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 \( [-1, 1] \). For \( f(x) = \cos^{-1}(x^2 - 4) \), the argument \( x^2 - 4 \) must be in this interval: \[ -1 \leq x^2 - 4 \leq 1 \] We solve this compound inequality: 1. \( x^2 - 4 \geq -1 \): \( x^2 \geq 3 \), which implies \( x \geq \sqrt{3} \) or \( x \leq -\sqrt{3} \). 2. \( x^2 - 4 \leq 1 \): \( x^2 \leq 5 \), which implies \( -\sqrt{5} \leq x \leq \sqrt{5} \). Combining these conditions yields the domain of \( f(x) \) as \( [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}] \).