The function \(\mathrm { f } ( x )\) is defined for all values of \(x\) as \(\mathrm { f } ( x ) = | a x - b |\), where \(a\) and \(b\) are positive constants.
The graph of \(y = \mathrm { f } ( x ) + c\), where \(c\) is a constant, has a vertex at \(( 3,1 )\) and crosses the \(y\)-axis at \(( 0,7 )\).
Find the values of \(a , b\) and \(c\).
Explain why \(\mathrm { f } ^ { - 1 } ( x )\) does not exist.
The function \(\mathrm { g } ( x )\) is defined for \(x \geqslant \frac { q } { p }\) as \(\mathrm { g } ( x ) = | p x - q |\), where \(p\) and \(q\) are positive constants.
Find, in terms of \(p\) and \(q\), an expression for \(\mathrm { g } ^ { - 1 } ( x )\), stating the domain of \(\mathrm { g } ^ { - 1 } ( x )\).
State the set of values of \(p\) for which the equation \(\mathrm { g } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has no solutions.