2021 SJI P2 Q7

Let \(\mathrm{f}(x)=\dfrac\pi2x-x\arctan x\).

1. Find the value of \(\mathrm{f}''(1)\).[2]
2. Using L’Hopital’s Rule, show that as \(x\to\infty,\ y\to1\).[4]
3. Explain why the inverse function \(\mathrm{f}^{-1}\) exists.[1]
4. Find the composite function \((\mathrm{f}\circ\mathrm{f}^{-1})(x)\), stating clearly its domain.[2]
5. Solve the equation \(\mathrm{f}(x)=\mathrm{f}^{-1}(x)\).[3]

The graph of \(y=\mathrm{f}(x)\) is mapped onto the graph of \(y=\mathrm{g}(x)\) through a sequence of transformations as follows:

I: a translation of magnitude \(1\) unit in the direction of the \(x\)-axis;

II: a stretch parallel to the \(x\)-axis by a factor of \(2\);

III: a stretch parallel to the \(y\)-axis by a factor of \(4\).

6. Point \(P\left(1,\dfrac\pi4\right)\) lies on the graph of \(y=\mathrm{f}(x)\). Find the image of P on the graph of \(y=\mathrm{g}(x)\).[3]