Two functions, two different conclusions

Determine the range of each of the following functions. State, with reasons, whether the inverse function exists in each case. Then, find the rule and domain of the inverse function if it exists.

(a) $h:x\mapsto\cos(x+1)$, $-1\leq x\leq-1+\frac{\pi}{2}$.

(b) $q:x\mapsto e^x(x-1)$, $x\in\mathbb R$.