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Question
The domain of the function
Options
(a) \[\left[ - \sqrt{3}, \sqrt{3} \right]\]
(b) \[\left[ - 1 - \sqrt{3}, - 1 + \sqrt{3} \right]\]
(c) [−2, 2]
(d) \[\left[ - 2 - \sqrt{3}, - 2 + \sqrt{3} \right]\]
Solution
(b) \[\left[ - 1 - \sqrt{3}, - 1 + \sqrt{3} \right]\]
\[ x^2 + 2x - 2 \leq 0\]
\[ \Rightarrow x^2 - 2x - 2 + 1 - 1 \leq 0\]
\[ \Rightarrow \left( x - 1 \right)^2 - \left( \sqrt{3} \right)^2 \leq 0\]
\[ \Rightarrow \left[ x - \left( - 1 - \sqrt{3} \right) \right]\left[ x - \left( - 1 + \sqrt{3} \right) \right] \leq 0\]
\[ \Rightarrow \left( - 1 - \sqrt{3} \right) \leq x \leq \left( - 1 + \sqrt{3} \right)\]
\[\text{ Thus, dom} (f) = \left[ - 1 - \sqrt{3}, - 1 + \sqrt{3} \right] . \]
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