Question

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

 

विकल्प

• (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]

\[f\left( x \right) = \sqrt{5 \left| x \right| - x^2 - 6}\]

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