Discussion
The domain of the function \(f(x) = \sin^{-1}(5 - 4x^{2}) + \cos^{-1}(5 - 4x^{2})\) is
Answer: A
The domain is \(\left[1, \sqrt{\tfrac{3}{2}}\right] \cup \left[-\sqrt{\tfrac{3}{2}}, -1\right]\).
Both \(\sin^{-1}\) and \(\cos^{-1}\) need their argument in \([-1, 1]\):
\[ \begin{aligned} -1 \leq 5 - 4x^{2} &\leq 1 \\ 1 \leq x^{2} &\leq \tfrac{3}{2} \end{aligned} \]
So \(1 \leq |x| \leq \sqrt{\frac{3}{2}}\), which gives the two intervals in the answer. Options with \(\sqrt{\frac{5}{2}}\), 2 or -2 as an endpoint come from solving one of the two inequalities wrongly.
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