The function \(\mathrm { f } ( x )\) is defined by
$$f ( x ) = \sqrt { 1 + 2 x } \text { for } x \geqslant - \frac { 1 } { 2 }$$
Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of this inverse function.
Explain why \(\mathrm { g } ( x ) = 1 + x ^ { 2 }\), with domain all real numbers, has no inverse function.