# How do you find the domain and range of sqrt [x/(x-6)]?

Apr 12, 2017

The domain is $x \in \left(- \infty , 0\right] \cup \left(6 , + \infty\right)$
The range is $\left[0 , 1\right) \cup \left(1 , + \infty\right)$

#### Explanation:

To find the domain, what's under the square root sign is $\ge 0$

So,

$\frac{x}{x - 6} \ge 0$

Let $p \left(x\right) = \frac{x}{x - 6}$

We need to build a sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a a a a a}$$6$$\textcolor{w h i t e}{a a a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 6$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$p \left(x\right)$$\textcolor{w h i t e}{a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$

Therefore,

$p \left(x\right) \ge 0$ when $x \in \left(- \infty , 0\right] \cup \left(6 , + \infty\right)$

The domain is $x \in \left(- \infty , 0\right] \cup \left(6 , + \infty\right)$

${\lim}_{x \to \pm \infty} p \left(x\right) = {\lim}_{x \to \pm \infty} \frac{x}{x} = 1$

When $x = 0$, $p \left(x\right) = 0$

${\lim}_{x \to {6}^{+}} p \left(x\right) = + \infty$

So,

the range is $\left[0 , 1\right) \cup \left(1 , + \infty\right)$

graph{sqrt(x/(x-6)) [-25.67, 25.65, -12.83, 12.84]}