2022 HCI J1 BT Q1

A piecewise function is given as

\({\mathrm{f}(x) = \begin{cases} 3 - x & \mathrm{for } \quad 1 \leq x \leq 3, \\ \sqrt{4 - (x - 5)^2} & \mathrm{for } \quad 3 < x \leq 5 \end{cases}}\)

And that \(\mathrm{f}\left( x \right)=\mathrm{f}\left( x-4 \right)\) for all real values of \(x\).

1. Find the exact values of \(\mathrm{f}\left( 8 \right)\) and \(\mathrm{f}\left( -10 \right)\).[2]

Another function \(\mathrm{g}\) is defined as \(\mathrm{g}\left( x \right)=\ln x\), \({{e}^{2}} < x < {{e}^{4}}\).

2. Show that \(\mathrm{fg}\) exists and find the exact range of \(\mathrm{fg}\).[3]