2025 IGCSE Additional Mathematics February/March 0606/22 Q7

It is given that \(\mathrm{f}(x)=2\mathrm{e}^x+a\) for \(x\ge0\), where \(a\) is an integer

and \(\mathrm{g}(x)=\sqrt{x-1}\) for \(x\ge1\).

1. Find the least value of \(a\) so that the function \(\mathrm{gf}\) exists for all \(x\ge0\).[2]
2. In the case where \(a=5\), solve the equation \(\mathrm{gf}(x)=3\).
   Give your answer correct to 3 decimal places.[3]