# How do you find the domain of f(x) = sqrt ( x- (3x^2))?

Mar 18, 2018

The domain of $f \left(x\right)$ is $x \in \left[0 , \frac{1}{3}\right]$

#### Explanation:

What's under the $\sqrt{}$ sign is $\ge 0$

Here,

$f \left(x\right) = \sqrt{x - 3 {x}^{2}}$

Therefore,

$x - 3 {x}^{2} \ge 0$

$x \left(1 - 3 x\right) \ge 0$

Let $g \left(x\right) = x \left(1 - 3 x\right)$

Build the 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 a a}$$0$$\textcolor{w h i t e}{a a a a a a a}$$\frac{1}{3}$$\textcolor{w h i t e}{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}$$0$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$1 - 3 x$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a}$color(white)(aaaa)+$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{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}$$0$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a a a}$$-$

Therefore,

$g \left(x\right) \le 0$, $\implies$, $x \in \left[0 , \frac{1}{3}\right]$

graph{sqrt(x-3x^2) [-0.589, 1.096, -0.307, 0.536]}