8 The curve with equation $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = \ln ( 2 x - 3 ) , x > \frac { 3 } { 2 }$, is sketched below.\\
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-08_630_1173_424_443}
\begin{enumerate}[label=(\alph*)]
\item The inverse of f is $\mathrm { f } ^ { - 1 }$.
\begin{enumerate}[label=(\roman*)]
\item Find $\mathrm { f } ^ { - 1 } ( x )$.
\item State the range of $\mathrm { f } ^ { - 1 }$.
\item Sketch, on the axes given on page 9 , the curve with equation $y = \mathrm { f } ^ { - 1 } ( x )$, indicating the value of the $y$-coordinate of the point where the curve intersects the $y$-axis.\\
(2 marks)
\end{enumerate}\item The function g is defined by

$$\mathrm { g } ( x ) = \mathrm { e } ^ { 2 x } - 4 , \text { for all real values of } x$$
\begin{enumerate}[label=(\roman*)]
\item Find $\mathrm { gf } ( x )$, giving your answer in the form $( a x - b ) ^ { 2 } - c$, where $a , b$ and $c$ are integers.
\item Write down an expression for $\mathrm { fg } ( x )$, and hence find the exact solution of the equation $\operatorname { fg } ( x ) = \ln 5$.\\
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-09_1079_1422_233_358}
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA C3 2013 Q8 [12]}}