Question:
The domain of the function $f(x)=\sqrt{5|x|-x^{2}-6}$ is
(a) (−3, − 2) ∪ (2, 3)
(b) [−3, − 2) ∪ [2, 3)
(c) [−3, − 2] ∪ [2, 3]
(d) None of these
Solution:
(c) [−3, − 2] ∪ [2, 3]
$f(x)=\sqrt{5|x|-x^{2}-6}$
For $f(x)$ to be defined, $5|x|-x^{2}-6 \geq 0$
$\Rightarrow 5|x|-x^{2}-6 \geq 0$
$\Rightarrow x^{2}-5|x|+6 \leq 0$For $x>0,|x|=x$
$\Rightarrow x^{2}-5 x+6 \leq 0$
$\Rightarrow(x-2)(x-3) \leq 0$
$\Rightarrow x \in[2,3] \quad \ldots \ldots \ldots(1)$
For $x<0,|x|=-x$
$\Rightarrow x^{2}+5 x+6 \leq 0$
$\Rightarrow(x+2)(x+3) \leq 0$
$\Rightarrow x \in[-3,-2] \quad \ldots \ldots(2)$
From (1) and (2),
$x \in[-3,-2] \cup[2,3]$
or, $\operatorname{dom}(f)=[-3,-2] \cup[2,3]$