11 Functions f and g are defined for \(x \in \mathbb { R }\) by
$$\begin{aligned}
& \mathrm { f } : x \mapsto 2 x + 1
& \mathrm {~g} : x \mapsto x ^ { 2 } - 2
\end{aligned}$$
- Find and simplify expressions for \(\mathrm { fg } ( x )\) and \(\mathrm { gf } ( x )\).
- Hence find the value of \(a\) for which \(\mathrm { fg } ( a ) = \mathrm { gf } ( a )\).
- Find the value of \(b ( b \neq a )\) for which \(\mathrm { g } ( b ) = b\).
- Find and simplify an expression for \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\).
The function h is defined by
$$\mathrm { h } : x \mapsto x ^ { 2 } - 2 , \quad \text { for } x \leqslant 0$$
- Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).