Question

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

(a) (−3, − 2) ∪ (2, 3)
(b) [−3, − 2) ∪ [2, 3)
(c) [−3, − 2] ∪ [2, 3]
(d) None of these

Answer 1

(c) [−3, − 2] ∪ [2, 3]

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