Tampines Meridian Junior College H2 Math papers, with video solutions

The function \(\mathrm{f}\) is defined by

\(\mathrm{f}:x\mapsto -\ln (x-2)\), \(x\in \mathbb{R}\), \(2 < x\le 3\).

1. Find \({{\mathrm{f}}^{-1}}(x)\) and state the domain and range of \({{\mathrm{f}}^{-1}}\).[4]
2. Sketch on the same diagram the graphs of \(y=\mathrm{f}(x)\), \(y={{\mathrm{f}}^{-1}}(x)\) and \(y={{\mathrm{f}}^{-1}}\mathrm{f}(x)\), giving the equations of any asymptotes and the exact coordinates of any points where the curves cross the \(x\)- and \(y\)- axes.[6]
3. Explain why the x-coordinate of the point of intersection of the graph of \(y=\mathrm{f}(x)\) and \(y={{\mathrm{f}}^{-1}}(x)\) satisfies the equation
   \(x+\ln (x-2)=0\),
   and find the value of this \(x\)-coordinate.[2]