2022 CJC J2 MYE P2 Q3

The greatest integer function denoted by \(\left[ x \right]\), is a function that gives the largest integer which is less than or equal to \(x\). For example, \(\left[ 1.5 \right]=1\), \(\left[ 3 \right]=3\) and \(\left[ -5.7 \right]=-6\).

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

\(\mathrm{f}:x\mapsto \left[ x \right]-x\), \(x\in \mathbb{R}\).

1. Sketch the graph of \(\mathrm{f}\) for \(-3\le x\le 3\). Find the range of \(\mathrm{f}\).[3]
2. Explain why \({{\mathrm{f}}^{-1}}\) does not exist.[1]

The functions \(\mathrm{g}\) and \(\mathrm{h}\) are defined by

\(\mathrm{g}:x\mapsto \left[ x \right]-x\), \(1\le x < 2\),

\(\mathrm{h}:x\mapsto {{\mathrm{e}}^{x}}\), \(x>0\).

3. Define \({{\mathrm{g}}^{-1}}\) in a similar form.[2]
4. Explain why \(\mathrm{h}{{\mathrm{g}}^{-1}}\) exists. Find the exact range of \(\mathrm{h}{{\mathrm{g}}^{-1}}\).[3]