Question

The domain of  \[\cos^{- 1} \left( x^2 - 4 \right)\] is

 

Options

• [3, 5]

• [−1, 1]

•  \[\left[ - \sqrt{5}, - \sqrt{3} \right] \cup \left[ \sqrt{3}, \sqrt{5} \right]\]

•  \[\left[ - \sqrt{5}, - \sqrt{3} \right] \cap \left[ \sqrt{3}, \sqrt{5} \right]\]

Answer 1

The domain of \[\cos^{- 1} \left( x \right)\] is [-1, 1]

\[\therefore - 1 \leq x^2 - 4 \leq 1\]
\[ \Rightarrow - 1 + 4 \leq x^2 - 4 + 4 \leq 1 + 4\]
\[ \Rightarrow 3 \leq x^2 \leq 5\]
\[ \Rightarrow \pm \sqrt{3} \leq x \leq \pm \sqrt{5}\]
\[ \Rightarrow x \in \left[ - \sqrt{5}, - \sqrt{3} \right] \cup \left[ \sqrt{3}, \sqrt{5} \right]\]

Hence, the correct answer is option (c).