IB AA HL Q9: Quadratic inverse and composition (paper unspecified)

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

\[\mathrm{f}(x)=x^2+x+3,\ x\geq a\]

where \(a\) is a real constant.

1. State the least value of \(a\) for which \(\mathrm{f}^{-1}\) exists.[1]
2. Using this value of \(a\), find \(\mathrm{f}^{-1}(x)\).[4]
3. Find the domain of \(\mathrm{f}^{-1}\).[2]
4. Sketch, on the same axes, the graphs of \(\mathrm{f}\) and \(\mathrm{f}^{-1}\), labelling clearly the coordinates of any endpoints.[3]

The function \(\mathrm{g}\) is defined by \(\mathrm{g}(x)=x^2+x+3\).

5. Find two possible linear functions \(\mathrm{h}\) such that \((\mathrm{g}\circ\mathrm{h})(x)=4x^2-14x+15\).[6]