The function \(\mathrm f\) is defined by \(\mathrm f(x)=\dfrac1{x^2-2x-8}\), where \(x\in\mathbb R\), \(x\neq p\), \(x\neq q\) and \(p<q\).
Find the value of \(p\) and of \(q\).[3]
Sketch the graph of \(y=\mathrm f(x)\), clearly indicating any asymptotes with their equations. State clearly the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.[5]
The function \(\mathrm g\) is defined by \(\mathrm g(x)=\dfrac1{x^2-2x-8}\), where \(x\in\mathbb R\), \(x>4\).
Find the inverse function of \(\mathrm g\) and state its domain.[7]
Sketch the graph of \(y=\mathrm g^{-1}(x)\) on the same sketch as the graph of \(y=\mathrm f(x)\).[3]
The function \(\mathrm h\) is defined by \(\mathrm h(x)=\arctan\dfrac x7\), where \(x\in\mathbb R\).
Find the value of \(m\) such that \((\mathrm h\circ\mathrm g)(m)=\dfrac\pi4\). Give your answer in the form \(a+\dfrac b7\sqrt c\), where \(a,b,c\in\mathbb Z^+\).[7]