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

Question

$\mathrm{f}: x \mapsto \dfrac{3x}{x + 2}, x \in \mathbb{R}, x > -2$ and $g: x \mapsto x^{2} + 1, x \in \mathbb{R}, x \le 5$. 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

$\mathrm{fg}$ and $\mathrm{gf}$ exist. $\mathrm{fg}\left( x \right)=\frac{3{{x}^{2}}+3}{{{x}^{2}}+3},{{D}_{\mathrm{fg}}}=(-\infty ,5],{{R}_{\mathrm{fg}}}=[1,3)$; $\mathrm{gf}\left( x \right)=\frac{10{{x}^{2}}+4x+4}{{{\left( x+2 \right)}^{2}}},{{D}_{\mathrm{gf}}}=(-2,\infty ),{{R}_{\mathrm{gf}}}=[1,\infty)$

Functions - Composite Functions

Functions - Composite Functions

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