The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \frac{1}{x+a}$

Question

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \frac{1}{x+a}$, $x\in \mathbb{R}$, $x\ne -a$ $\mathrm{g}:x\mapsto {{x}^{2}}+2$,$x\in \mathbb{R}$, where $a$ is a positive constant. Show that ${{\mathrm{f}}^{-1}}$ exists and define ${{\mathrm{f}}^{-1}}$ in a similar form.[3] Show that the composite function $\mathrm{gf}$ exists and find its exact range.[3]

Accepted Answer

(a)(i) $\mathrm f^{-1}$ exists and $\mathrm f^{-1}:x\mapsto\dfrac1x-a$, $x\in\mathbb R$, $x\ne0$ (a)(ii) $\mathrm{gf}$ exists and $R_{\mathrm{gf}}=(2,\infty)$

2023 DHS Promo Q9

2023 DHS Promo Q9

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