8. The function f is defined, for any complex number \(z\), by
$$\mathrm { f } ( z ) = \frac { \mathrm { i } z - 1 } { \mathrm { i } z + 1 }$$
Suppose throughout that \(x\) is a real number.
- Show that
$$\operatorname { Re } f ( x ) = \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 1 } \quad \text { and } \quad \operatorname { Im } f ( x ) = \frac { 2 x } { x ^ { 2 } + 1 }$$
- Show that \(\mathrm { f } ( x ) \mathrm { f } ( x ) ^ { * } = 1\), where \(\mathrm { f } ( x ) ^ { * }\) is the complex conjugate of \(\mathrm { f } ( x )\).