Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\begin{matrix} &\mathrm{f}:x\mapsto 1+\frac{

Question

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\begin{matrix} &\mathrm{f}:x\mapsto 1+\frac{2}{x-1},x\in \mathbb{R},x$\begin{matrix} & \mathrm{g}:x\mapsto \ln x,x\in \mathbb{R},0 Explain why the composite function $\mathrm{gf}$ does not exist. [1] Find an expression for $\mathrm{fg}\left( x \right)$. Hence or otherwise, find ${{\left( \mathrm{fg} \right)}^{-1}}\left( 0 \right)$. [4] Find an expression for $\mathrm{h}\left( x \right)$ for each of the following cases: $\mathrm{gh}\left( x \right)=x$, [1] $\mathrm{hg}\left( x \right)={{x}^{2}}+1$. [2]

Accepted Answer

(i) $R_{\mathrm{f}} \not\subseteq D_{\mathrm{g}}$; (ii) $\dfrac{\ln x+1}{\ln x-1}$, $(fg)^{-1}(0) = e^{-1}$; (iii)(a) $e^x$ (b) $e^{2x}+1$

2018 DHS P1 Q5

2018 DHS P1 Q5

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