Step-by-Step Solution

1. The domain of \(\cos^{-1}(x)\) is \([-1, 1]\). Therefore, for \(f(x) = \cos^{-1}(x^2 - 4)\) to be defined, we must have \(-1 \le x^2 - 4 \le 1\).
2. We can split this into two inequalities: \begin{enumerate}
3. \(x^2 - 4 \ge -1\)
4. \(x^2 - 4 \le 1\)
\end{enumerate}
5. Solving the first inequality: \(x^2 - 4 \ge -1\) \(x^2 \ge 3\) \(x \le -\sqrt{3}\) or \(x \ge \sqrt{3}\)
6. Solving the second inequality: \(x^2 - 4 \le 1\) \(x^2 \le 5\) \(-\sqrt{5} \le x \le \sqrt{5}\)
7. Combining the two solutions, we have: \(-\sqrt{5} \le x \le -\sqrt{3}\) or \(\sqrt{3} \le x \le \sqrt{5}\)
8. Therefore, the domain of the function is \([-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]\).