# 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

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

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

Now for the range, we have

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

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

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

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

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

Now for the range, we have

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

Thus, the range of the function is $f = R - \left\{ 0 \right\}$