# Question 9738d

Jan 10, 2017

The domain is x in ] 10 ,12 [ 

#### Explanation:

What is under the sqrt is $\ge 0$

But the sqrt is in the denominator

So,

$\sqrt{\left(x - 10\right) \left(12 - x\right)} > 0$

Let $f \left(x\right) = \sqrt{\left(x - 10\right) \left(12 - x\right)}$

Construct 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 a}$$10$$\textcolor{w h i t e}{a a a a a a a}$$12$$\textcolor{w h i t e}{a a a a}$$- \infty$

$\textcolor{w h i t e}{a a a a}$$x - 10$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$color(red)(∣∣)$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$color(red)(∣∣)$\textcolor{w h i t e}{a a}$$+$

$\textcolor{w h i t e}{a a a a}$$12 - x$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$color(red)(∣∣)$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$color(red)(∣∣)$\textcolor{w h i t e}{a a}$$-$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$color(red)(∣∣)$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$color(red)(∣∣)$\textcolor{w h i t e}{a a}$$-$

Therefore,

$f \left(x\right) > 0$, when x in ] 10 ,12 [ #