Worked Example 2

$\text{g}:x\mapsto \ln \left( 1+x \right),x\in \mathbb{R},x>-1$ and $\text{h}:x\mapsto 2+{{e}^{-x}},x\in {{\mathbb{R}}^{+}}$.

Determine, with reasons, whether $\text{gh}$ and $\text{hg}$ exist. If the composite function exists, give the rule, domain and range in exact form.