Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \frac{2x+7}{x-2}$,

Question

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \frac{2x+7}{x-2}$, $x\in \mathbb{R},$$x\ne 2$, $\mathrm{g}:x\mapsto {{\left( x-1 \right)}^{2}}-2$, $x\in \mathbb{R},$$0 < x < 3$ Show that the composite function $\mathrm{fg}$ exists.[2] Define $\mathrm{fg}$ in a similar form and find its range.[3]

Accepted Answer

(i) $R_{\mathrm g}=[-2,2)\subset D_{\mathrm f}$, so $\mathrm{fg}$ exists (ii) $\mathrm{fg}:x\mapsto\dfrac{2x^2-4x+5}{x^2-2x-3}$, $0<x<3$, with range $\left(-\infty,-\dfrac34\right]$

2023 TMJC Promo Q2

2023 TMJC Promo Q2

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