2024 IGCSE Additional Mathematics October/November 0606/23 Q2

The function \(\mathrm{f}\) is defined by \(\mathrm{f}(x)=1-4x-x^2\) for all real values of \(x\).

1. Write \(\mathrm{f}(x)\) in the form \(a-(x+b)^2\), where \(a\) and \(b\) are constants.[2]
2. Find the range of \(\mathrm{f}\).[1]

The function \(\mathrm{g}\) is defined by \(\mathrm{g}(x)=1-4x-x^2\) for \(x\geq k\), where \(k\) is a constant.

3. State the least possible value of \(k\) such that \(\mathrm{g}\) has an inverse.[1]
4. Using your value of \(k\), find \(\mathrm{g}^{-1}(x)\), stating its domain and range.[5]