N2008 P2 Q4

The function \(\mathrm{f}\) is defined by \(\mathrm{f}:x\mapsto {{(x-4)}^{2}}+1\) for \(x\in \mathbb{R},x>4\).

1. Sketch the graph of \(y=\mathrm{f}(x)\). Your sketch should indicate the position of the graph in relation to the origin.[2]
2. Find \({ {\mathrm{f}}^{-1}}(x)\), stating the domain of \({{\mathrm{f}}^{-1} }\).[3]
3. On the same diagram as in part (i), sketch the graph of \(y={{\mathrm{f}}^{-1}}(x)\).[1]
4. Write down the equation of the line in which the graph of \(y=\mathrm{f(}x\mathrm{)}\) must be reflected in order to obtain the graph of \(y={{\mathrm{f}}^{-1}}(x)\), and hence find the exact solution of the equation \(\mathrm{f}(x)={{\mathrm{f}}^{-1}}(x)\).[5]