Domain and range

The range is the set of all possible outputs (” $y$ “-values) of a function.

We denote the range of a function $f$ by $R_f$ .

Graphs are especially useful to determine the range of a function.
End points, turning points and asymptotes are important in determining the range.

Examples

Use the following to generate functions and observe how their range can be determined from the graph.

{f: x \mapsto - 3 - \frac{5}{x - 3}} \allowbreak \quad \allowbreak {\textrm{for } x \in \mathbb{R}, x \neq 3.}

\begin{aligned} D_f &= \left(-\infty, 3 \right) \cup \left(3, \infty\right). \\ R_f &= \left(-\infty, -3\right) \cup \left(-3, \infty\right). \end{aligned}