6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{032d2541-9905-4570-9584-9a144b02fde5-14_768_1006_251_532}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of the curve \(C\) with parametric equations
$$x = 1 + 3 \tan t \quad y = 2 \cos 2 t \quad - \frac { \pi } { 6 } \leqslant t \leqslant \frac { \pi } { 3 }$$
The curve crosses the \(x\)-axis at point \(P\), as shown in Figure 3.
- Find the equation of the tangent to \(C\) at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where f is a function with domain \([ k , 1 + 3 \sqrt { 3 } ]\)
- Find the exact value of the constant \(k\).
- Find the range of f.