Yishun Innova Junior College H2 Math papers, with video solutions

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$\begin{matrix} & \mathrm{f}:x\mapsto {{x}^{2}}+3x+1,x\in \mathbb{R},x\le -\frac{3}{2}, \\ & \mathrm{g}:x\mapsto 3+{{\mathrm{e}}^{-x}},x\in \mathbb{R}. \\ \end{matrix}$

1. Show that $\mathrm{f}$ has an inverse.

   [1]

2. Give a reason why the composite function ${{\mathrm{f}}^{-1}}\mathrm{g}$ exists. Find ${{\mathrm{f}}^{-1}}\mathrm{g}\left( x \right)$ and state the domain of ${{\mathrm{f}}^{-1}}\mathrm{g}$.

   [6]

The function $\mathrm{h}$ is defined as follows.

$\mathrm{h}:x\mapsto \ln \left( x-k+1 \right),x\in \mathbb{R},x\ge k,$ where $k<0$.

3. Find the value of $k$ such that the range of $\mathrm{hg}$ is given by $\left( \ln 5,\infty \right)$.

   [2]