Inverse of a quadratic function on a restricted domain, its graph, and linear functions giving a composite quadratic

A function \(\mathrm{f}\) is given by

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

in which \(a\) is a real constant.

1. State the smallest value of \(a\) such that \(\mathrm{f}^{-1}\) exists.[1]
2. Taking \(a\) to be this value, find an expression for \(\mathrm{f}^{-1}(x)\).[4]
3. Find the set of values of \(x\) for which \(\mathrm{f}^{-1}\) is defined.[2]
4. On a single set of axes, sketch the graphs of \(\mathrm{f}\) and \(\mathrm{f}^{-1}\), and label the coordinates of the endpoint of each graph.[3]

A second function \(\mathrm{g}\) is given by \(\mathrm{g}(x)=x^2+x+3\), for all real \(x\).

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