# How do you find the domain and range of g(x)=2/x?

Jun 10, 2018

Domain: $\left(- \infty , 0\right) \cup \left(0 , + \infty\right)$ Range: $\left(- \infty , + \infty\right)$

#### Explanation:

$g \left(x\right) = \frac{2}{x}$

$g \left(x\right)$ is defined $\forall x \ne 0$

Hence, the domain of $g \left(x\right)$ is : $\left(- \infty , 0\right) \cup \left(0 , + \infty\right)$

Now consider the limit of $g \left(x\right)$ as $x \to 0$ from below and from above.

${\lim}_{x \to {0}^{-}} \frac{2}{x} \to - \infty$

${\lim}_{x \to {0}^{+}} \frac{2}{x} \to + \infty$

Thus $g \left(x\right)$ has a vertical asymptote at $x = 0$.

The range of $g \left(x\right)$ is therefore $\left(- \infty , + \infty\right)$

We can visualise these results from the graph of $g \left(x\right)$ below.

graph{2/x [-10, 10, -5, 5]}