N2011 P2 Q3

The function \(f\) is defined by \(f:x\mapsto\ln(2x+1)+3\), \(x\in\mathbb R\), \(x>-\frac12\).

1. Find \(f^{-1}(x)\) and write down the domain and range of \(f^{-1}\).[4]
2. Sketch on the same diagram the graphs of \(y=f(x)\) and \(y=f^{-1}(x)\), giving the equations of any asymptotes and the exact coordinates of any points where the curves cross the \(x\)- and \(y\)-axes.[4]
3. Explain why the \(x\)-coordinates of the points of intersection of the curves in part (ii) satisfy the equation \(\ln(2x+1)=x-3\), and find the values of these \(x\)-coordinates, correct to 4 significant figures.[3]