Definition of range of a function

Definition (Range of a function).

Let \(\mathrm{f} : A \to B\) be a function from \(A\) to \(B\). The set \(R_{\mathrm{f}}\) of all possible images \(\mathrm{f}(x)\) as \(x \in A\) varies over all of the set \(A\) is called the range of \(\mathrm{f}\), i.e.,

\[R_{\mathrm{f}} = \{\mathrm{f}(x) : x \in A\}.\]

In terms of the graphical representation of the function \(\mathrm{f}\), to obtain the range of \(\mathrm{f}\), we imagine horizontal rays emitting from the points on the graph of \(y = \mathrm{f}(x)\), which then light up portions (or all of) the \(y\)-axis.