Find the Domain of the Function
The domain of the function
\[ f(x)=\sqrt{5|x|-x^2-6} \]
is
(a) \(( -3,-2)\cup(2,3)\)
(b) \([ -3,-2]\cup[2,3)\)
(c) \([ -3,-2]\cup[2,3]\)
(d) none of these
For square root function,
\[ 5|x|-x^2-6\ge0 \]
Case I: \(x\ge0\)
\[ 5x-x^2-6\ge0 \]
\[ x^2-5x+6\le0 \]
\[ (x-2)(x-3)\le0 \]
\[ 2\le x\le3 \]
Case II: \(x<0\)
\[ -5x-x^2-6\ge0 \]
\[ x^2+5x+6\le0 \]
\[ (x+2)(x+3)\le0 \]
\[ -3\le x\le-2 \]
Therefore,
\[ \boxed{[-3,-2]\cup[2,3]} \]
\[ \boxed{\text{Correct Answer: (c)}} \]