- The functions \(f\) and \(g\) are defined by
$$\begin{array} { l l }
\mathrm { f } : x \rightarrow \mathrm { e } ^ { 2 x } - 5 , & x \in \mathbb { R }
\mathrm {~g} : x \rightarrow \ln ( 3 x - 1 ) , & x \in \mathbb { R } , x > \frac { 1 } { 3 }
\end{array}$$
- Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
- Find \(\mathrm { fg } ( 3 )\), giving your answer in its simplest form.
(ii) (a) Sketch the graph with equation
$$y = | 4 x - a |$$
where \(a\) is a positive constant. State the coordinates of each point where the graph cuts or meets the coordinate axes.
Given that
$$| 4 x - a | = 9 a$$
where \(a\) is a positive constant, - find the possible values of
$$| x - 6 a | + 3 | x |$$
giving your answers, in terms of \(a\), in their simplest form.