Functions

So, what's a function?

Definition. [Function] A function is a relation from set $D$ to set $R$ that assigns to each element in $D$ exactly one element in $R$.

Example: Suppose that $3 \in D$. A function from $D$ to $R$ will contain one pair of the form $(3, y)$, where $y$ is exactly one element in $R$. For example, the function might have $(3, 2)$. But there won't be additional pairs of the form $(3, y)$ in the function in which $y \neq 2$. For example, a function can't have both $(3, 2)$ and $(3, 0)$ in it.

Definition. [Domain] The set $D$ of inputs into a function is called the domain: this is the set of first coordinates $x$ of the pairs $(x, y)$ in the function.

Definition. [Range] The set $R$ of outputs from a function is called the range: this is the set of second coordinates $y$ of the pairs $(x, y)$ in the function.