The diagrams show four different relations.

(b) State whether each relation is one-one, many-one, and whether it is its own inverse.

(c) The functions are given by \(\mathrm{f}:x\mapsto\sin x\), for \(30^\circ\leqslant x\leqslant a^\circ\), and \(\mathrm{g}:x\mapsto\sqrt{x-\frac12}\), for \(x\geqslant\frac12\). Given that \(\mathrm{g}\mathrm{f}\) exists, find the largest possible value of \(a\), the range of \(\mathrm{g}\mathrm{f}\), and explain why \(\mathrm{g}^2\) does not exist.