2022 RI P1 Q9
The functions $\text{f}$, $\text{g}$ and $\text{h}$ are defined as follows:
$\text{f}:x\mapsto {{x}^{3}}-1$, $x\in \mathbb{R}$,
$\text{g}:x\mapsto {{\text{e}}^{-3x}}$, $x\in \mathbb{R}$, $x<0$,
$\text{h}:x\mapsto \frac{x+1}{x-1}$, $x\in \mathbb{R}$, $x\ne 1$.
(i)
Define in a similar form, the inverse functions ${{\text{g}}^{-1}}$ and ${{\text{h}}^{-1}}$.
[4]
(ii)
Write down the rule of the composite function $\text{ff}$.
[1]
(iii)
It is known that the equation $\text{ff}\left( x \right)=0$ has only one real root $\alpha $.
Find the exact value of $\alpha $ and hence show that ${{\text{g}}^{-1}}\left( \alpha \right)=k\,\,\ln 2$, where $k$ is a constant to be found.
[2]
(iv)
Show that the composite function $\text{hg}$ exists and write down the range of this function.
[2]
(v)
Denoting composite functions $\text{hh}$ as ${{\text{h}}^{2}}$, $\text{hhh}$ as ${{\text{h}}^{3}}$ and so on, find the value of $x$ for which $\text{h}\,”’\left( x \right)={{\text{h}}^{-1}}\left( -1 \right)$, where $m$ is a positive even integer.
[2]
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