2020 RI P2 Q3

The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by

\(\mathrm{f}(x)=1+3\mathrm{e}^{-x}, \quad\quad\quad\, x \in \mathbb{R}, x>0,\)
\(\mathrm{g}(x)=|x-1|(x-3), \quad x \in \mathbb{R}, x<c \text{ where}\hspace{0.5em} c \text{ is a real constant.}\)

1. Given that \(c=4\), determine if the composite functions \(\mathrm{fg}\) and \(\mathrm{gf}\) exist, justifying your answers. Find the range of the composite function that exists.[4]
2. Given that \(\mathrm{g}^{-1}\) exists, state the largest possible value of \(c\). Using this value of \(c\) find \(\mathrm{g}^{-1}(x)\).[4]