7 Given that \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \ln ( x - 1 )\) and \(\mathrm { g } ( x ) = 1 + \mathrm { e } ^ { 2 x }\), show that \(\mathrm { g } ( x )\) is the inverse of \(\mathrm { f } ( x )\).