The functions $\mathrm{g}$ and $\text{h}$ are defined by \(\begin{align} & \mathrm{g}:x\mapsto \

Question

The functions $\mathrm{g}$ and $\text{h}$ are defined by $\begin{align} & \mathrm{g}:x\mapsto \ln (1+x),\text{}\hspace{0.5em}x>-1 \ & \mathrm{h}:x\mapsto 2+{{e}^{-x}},\text{}\hspace{0.5em}x\in {{\mathbb{R}}^{+}} \ \end{align}$ Determine, with reasons, whether $\mathrm{gh}$ and $\mathrm{hg}$ exist. If the composite function exists, give the rule, domain and range in exact form.

Accepted Answer

$	ext{gh}\left( x ight)$exists. $	ext{hg}\left( x ight)$ does not exist; $	ext{gh}:x\mapsto \ln (3+{{e}^{-x}}),x>0$; range of $	ext{gh}$ is $(\ln 3,2\ln 2)$

Functions - Composite Functions

Functions - Composite Functions

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