2016 NYJC P1 Q10

Express \(\sin x+\sqrt{3}\cos x\) in the form \(R\sin \left( x+\alpha \right)\) where \(R\) and \(\alpha \) are exact positive constants to be found.[1]

The function \(\mathrm{f}\) is defined by \(\mathrm{f}:x\mapsto \sin x+\sqrt{3}\cos x,\) \(\frac{\pi }{6}\le x\le k\). Find the largest exact value of \(k\) such that \(\mathrm{f}\) has an inverse. Hence define \({{\mathrm{f}}^{-1}}\) in similar form and write down the set of values of \(x\) for which \(\mathrm{f}{{\mathrm{f}}^{-1}}\left( x \right)={{\mathrm{f}}^{-1}}\mathrm{f}\left( x \right)\).[5]