Let \(\mathrm f(x)=x^2+1\), where \(x\in\mathbb R\).
The graph of \(y=\mathrm g(x)\) is drawn below.
Find the value of \((\mathrm f\circ\mathrm g)(3)\).
Find the value of \((\mathrm g\circ\mathrm f)(\sqrt2)\).
Sketch the graph of \(y=(\mathrm f\circ\mathrm g)(x)\), showing the coordinates of any intercepts with the axes and the coordinates of the turning points.[9]
If the domain of \(\mathrm f\) is further restricted to \(x\leq k\), state with a reason the largest value of \(k\) for which the function \(\mathrm f^{-1}\) exists. Hence, find \(\mathrm f^{-1}(x)\) and the domain of \(\mathrm f^{-1}\).[6]
Describe a sequence of transformations which transform the graph of \(y=\mathrm f(x)\) to the graph of \(y=3\mathrm f(2x-1)\).[3]