The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\begin{matrix} & \mathrm{f}:x\mapsto {{x

Question

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\begin{matrix} & \mathrm{f}:x\mapsto {{x}^{2}}+3x+1,x\in \mathbb{R},x\le -\frac{3}{2}, \ & \mathrm{g}:x\mapsto 3+{{\mathrm{e}}^{-x}},x\in \mathbb{R}. \ \end{matrix}$ Show that $\mathrm{f}$ has an inverse. [1] Give a reason why the composite function ${{\mathrm{f}}^{-1}}\mathrm{g}$ exists. Find ${{\mathrm{f}}^{-1}}\mathrm{g}\left( x ight)$ and state the domain of ${{\mathrm{f}}^{-1}}\mathrm{g}$. [6] The function $\mathrm{h}$ is defined as follows. $\mathrm{h}:x\mapsto \ln \left( x-k+1 ight),x\in \mathbb{R},x\ge k,$ where $kFind the value of $k$ such that the range of $\mathrm{hg}$ is given by $\left( \ln 5,\infty ight)$. [2]

Accepted Answer

$k = -1$

2020 YIJC P1 Q7

2020 YIJC P1 Q7

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