Question

The domain of the function $f\left(x\right)=\sqrt{5|x|-x^2-6}$ is

• A. $(-3,-2)\cup (2,3)$
• B. $(-3,-2)\cup (2,3)$
• C. $[-3,-2]\cup [2,3]$
• D. None of these

Answer 1

The correct option is C

$[-3,-2]\cup [2,3]$

$f\left(x\right)=\sqrt{5|x|-x^2-6}$
For f(x) to be defined, $5|x|-x^2-6\ge 0$
$\Rightarrow 5|x|-x^2-6\ge 0\Rightarrow x^2-5|x|+6\le 0$
For $x>0,|x|=x$
$\Rightarrow x^2-5x+6\le 0\Rightarrow (x-2)(x-3)\le 0\Rightarrow ϵ[2,3]\dots \dots \left(1\right)$
For x < 0, |x| = - x
$\Rightarrow x^2+5x+6\le 0\Rightarrow (x+2)(x+3)\le 0\Rightarrow xϵ[-3,-2]\dots \dots \left(2\right)$
From (1) and (2),
$xϵ[-3,-2]\cup [2,3]$
or $dom\left(f\right)=[-3,-2]\cup [2,3]$