2020 NJC P1 Q9

The diagram shows the curve with equation \(y=\mathrm{f}\left( x \right)\), for \(x\in \mathbb{R},x\ne -\frac{1}{2},x\ne 2\). The curve crosses the axes at \(x=-2,x=4\) and \(y=9\), and has turning points with coordinates \(\left( -4,7 \right)\) and \(\left( 0.75,7 \right)\). The curve has asymptotes with equations \(x=-0.5,x=2,y=0\) and \(y=5\).

1. Sketch the curve \(y=\frac{1}{\mathrm{f}\left( x \right)}\), labelling any axial intercepts and turning points, and the equations of any asymptotes.[4]

For the remainder of this question, the domain of \(\mathrm{f}\)is \(\left[ k,2 \right)\cup \left( 2,\infty \right)\), where \(k\) is a constant.

2. State the least value of \(k\) for which the function \({{\mathrm{f}}^{-1}}\) exists.[1]

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

\(\mathrm{g}:x\mapsto \frac{2x+a}{x-b}\) for \(x\in \mathbb{R},x>b\),

where \(a,b\in {{\mathbb{R}}^{+}}\).

3. Find the range of \(\mathrm{g}\).[2]
4. Find \({{\left( \mathrm{fg} \right)}^{-1}}\left( 0 \right)\). Leave your answer in terms of \(a\) and \(b\).[3]