2020 ACJC P1 Q9

Given that \(\sin x+\sqrt{3}\cos x=R\sin \left( x+\frac{\pi }{3} \right)\), find \(R\), where \(R>0\).[1]

The function \(\mathrm{f}\) is defined by

\(\mathrm{f}:x\mapsto \sin x+\sqrt{3}\cos x,\,\,\,\,\,x\in \mathbb{R},\,\,\,\,\,0\le x\le \pi .\)

Using a graphical method, explain why \({{\mathrm{f}}^{-1}}\) does not exist.[2]

the function \(\mathrm{g}\) is defined by

\(\mathrm{g}:x\mapsto \mathrm{f }(x+a),\,\,\,\,\,\,x\in \mathbb{R},\,\,\,\,\,0\le x\le \pi ,\) where \(a>0.\)

State in exact form the minimum value of \(a\) such that \({{\mathrm{g}}^{-1}}\) exists.[1]

Using this value of \(a\), sketch on the same diagram the graphs of \(y=\mathrm{g}(x)\) and \(y={{\mathrm{g}}^{-1}}(x)\), indicating the equation of the line of symmetry and the coordinates of the endpoints.[3]

Determine also the solution to \(\mathrm{g}{{\mathrm{g}}^{-1}}(x)={{\mathrm{g}}^{-1}}\mathrm{g}(x)\).[1]

The function \(\mathrm{h}\) is defined by

\(\mathrm{h}:x\mapsto x\,{{\mathrm{e}}^{{{x}^{2}}}},\,\,\,\,\,x\in \mathbb{R}\).

Show that \(\mathrm{h}\) is an increasing function. Hence, using the value of \(a\) found in (iii), find the range of values of \(x\) for which \({{\mathrm{h}}^{-1}}\mathrm{g}(x)\le 0.5\).[4]