Question

Find the domain of the function \( f(x) = \cos^{-1}(x^2 - 4) \).

Hint:

When solving for the domain of a function involving inverse trigonometric functions, ensure the argument lies within the valid range for that function.

Answer 1

The domain of the inverse cosine function \( \cos^{-1}(x) \) is restricted to \( -1 \leq x \leq 1 \). Therefore, for \( f(x) = \cos^{-1}(x^2 - 4) \), the argument \( x^2 - 4 \) must lie within this range. \[ -1 \leq x^2 - 4 \leq 1 \] Now, let's solve the inequality: 1. Solve \( x^2 - 4 \geq -1 \): \[ x^2 \geq 3 \quad \Rightarrow \quad x \geq \sqrt{3} \quad \text{or} \quad x \leq -\sqrt{3} \] 2. Solve \( x^2 - 4 \leq 1 \): \[ x^2 \leq 5 \quad \Rightarrow \quad -\sqrt{5} \leq x \leq \sqrt{5} \] Thus, combining the two results, we find the domain of \( f(x) \) is: \[ [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}] \]