OCR MEI C3 (Core Mathematics 3)

2 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } - 1 , x \in \mathbb { R } .$$ The curve crosses the \(x\)-axis at O and P , and has a turning point at Q . \begin{figure}[h] \end{figure}

1. Find the exact \(x\)-coordinate of P .
2. Show that the \(x\)-coordinate of Q is \(\ln 2\) and find its \(y\)-coordinate.
3. Find the exact area of the region enclosed by the curve and the \(x\)-axis. The domain of \(\mathrm { f } ( x )\) is now restricted to \(x \geqslant \ln 2\).
4. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\). Write down its domain and range, and sketch its graph on the copy of Fig. 9.

3 Fig. 7 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = 1 + 2 \arctan x , x \in \mathbb { R }\). The scales on the \(x\) - and \(y\)-axes are the same. \begin{figure}[h] \end{figure}

1. Find the range of f , giving your answer in terms of \(\pi\).
2. Find \(\mathrm { f } ^ { - 1 } ( x )\), and add a sketch of the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) to the copy of Fig. 7.

4 Given that \(\mathrm { f } ( x ) = 2 \ln x\) and \(\mathrm { g } ( x ) = \mathrm { e } ^ { x }\), find the composite function \(\mathrm { gf } ( x )\), expressing your answer as simply as possible.

5 Write down the conditions for \(\mathrm { f } ( x )\) to be an odd function and for \(\mathrm { g } ( x )\) to be an even function.
Hence prove that, if \(\mathrm { f } ( x )\) is odd and \(\mathrm { g } ( x )\) is even, then the composite function \(\mathrm { gf } ( x )\) is even.

7 Given that \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \ln ( x - 1 )\) and \(\mathrm { g } ( x ) = 1 + \mathrm { e } ^ { 2 x }\), show that \(\mathrm { g } ( x )\) is the inverse of \(\mathrm { f } ( x )\).

8 Sketch the curve \(y = 2 \arccos x\) for \(- 1 \leqslant x \leqslant 1\).