2019 SAJC Promo Q5

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

\(\begin{matrix} & \mathrm{f}:x\mapsto -\ln \left( x+a \right),x\in \mathbb{R},x>-a \\ & \mathrm{g}:x\mapsto {{\left( x-\sqrt{a} \right)}^{2}},x\in \mathbb{R},x>0 \\ \end{matrix}\)

where \(a\) is a positive constant such that \(1 < a\le 3\).

1. Sketch the graph of \(y=\mathrm{f}\left( x \right)\) and show that \(\mathrm{f}\) has an inverse.[4]
2. On the same diagram in part (i), sketch the graph of \(y={{\mathrm{f}}^{-1}}\left( x \right)\) and \(y={{\mathrm{f}}^{-1}}\mathrm{f}\left( x \right)\).[2]
3. Find the range of the composite function \(\mathrm{fg}\).[2]
4. Without finding \({{\mathrm{f}}^{-1}}\left( x \right)\), find \(x\) given that \({{\mathrm{f}}^{-1}}\mathrm{g}\left( x \right)={{\mathrm{e}}^{-3}}-a\).[3]