Question
Find the domain of:
\[ \cos^{-1}(x^2 – 4) \]
Solution
For \( \cos^{-1}(t) \) to be defined:
\[ -1 \le t \le 1 \]
So,
\[ -1 \le x^2 – 4 \le 1 \]
Add 4:
\[ 3 \le x^2 \le 5 \]
Thus,
\[ \sqrt{3} \le |x| \le \sqrt{5} \]
So domain is:
\[ -\sqrt{5} \le x \le -\sqrt{3} \quad \text{or} \quad \sqrt{3} \le x \le \sqrt{5} \]
Final Answer:
\[ \boxed{[-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]} \]
Key Concept
Argument of cos⁻¹ must lie in [−1, 1].