Question

Given that \( f(x) = \sin x + \cos x \) and \( g(x) = x^2 - 1 \), find the conditions under which \( g(f(x)) \) is invertible.

• A. \( -\frac{\pi}{4} \leq x \leq \frac{\pi}{4} \)
• B. \( 0 \leq x \leq \pi \)
• C. \( -\frac{\pi}{4} \leq x \leq \pi \)
• D. \( 0 \leq x \leq \frac{\pi}{2} \)

Hint:

Invertibility requires the function to be monotonic (either strictly increasing or strictly decreasing) over the interval.