2019 TMJC P1 Q7

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

\(\mathrm{f}\left( x \right)=\left\{ \begin{matrix}<br />n-x\mathrm{ for }n\le x<n+1\mathrm{, where }n\mathrm{ is any positive odd integer,} \\<br />x-\frac{n}{2}\mathrm{ for }n\le x<n+1\mathrm{, where }n\mathrm{ is any positive even integer}\mathrm{.} \\<br />\end{matrix} \right.\)

1. Show that \(\mathrm{f}\left( 1.5 \right)=-0.5\) and find \(\mathrm{f}\left( 2.5 \right)\).[2]
2. Sketch the graph of \(y=\mathrm{f}\left( x \right)\) for \(1\le x<5\).[2]
3. Does \(\mathrm{f}\) have an inverse for \(1\le x<5\)? Justify your answer.[2]
4. The function \(\mathrm{g}\) is defined by \(\mathrm{g}:x\mapsto \frac{2x-1}{x+1},x\in \mathbb{R},x\ne -1\). For \(2\le x<3\), find an expression for \(\mathrm{gf}\left( x \right)\) and hence, or otherwise, find \({{\left( \mathrm{gf} \right)}^{-1}}\left( \frac{2}{3} \right)\).[4]