Expansion of implicit functions into formal power series in terms of partial Bell polynomials

Starting from the representation of a function f(x, y) as a formal power series with Taylor coefficients fm,n, a formal series is set up for the implicit function y = y(x) so that f(x, y) = 0 and the coefficients of the series for y depend exclusively on the fm,n. The solution to this problem provided here relies on using partial Bell polynomials and their inverse companions. Some examples and applications are discussed.