The function \(\mathrm{f}\) is defined by \[\mathrm{f}:x\mapsto-4x^2-2x+2,\qquad x\in\mathbb{R},\quad -2\leq x<2.\]
Sketch the graph of \(y=\mathrm{f}(x)\), stating clearly the coordinates of the axial intercepts and end points.[2]
Explain why \(\mathrm{f}^{-1}\) does not exist.[1]
The function \(\mathrm{g}\) is defined by \[\mathrm{g}:x\mapsto-4x^2-2x+2,\qquad x\in\mathbb{R},\quad -2\leq x<-\frac14.\]
Find \(\mathrm{g}^{-1}\) in a similar form.[4]
The functions \(\mathrm{h}\) and \(\mathrm{k}\) are defined by \[\mathrm{h}:x\mapsto-\ln(x+1),\qquad x\in\mathbb{R},\quad x>-1,\] \[\mathrm{k}:x\mapsto|4x^2+2x-2|,\qquad x\in\mathbb{R},\quad -1\leq x<0.\]
Show that \(\mathrm{hk}\) exists and find the exact range of the composite function \(\mathrm{hk}\).[2]