Step 1: Compute \( g[f(x)] \)
Given \( f(x) = \sin x + \cos x \) and \( g(x) = x^2 - 1 \), we compute \( g[f(x)] \). This involves substituting \( f(x) \) into \( g(x) \).
\[ g[f(x)] = g(\sin x + \cos x) = (\sin x + \cos x)^2 - 1 \]Step 2: Simplify \( g[f(x)] \)
Expand the expression and apply trigonometric identities.
\[ g[f(x)] = (\sin^2 x + 2\sin x \cos x + \cos^2 x) - 1 \]Using \( \sin^2 x + \cos^2 x = 1 \) and \( 2\sin x \cos x = \sin 2x \):
\[ g[f(x)] = (1 + \sin 2x) - 1 = \sin 2x \]Step 3: Analyze Monotonicity
The function \( \sin 2x \) must be monotonic (strictly increasing or decreasing) for invertibility. A sinusoidal function's monotonicity is determined by the interval of its argument.
Step 4: Determine Invertibility Interval
The sine function is strictly increasing on \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). For \( \sin 2x \) to be strictly increasing, its argument \( 2x \) must lie within this interval.
\[ -\frac{\pi}{2} \le 2x \le \frac{\pi}{2} \]Dividing by 2 yields the interval for \( x \):
\[ -\frac{\pi}{4} \le x \le \frac{\pi}{4} \]Within \( [-\frac{\pi}{4}, \frac{\pi}{4}] \), \( \sin 2x \) is strictly increasing and thus invertible.
Conclusion: \( g[f(x)] = \sin 2x \). This function is invertible on the interval \( [-\frac{\pi}{4}, \frac{\pi}{4}] \).