# How do you find the range of f(x)= x/absx?

Sep 21, 2015

Use $\left\mid x \right\mid = \left\{\begin{matrix}x & \mathmr{if} & x \ge 0 \\ - x & \mathmr{if} & x < 0\end{matrix}\right.$

#### Explanation:

$\left\mid x \right\mid = \left\{\begin{matrix}x & \mathmr{if} & x \ge 0 \\ - x & \mathmr{if} & x < 0\end{matrix}\right.$

So for $x > 0$, $f \left(x\right) = \frac{x}{x} = 1$

For $x = 0$, $f \left(x\right)$ is not defined#

For $x < 0$, $f \left(x\right) = \frac{x}{-} x = - 1$

We see that

$f \left(x\right) = \frac{x}{\left\mid x \right\mid} = \left\{\begin{matrix}1 & \mathmr{if} & x > 0 \\ - 1 & \mathmr{if} & x < 0\end{matrix}\right.$

The range of $f$ is $\left\{- 1 , 1\right\}$ (not the interval, just the set of those two numbers.)

Here is the graph. There should be open circles at $\left(0 , 1\right)$ and $\left(0 , - 1\right)$

graph{y=x/absx [-10, 10, -5, 5]}