Piecewise Function – Advanced Level

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

\(\mathrm{f}:x\mapsto \frac{{{e}^{x}}-1}{e-1}\) for \(x\in\mathbb{R}\).

Sketch the graph of \(y=\mathrm{f}\left( x \right)\) and state the range of \(\mathrm{f}\).

Another function \(\mathrm{h}\) is defined by

\(\mathrm{h}:x\mapsto \left\{ \begin{aligned}& {{\left( x-1 \right)}^{2}}+1,\,\,\mathrm{for }x\le 1 \\<br />& 1-\frac{\left| 1-x \right|}{2},\,\,\mathrm{for }\,1<x\le 4 \\<br />\end{aligned} \right.\)

Sketch the graph of \(y=\mathrm{h}\left( x \right)\) for \(x\le 4\) and explain why the composite function \({{\mathrm{f}}^{-1}}\mathrm{h}\) exists.

Hence find the exact value of \({{\left( {{\mathrm{f}}^{-1}}\mathrm{h} \right)}^{-1}}\left( 3 \right)\).