Consider the piecewise-defined function
g(x)=\begin{cases}
\sin x, & -\dfrac{\pi}{2}\le x\le \dfrac{\pi}{2},
x-1, & x\ge \dfrac{\pi}{2}.
\end{cases}
(a) Prove that g is continuous on its entire domain [−2π,∞).
(b) Identify all sub-intervals of the domain on which g is one-to-one. Justify each selection with an analysis of monotonicity.
(c) For each sub-interval found in part (b), justify that the corresponding inverse function g−1 is continuous throughout its domain, including points that correspond to x=2π in the original function.
(d) For the branch of g−1 whose range contains 1, compute (g−1)′(1) if it exists, and justify your result.