Functions Revision Set Tutorial Q2
The functions $\text{f}$ and $\text{g}$ are defined as follows:
$\begin{matrix}
& \text{f}:x\mapsto {{x}^{3}}+1,x\in \mathbb{R}, \\
& \text{g}:x\mapsto {{e}^{-2x}},x\in \mathbb{R} \\
\end{matrix}$
(i)
Define in a similar manner, the inverse functions ${{\text{f}}^{-1}}$ and ${{\text{g}}^{-1}}$.
(ii)
Write down the rule and state the range for each of the functions $\text{fg}$ and $\text{ff}$.
(iii)
Show that the equation $\text{ff}\left( x \right)=0$ has exactly one real root.
(iv)
Denoting this root by $\alpha $, show that $\text{f}(\alpha )=-1$ and ${{\text{g}}^{-1}}(-\alpha )=-\frac{1}{6}\ln 2$ .
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