2015 HCI P1 Q5

The function \(\mathrm{f}\) is defined by \(\mathrm{f}:x\mapsto \ln {{\left( 2-x \right)}^{2}}-2,\mathrm{ }x\in \mathbb{R},\mathrm{ }x\ne 2.\)

1. By considering the graph of \(y=\mathrm{f}\left( x \right),\) give a reason why \({{\operatorname{f}}^{-1}}\) does not exist.[2]

The function \(\mathrm{h}\) is defined by \(\mathrm{h}:x\mapsto \mathrm{f}\left( x \right),\mathrm{ }x\ge a,\mathrm{ }a\in \mathbb{Z}.\)

2. Find the largest possible domain of \(\mathrm{h}\) such that \({{\mathrm{h}}^{-1}}\) exists.[1]
3. Define \({{\mathrm{h}}^{-1}}\) in a similar form.[3]
4. Find the set of values of \(x\) which satisfies the equation \(\mathrm{h}{{\mathrm{h}}^{-1}}\left( x \right)={{\mathrm{h}}^{-1}}\mathrm{h}\left( x \right)\).[2]