Find the Domain of the Function
The domain of definition of
\[ f(x)=\sqrt{\frac{x+3}{(2-x)(x-5)}} \]
is
(a) \((-\infty,-3]\cup(2,5)\)
(b) \((-\infty,-3)\cup(2,5)\)
(c) \((-\infty,-3]\cup[2,5]\)
(d) none of these
For square root function,
\[ \frac{x+3}{(2-x)(x-5)}\ge0 \]
Critical points are
\[ x=-3,\;2,\;5 \]
Using sign analysis,
\[ \frac{x+3}{(2-x)(x-5)}\ge0 \]
for
\[ (-\infty,-3]\cup(2,5) \]
Note that \(x=2\) and \(x=5\) are excluded because denominator becomes zero.
Therefore,
\[ \boxed{(-\infty,-3]\cup(2,5)} \]
\[ \boxed{\text{Correct Answer: (a)}} \]