The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \frac{2x}{x-2}$, for \(x\in \m

Question

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \frac{2x}{x-2}$, for $x\in \mathbb{R}$, $x\ne 2$. Sketch the graph of $\mathrm{f}$ and find its range.[3] Another function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto 3+\left| x+2 \right|$, for $x\in \mathbb{R}$. Show that the composite function $\mathrm{fg}$ exists. Find $\mathrm{fg}(x)$ and state the domain and range of $\mathrm{fg}$.[5]

Accepted Answer

(a) Vertical asymptote $x=2$, horizontal asymptote $y=2$, and $\operatorname{range}(f)=(-\infty,2)\cup(2,\infty)$ (b) $fg$ exists, $fg(x)=\frac{6+2|x+2|}{1+|x+2|}$, $\operatorname{domain}(fg)=\mathbb R$, and $\operatorname{range}(fg)=(2,6]$

2024 RI P2 Q1

2024 RI P2 Q1

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