The graph of $y=2\ln x-{{\left( \ln x \right)}^{2}}$ where $x0$ is shown in the diagram below. The

Question

The graph of $y=2\ln x-{{\left( \ln x \right)}^{2}}$ where $x>0$ is shown in the diagram below. The graph cuts the $x$-axis at $x=1$ and $x={{e}^{2}}$. The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto 2\ln x-{{\left( \ln x \right)}^{2}},$$0 where $a$ is a constant. Find the greatest possible value of $a$ for which ${{\mathrm{f}}^{-1}}$ exists, giving your answer in exact form. Hence, find ${{\mathrm{f}}^{-1}}$ in similar form. [6] The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto \frac{1}{3}{{e}^{x}}$, $x\in \mathbb{R}$ . Find the exact range of the composite function $\mathrm{g}{{\mathrm{f}}^{-1}}$. [2]

Accepted Answer

$a = \mathrm{e}$; $\mathrm{f}^{-1}(x) = \mathrm{e}^{1-\sqrt{1-x}}$, $x \leq 1$; $R_{\mathrm{gf}^{-1}} = \left(\dfrac{1}{3}, \dfrac{1}{3}\mathrm{e}^{\mathrm{e}}\right]$

2018 SAJC MYE BT2 P1 Q5

2018 SAJC MYE BT2 P1 Q5

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