8. The functions \(f\) and \(g\) are defined by
$$\begin{array} { l l }
\mathrm { f } : x \rightarrow 2 x + \ln 2 , & x \in \mathbb { R } ,
\mathrm {~g} : x \rightarrow \mathrm { e } ^ { 2 x } , & x \in \mathbb { R } .
\end{array}$$
- Prove that the composite function gf is
$$\operatorname { gf } : x \rightarrow 4 \mathrm { e } ^ { 4 x } , \quad x \in \mathbb { R }$$
- In the space provided on page 19, sketch the curve with equation \(y = \operatorname { gf } ( x )\), and show the coordinates of the point where the curve cuts the \(y\)-axis.
- Write down the range of gf.
- Find the value of \(x\) for which \(\frac { \mathrm { d } } { \mathrm { d } x } [ \operatorname { gf } ( x ) ] = 3\), giving your answer to 3 significant figures.