2026 IGCSE Additional Mathematics May/June 0606/11 Q6

The function \(\mathrm{f}\) is such that \(\mathrm{f}(x)=3x^2+8x\) for \(x\ge k\), where \(k\) is a constant.

It is given that \(\mathrm{f}^{-1}\) exists.

\(k\) has the least possible value for which \(\mathrm{f}^{-1}\) exists.

1. Find the value of \(k\).[2]
2. Find the range of \(\mathrm{f}\).[2]
3. Find an expression for \(\mathrm{f}^{-1}\).[3]
4. The function \(\mathrm{g}\) is such that \(\mathrm{g}(x)=3^x\) for \(x>-6\).
   1. Write down the domain of \(\mathrm{gf}\).[1]
   2. Solve the equation \(\mathrm{gf}(x)=27\).[4]