f|A_0$, which is read "$f$ restricted to $A_0$".
A function $f : A \to B$ is said to be injective if for each pair of distinct points of $A$, their images under $f$ are distinct. It is said to be surjective if every element of $B$ is the image of some element of $A$ under the function $f$. If $f$ is both injective and surjective, it is said to be bijective. To remember this, just picture injecting someone with fluid. You don't expect to hit everything; but you don't want to hit two veins twice.