Find the Domain of \( f(x)=\sqrt{2-2x-x^2} \)
Question:
The domain of the function
\[ f(x)=\sqrt{2-2x-x^2} \]
is
(a) \([-\sqrt3,\sqrt3]\)
(b) \([-1-\sqrt3,-1+\sqrt3]\)
(c) \([-2,2]\)
(d) \([-2-\sqrt3,-2+\sqrt3]\)
Solution:
For square root function,
\[ 2-2x-x^2\ge0 \]
\[ x^2+2x-2\le0 \]
Solving,
\[ x=\frac{-2\pm\sqrt{4+8}}{2} \]
\[ x=-1\pm\sqrt3 \]
Since quadratic expression is \(\le0\),
\[ -1-\sqrt3\le x\le -1+\sqrt3 \]
Therefore,
\[ \boxed{[-1-\sqrt3,\,-1+\sqrt3]} \]
\[ \boxed{\text{Correct Answer: (b)}} \]