# How do you find the domain and range of f(x) = 1/x?

Jun 9, 2018

Domain: $\left\{x | x \in \mathbb{R} : x \ne 0\right\}$
Domain interval notation $\left(- \infty , 0\right) \cup \left(0 , \infty\right)$

Range: $\left\{f \left(x\right) | f \left(x\right) \in \mathbb{R}\right\}$
Range interval notation $\left(- \infty , \infty\right)$

#### Explanation:

The only value in the domain excluded is when the function is undefined: $\frac{1}{0}$, that is when $x = 0$

Domain: $\left\{x | x \in \mathbb{R} : x \ne 0\right\}$ in interval notation $\left(- \infty , 0\right) \cup \left(0 , \infty\right)$

Now for the range, the function $f \left(x\right) = \frac{1}{x}$ is continuous across the domain above.

$x \to \pm \infty , f \left(x\right) \to 0$

$x \to {4}^{+} , f \left(x\right) \to \infty$

$x \to {4}^{-} , f \left(x\right) \to - \infty$

so the range is:

Range : $\left\{f \left(x\right) | f \left(x\right) \in \mathbb{R}\right\}$ or in interval notation $\left(- \infty , \infty\right)$

graph{1/x [-10, 10, -5, 5]}