Extension

Suppose we have sets $X$ and $Y$, a subset $A$ of $X$, and a function $f: A → Y$ on $A$. Then an extension function $F: X → Y$ of $f$ is any function that satisfies $$F(x) = f(x)$$ for all $x∈A$.

In that case, the original function $f$ is the restriction $f=F|_A$ of $F$ to $A$.