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The domain of the function f(x) =

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, $\operatorname{dom}(f)=[-3,-2] \cup[2,3]$

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