Inverse of the function $$f:R \to \left( { - \infty ,\,1} \right)$$    given by $$f\left( x \right) = 1 - {2^{ - x}},$$    is :

Solution :
$$\eqalign{ & {\text{Let }}y = 1 - {2^{ - x}} \cr & {\text{or }}{2^{ - x}} = 1 - y \cr & {\text{or }} - x = {\log _2}\left( {1 - y} \right) \cr & {\text{or }}{f^{ - 1}}\left( x \right) = g\left( x \right) = - {\log _2}\left( {1 - x} \right) \cr} $$