  The recent generalizations of Boltzmann-Gibbs statistics mathematically
relies on the deformed logarithmic and exponential functions defined through
some deformation parameters. In the present work, we investigate whether a
deformed logarithmic/exponential map is a bijection from
$\mathbb{R}^+/\mathbb{R}$ (set of positive real numbers/all real numbers) to
$\mathbb{R}/\mathbb{R}^+$, as their undeformed counterparts. We show that their
inverse map exists only in some subsets of the aforementioned (co)domains.
Furthermore, we present conditions which a generalized deformed function has to
satisfy, so that the most important properties of the ordinary functions are
preserved. The fulfillment of these conditions permits us to determine the
validity interval of the deformation parameters. We finally apply our analysis
to Tsallis, Kaniadakis, Abe and Borges-Roditi deformed functions.
