3 Fig. 9 shows the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). The function \(y = \mathrm { f } ( x )\) is given by
$$f ( x ) = \ln \left( \frac { 2 x } { 1 + x } \right) , x > 0$$
The curve \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis at P , and the line \(x = 2\) at Q .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{75eebbfb-7bfa-4382-a6d7-1c5a7f3f419a-3_559_644_622_745}
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\caption{Fig. 9}
\end{figure}
- Verify that the \(x\)-coordinate of P is 1 .
Find the exact \(y\)-coordinate of Q .
- Find the gradient of the curve at P. [Hint: use \(\ln \frac { a } { b } = \ln a - \ln b\).]
The function \(\mathrm { g } ( x )\) is given by
$$\mathrm { g } ( x ) = \frac { \mathrm { e } ^ { x } } { 2 - \mathrm { e } ^ { x } } , \quad x < \ln 2 .$$
The curve \(y = \mathrm { g } ( x )\) crosses the \(y\)-axis at the point R .
- Show that \(\mathrm { g } ( x )\) is the inverse function of \(\mathrm { f } ( x )\).
Write down the gradient of \(y = \mathrm { g } ( x )\) at R .
- Show, using the substitution \(u = 2 - \mathrm { e } ^ { x }\) or otherwise, that \(\int _ { 0 } ^ { \ln \frac { 4 } { 3 } } \mathrm {~g} ( x ) \mathrm { d } x = \ln \frac { 3 } { 2 }\).
Using this result, show that the exact area of the shaded region shown in Fig. 9 is \(\ln \frac { 32 } { 27 }\). [Hint: consider its reflection in \(y = x\).]