8 The function f is defined by \(\mathrm { f } : x \mapsto \frac { 1 } { 2 - 2 x }\) for \(x \in \mathbb { R } , x \neq 0 , x \neq \frac { 1 } { 2 } , x \neq 1\). The function g is defined by \(\mathrm { g } ( x ) = \mathrm { ff } ( x )\).

1. Show that \(\mathrm { g } ( x ) = \frac { 1 - x } { 1 - 2 x }\) and that \(\operatorname { gg } ( x ) = x\). It is given that f and g are elements of a group \(K\) under the operation of composition of functions. The element e is the identity, where e : \(x \mapsto x\) for \(x \in \mathbb { R } , x \neq 0 , x \neq \frac { 1 } { 2 } , x \neq 1\).
2. State the orders of the elements f and g .
3. The inverse of the element f is denoted by h . Find \(\mathrm { h } ( x )\).
4. Construct the operation table for the elements e, f, g, h of the group \(K\).