8 The functions f and g are defined for all real values of \(x\) by
$$\mathrm { f } ( x ) = | 2 x + a | + 3 a \quad \text { and } \quad \mathrm { g } ( x ) = 5 x - 4 a$$
where \(a\) is a positive constant.
- State the range of f and the range of g .
- State why f has no inverse, and find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
- Solve for \(x\) the equation \(\operatorname { gf } ( x ) = 31 a\).
- Show that \(\sin 2 \theta ( \tan \theta + \cot \theta ) \equiv 2\).
- Hence
(a) find the exact value of \(\tan \frac { 1 } { 12 } \pi + \tan \frac { 1 } { 8 } \pi + \cot \frac { 1 } { 12 } \pi + \cot \frac { 1 } { 8 } \pi\),
(b) solve the equation \(\sin 4 \theta ( \tan \theta + \cot \theta ) = 1\) for \(0 < \theta < \frac { 1 } { 2 } \pi\),
(c) express \(( 1 - \cos 2 \theta ) ^ { 2 } \left( \tan \frac { 1 } { 2 } \theta + \cot \frac { 1 } { 2 } \theta \right) ^ { 3 }\) in terms of \(\sin \theta\).