ASRJC Functions Tutorial Q9

The function \(\mathrm{f}\) is defined by \(\mathrm{f}:x\mapsto \left| 3+x-2{{x}^{2}} \right|\) for \(x\in \mathbb{R}\), \(x\le k\).

1. Find the greatest value of \(k\) such that \(\mathrm{f}\) has an inverse.[1]
2. Using the value of \(k\) in part (i), find the inverse function \({{\mathrm{f}}^{-1}}\).[3]
3. Sketch both \(\mathrm{f}\) and \({{\mathrm{f}}^{-1}}\) on the same diagram. Is there any solution for \(\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)\)?[3]

Another function \(\mathrm{g}\) is given by \(\mathrm{g}:x\mapsto {{\mathrm{e}}^{-x}}\), \(x\in \mathbb{R}\).

4. Show that \({{\mathrm{f}}^{-1}}\mathrm{g}\) exists and find the range of this composite function.[3]