2026 RI Promo Q10

The function \(\mathrm{f}\) is defined by \(\mathrm{f}:x\mapsto\ln|4-ax|\), for \(x\in\mathbb R\), \(x\ne k\), where \(a\) and \(k\) are constants with \(a>0\).

1. Sketch the graph of \(y=\mathrm{f}(x)\), giving in terms of \(a\), the equations of any asymptotes and the coordinates of the points where the curve meets the axes.[3]
2. If the domain of \(\mathrm{f}\) is restricted to \(x>b\), state the least value of \(b\) in terms of \(a\), such that the function \(\mathrm{f}^{-1}\) exists. Hence find \(\mathrm{f}^{-1}(x)\) and state its domain.[5]

The function \(\mathrm{g}\) is defined by \(\mathrm{g}:x\mapsto\dfrac{2x+1}{4-3x}\), for \(x\in\mathbb R\) and \(x\ne\dfrac43\).

3. Explain why the function \(\mathrm{gf}\) does not exist.[1]

It is now given that \(a=3\).

4. Using the restricted domain found in part (b), find the value of \(x\) for which \(\mathrm{g}\mathrm{f}^{-1}(x)=-1\).[2]