Function

Let $A$ and $B$ be any sets. A function from $A$ to $B$ is a rule $f$ that associates each element of $A$ with exactly one element of $B$.

If $f$ associates $x\in A$ with $y\in B$, then we write $\boxed{f(x)=y}$ and say ”$f$ of $x$ is $y$” or ”$f$ maps $x$ to $y$” or the “value of $f$ at $x$ is $y$”.

If $y$ is a function from $A$ to $B$, then we write: $\boxed{f:A\to B}$.

Function Rules

We call $A$ the domain of $f$ and $B$ the codomain of $f$.

• every element of the domain has to be mapped somewhere in the codomain (but this does not have to apply in reverse)
• one element cannot be mapped to 2 different places (but it can happen that $2$ different elements are mapped to the same place)