$\mathrm{f}:x\mapsto \cos x,x\in \mathbb{R},0\le x

Question

$\mathrm{f}:x\mapsto \cos x,x\in \mathbb{R},0\le x < \frac{\pi }{2}$ and $\mathrm{g}:x\mapsto \ln \left( {{x}^{2}} \right),x\in \mathbb{R},x\ne 0$. 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

$\mathrm{gf}$ exists. $\mathrm{gf}\left( x \right)=\ln \left( {{\cos }^{2}}x \right),{{D}_{\mathrm{gf}}}=[0,\frac{\pi }{2}),{{R}_{\mathrm{gf}}}=(-\infty ,0]$

Functions - Composite Functions

Functions - Composite Functions

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