Invert a piecewise function with a gap in its range

The function $g$ is defined by

$g:x\mapsto\begin{cases}x-1,&x\leq2,\\\sqrt{x+1},&x>2.\end{cases}$

(i) Sketch the graph of $y=g(x)$.2 marks

(ii) Explain why the inverse function, $g^{-1}$, exists. Find $g^{-1}$ in the similar form.3 marks

Another function $h$ is defined by

$h:x\mapsto e^{x+1}$, $x\in\mathbb R$.

(iii) Find $hg(x)$ for $x<0$.1 mark

(iv) Find the range of the composite function $hg$ for $x<0$.1 mark