Examples of Functions

Example:
Find the range of the function $f\left( {\text{x}} \right) = \frac{{{\text{x}} + 1}}{{{\text{x}} - 1}}$.

Solution:
We have

$$f\left( {\text{x}} \right) = \frac{{{\text{x}} + 1}}{{{\text{x}} - 1}}$$

Put ${\text{x}} = 1$

$$f\left( {\text{1}} \right) = \frac{{{\text{1}} + 1}}{0} = \infty$$

Thus, the domain is $\forall \;{\text{x}} \in \mathbb{R} - \left\{ 1 \right\}.$

Now for the range, we have

$$\begin{gathered} f\left( {\text{x}} \right) = \frac{{{\text{x}} + 1}}{{{\text{x}} - 1}} \\ \therefore {\text{y}} = \frac{{{\text{x}} + 1}}{{{\text{x}} - 1}} \\ \Rightarrow {\text{xy}} - {\text{y}} = {\text{x}} + 1 \\ \Rightarrow {\text{xy}} - {\text{x}} = {\text{y}} + 1 \\ {\text{x}} = \frac{{{\text{y}} + 1}}{{{\text{y}} - 1}} \\ \end{gathered}$$

For ${\text{y}} = 1$

$${\text{x}} = \frac{{{\text{1}} + 1}}{0} = \infty$$

So, the range of the function $f$is $\left\{ {{\text{y}}:{\text{y}} \ne 1} \right\} = \left] { - \infty ,1} \right[\;\; \cup \;\;\left] {1,\infty } \right[$.

Example:
Let $f\left( {\text{x}} \right) = \frac{{\text{x}}}{{{{\text{x}}^2} - 16}}$. Find the domain and range of $f$.

Solution:
We have

$$f\left( {\text{x}} \right) = \frac{{\text{x}}}{{{{\text{x}}^2} - 16}}$$

For ${\text{x}} = 4$

$$f\left( 4 \right) = \frac{4}{{16 - 16}} = \infty$$

For ${\text{x}} = - 4$

$$f\left( { - 4} \right) = \frac{{ - 4}}{{16 - 16}} = \infty$$

Thus, the domain is $\forall \;{\text{x}} \in \mathbb{R} - \left\{ {4, - 4} \right\}$.

Now for the range, we have

$$\begin{gathered} f\left( {\text{x}} \right) = \frac{{\text{x}}}{{{{\text{x}}^2} - 16}} \\ \Rightarrow {\text{y}} = \frac{{\text{x}}}{{{{\text{x}}^2} - 16}} \\ \Rightarrow {\text{y}}\left( {{{\text{x}}^2} - 16} \right) = {\text{x}} \\ \Rightarrow {\text{y}}{{\text{x}}^2} - {\text{x}} - 16{\text{y}} = 0 \\ \Rightarrow {\text{x}} = \frac{{ - \left( { - 1} \right) \pm \sqrt {{{\left( { - 1} \right)}^2} - 4\left( {\text{y}} \right)\left( { - 16{\text{y}}} \right)} }}{{2\left( {\text{y}} \right)}} \\ \Rightarrow {\text{x}} = \frac{{1 \pm \sqrt {1 - 64{{\text{y}}^2}} }}{{2{\text{y}}}} \\ \end{gathered}$$

For ${\text{y}} = 0$

$${\text{x}} = \frac{{1 \pm \sqrt {1 + 0} }}{0} = \infty$$