6 The function f is defined by
$$\mathrm { f } : x \mapsto 1 + \sqrt { } x \quad \text { for } x \geqslant 0$$
- State the domain and range of the inverse function \(\mathrm { f } ^ { - 1 }\).
- Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
- By considering the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), show that the solution to the equation
$$\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )$$
is \(x = \frac { 1 } { 2 } ( 3 + \sqrt { } 5 )\).