2025 TGM P2 Q2

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

\(\mathrm{f} : x \mapsto \ln(1+x), \ x \in \mathbb{R}, \ x > -1,\)
\( \!\!\!\!\!\!\!\!\!\!\!\! \mathrm{g} : x \mapsto e^{-2x+1}, \ x \in \mathbb{R}, \ x \le 0.\)

1. Show that the composite function of \(\mathrm{fg}\) exists.[2]
2. Find \(\mathrm{fg}(x)\) and state its domain and range.[4]