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Question
The domain of the function \[f\left( x \right) = \sqrt{5 \left| x \right| - x^2 - 6}\] is
Options
(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]
\[\text{ For f(x) to be defined,} 5\left| x \right| - x^2 - 6 \geq 0\]
\[ \Rightarrow 5\left| x \right| - x^2 - 6 \geq 0\]
\[ \Rightarrow x {}^2 - 5\left| x \right| + 6 \leq 0\]
\[\text{ For } x > 0, \left| x \right| = x\]
\[ \Rightarrow x {}^2 - 5x + 6 \leq 0\]
\[ \Rightarrow (x - 2)(x - 3) \leq 0\]
\[ \Rightarrow x \in [2, 3] . . . . . . . . (1)\]
\[\text{ [For } x < 0, \left| x \right| = - x\]
\[ \Rightarrow x {}^2 + 5x + 6 \leq 0\]
\[ \Rightarrow (x + 2)(x + 3) \leq 0\]
\[ \Rightarrow x \in [ - 3, - 2] . . . . . . . (2)\]
\[\text{ From (1) and (2) } , \]
\[x \in [ - 3, - 2] \cup [2, 3] \]
\[\text{ or, dom } (f) = [ - 3, - 2] \cup [2, 3]\]
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