2014 VJC Promo Q8

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

\(\mathrm{f}:x\mapsto \frac{-x-2}{5x+1},\)                      \(x\in \mathbb{R},x\ne -\frac{1}{5}\).

1. Explain why both the function \({{\mathrm{f}}^{-1}}\) and composite function \({{\mathrm{f}}^{2}}\) exist.[4]
2. Find \({{\mathrm{f}}^{2}}\left( x \right)\) and state the range of \({{\mathrm{f}}^{2}}\).[3]
3. Determine the solution of the equation \(\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)\).[2]

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

\(\mathrm{g}:x\mapsto \frac{2x-2}{5x+4},\)           \(x\in \mathbb{R},x\ne -\frac{4}{5},x\ne -\frac{1}{5},x\ne -\frac{2}{5}\).

4. Verify that \({{\mathrm{g}}^{2}}\left( x \right)=\mathrm{f}\left( x \right)\).[1]
5. Given that \(k\) is an integer, find \({{\mathrm{g}}^{49}}\left( k \right)\), giving your answer in terms of \(k\).[3]