Consider \(\mathrm f(x)=\ln\left(\dfrac{\sqrt{x^2-1}}{\mathrm e}\right)\), \(x\in D\).
Find the largest possible domain \(D\) for \(\mathrm f\) to be a function.[2]
Sketch the graph of \(y=\mathrm f(x)\). Label clearly the equations of the asymptotes and the coordinates of any intercepts with the axes.[3]
Determine whether the inverse function \(\mathrm f^{-1}\) exists.[2]
The function \(\mathrm g\) is defined by \(\mathrm g(x)=\ln\left(\dfrac{\sqrt{x^2-1}}{\mathrm e}\right)\), \(x\in(1,\infty)\).
Find the inverse function \(\mathrm g^{-1}\) and state its domain.[5]
State the rule and domain of the composite function \(\mathrm g^{-1}\circ\mathrm f\).[2]
Find \(\mathrm g'(x)\).[3]
Solve \(\mathrm g'(x)=0\).[2]
Prove that there are no solutions to \(\dfrac{\mathrm d}{\mathrm dx}(\mathrm g^{-1}(x))=0\).[2]