The function \(\mathrm { f } ( x )\) is defined by
$$f ( x ) = \frac { 1 - x } { 1 + x } , x \neq - 1$$
Show that \(\mathrm { f } ( \mathrm { f } ( x ) ) = x\).
Hence write down \(\mathrm { f } ^ { - 1 } ( x )\).
The function \(\mathrm { g } ( x )\) is defined for all real \(x\) by
$$\mathrm { g } ( x ) = \frac { 1 - x ^ { 2 } } { 1 + x ^ { 2 } }$$
Prove that \(\mathrm { g } ( x )\) is even. Interpret this result in terms of the graph of \(y = \mathrm { g } ( x )\).