$\mathrm{f}:x\mapsto {{e}^{-2x}},x\in \mathbb{R},x0$ and \(\mathrm{g}:x\mapsto {{x}^{3}}+1,x\i

Question

$\mathrm{f}:x\mapsto {{e}^{-2x}},x\in \mathbb{R},x>0$ and $\mathrm{g}:x\mapsto {{x}^{3}}+1,x\in \mathbb{R}$. Determine, with reasons, whether $\mathrm{fg}$ and $\mathrm{gf}$ exist. If the composite function exists, give the rule, domain and range in exact form.

Accepted Answer

$	ext{fg}$ doesn’t exist; $	ext{gf}$ exists. $	ext{gf}\left( x ight)={{e}^{-6x}}+1$, ${{D}_{	ext{gf}}}=(0,\infty )$, ${{R}_{	ext{gf}}}=\left( 1,2 ight)$

Functions - Composite Functions

Functions - Composite Functions

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