The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows: $\begin{matrix} & \mathrm{f}

Question

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

Accepted Answer

(i) $f^{-1}(x)=(x-1)^{1/3}$, $g^{-1}(x)=-\frac12\ln x$; (ii) $fg(x)=e^{-6x}+1$, $ff(x)=(x^3+1)^3+1$; (iii) one real root; (iv) $-\frac16\ln2$

Functions Revision Set Tutorial Q2

Functions Revision Set Tutorial Q2

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