The function $\mathrm{g}$ is given by $\mathrm{g}:x\mapsto \ln \left( \frac{{{e}^{x}}+5}{{{e}^{x}}-1

Question

The function $\mathrm{g}$ is given by $\mathrm{g}:x\mapsto \ln \left( \frac{{{e}^{x}}+5}{{{e}^{x}}-1} \right)$, for $x\in \mathbb{R}$, $x>0$. Find ${{\mathrm{g}}^{-1}}\left( x \right)$ and states its domain. The function $\mathrm{h}$ is defined by $\mathrm{h}:x\mapsto 1+\sqrt{9-{{\left( x-2 \right)}^{2}}}$, for $x\in \mathbb{R}$, $-1\le x\le 5$. Find the exact solutions of ${{\mathrm{g}}^{2021}}\mathrm{h}\left( x \right)=\ln 2$, giving your answer in its simplest form.

Accepted Answer

$x=2\pm\sqrt{(4-\ln 7)(2+\ln 7)}$

2021 NYJC P1 Q5

2021 NYJC P1 Q5

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