10 Functions \(f\) and \(g\) are defined as follows:
$$\begin{array} { l l }
\mathrm { f } ( x ) = \frac { 2 x + 1 } { 2 x - 1 } & \text { for } x \neq \frac { 1 } { 2 }
\mathrm {~g} ( x ) = x ^ { 2 } + 4 & \text { for } x \in \mathbb { R }
\end{array}$$
\includegraphics[max width=\textwidth, alt={}, center]{bb7595c9-93ae-49e8-9cc5-9ecc802e6060-16_773_1182_555_511}
The diagram shows part of the graph of \(y = \mathrm { f } ( x )\).
State the domain of \(\mathrm { f } ^ { - 1 }\).- Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
- Find \(\mathrm { gf } ^ { - 1 } ( 3 )\).
- Explain why \(\mathrm { g } ^ { - 1 } ( x )\) cannot be found.
- Show that \(1 + \frac { 2 } { 2 x - 1 }\) can be expressed as \(\frac { 2 x + 1 } { 2 x - 1 }\). Hence find the area of the triangle enclosed by the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = 1\) and the \(x\) - and \(y\)-axes.