# How do you find the domain of the function f(x)=sqrt(x^3-64 x) ?

May 19, 2017

The domain of $f \left(x\right)$ is $x \in \left[- 8 , 0\right] \cup \left[8 , + \infty\right)$

#### Explanation:

What's under the square root sign is $\ge 0$

So,

${x}^{3} - 64 x \ge 0$

${x}^{3} - 64 x = x \left({x}^{2} - 64\right) = x \left(x + 8\right) \left(x - 8\right)$

Let $g \left(x\right) = x \left(x + 8\right) \left(x - 8\right) \ge 0$

We 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}$$- 8$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a a}$$+ 8$$\textcolor{w h i t e}{a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 8$$\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}$$+$$\textcolor{w h i t e}{a a a a a}$$+$

$\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 a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 8$$\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}$$-$$\textcolor{w h i t e}{a a a a a}$$+$

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

Therefore,

$g \left(x\right) \ge 0$ when $x \in \left[- 8 , 0\right] \cup \left[8 , + \infty\right)$ graph{sqrt(x^3-64x) [-27.54, 30.2, -4.14, 24.74]}