2022 RI P1 Q9

The functions \(\mathrm{f}\), \(\mathrm{g}\) and \(\mathrm{h}\) are defined as follows:

\(\mathrm{f}:x\mapsto {{x}^{3}}-1\), \(x\in \mathbb{R}\),

\(\mathrm{g}:x\mapsto {{\mathrm{e}}^{-3x}}\), \(x\in \mathbb{R}\), \(x<0\),

\(\mathrm{h}:x\mapsto \frac{x+1}{x-1}\), \(x\in \mathbb{R}\), \(x\ne 1\).

1. Define in a similar form, the inverse functions \({{\mathrm{g}}^{-1}}\) and \({{\mathrm{h}}^{-1}}\).[4]
2. Write down the rule of the composite function \(\mathrm{ff}\).[1]
3. It is known that the equation \(\mathrm{ff}\left( x \right)=0\) has only one real root \(\alpha \).
   Find the exact value of \(\alpha \) and hence show that \({{\mathrm{g}}^{-1}}\left( \alpha \right)=k\,\,\ln 2\), where \(k\) is a constant to be found.[2]
4. Show that the composite function \(\mathrm{hg}\) exists and write down the range of this function.[2]
5. Denoting composite functions \(\mathrm{hh}\) as \({{\mathrm{h}}^{2}}\), \(\mathrm{hhh}\) as \({{\mathrm{h}}^{3}}\) and so on, find the value of \(x\) for which \(\mathrm{h}\,'''\left( x \right)={{\mathrm{h}}^{-1}}\left( -1 \right)\), where \(m\) is a positive even integer.[2]