Definition of function

In mathematics, we like to restrict our attention to a special class of relations called functions.

Definition (Function).

Let \(A\) and \(B\) be given sets. We say that \(\mathrm{f}\) is a function from \(A\) to \(B\), denoted by \(\mathrm{f}:A \to B\), if it is a relation between \(A\) and \(B\) that satisfies the following two conditions:

1. For every element \(a \in A\) there is an element \(b \in B\) for which \((a,b) \in \mathrm{f}\). We use the symbol \(\mathrm{f}(a) = b\) to represent the situation that the element \(a\) is mapped to the element \(b\) by \(\mathrm{f}\). We also say that \(b\) is the image of \(a\) under \(\mathrm{f}\) if \(\mathrm{f}(a) = b\).
2. This element \(b \in B\) mentioned in (\(1\)) is unique, i.e., it is the only element such that \((a,b) \in \mathrm{f}\). This explains why \(\mathrm{f}(a) = b\) is a good notation. Sometimes, we also write \(\mathrm{f}:a \mapsto b\) in the case when \(\mathrm{f}(a) = b\).

In this case, the left-hand set \(A\) from which \(\mathrm{f}\) draws its inputs is called the domain of \(\mathrm{f}\), and we denote it by \(D_{\mathrm{f}}\).

In the IB HL Math syllabus, the right-hand set \(B\) does not have a name. However, mathematicians called this set \(B\) the codomain of \(\mathrm{f}\).