2023 ACJC P1 Q7

The graph below shows the graph of \(y=\mathrm{f}\left( x \right)\).

1. Sketch on the same diagram the graph of \(y={{\mathrm{f}}^{-1}}\left( x \right)\), clearly stating the equations of any asymptotes and the coordinates of the points where the curve crosses the axes. State also the geometrical relationship between these two graphs.[3]

The function \(\mathrm{g}\) is defined by \(\mathrm{g}:x\mapsto {{x}^{2}}+6x+5\), \(x\in \mathbb{R}\).

2. Find the range of the composite function \(\mathrm{fg}\).[2]
3. It is given that \(\mathrm{fg}\left( a \right)=0\). Find the value of \(\mathrm{g}\left( a \right)\), where \(a\) is a negative constant.[1]

The function \(\mathrm{h}\) is defined by \(\mathrm{h}:x\mapsto {{x}^{2}}+6x+5\), \(x\in \mathbb{R}\), \(x<-4\).

4. Find \({{\mathrm{h}}^{-1}}\) and state the domain of \({{\mathrm{h}}^{-1}}\).[3]