  In this paper we establish the coefficient bodies for a wide class of
families of inverse functions. We also completely describe those functions that
provide boundary points of that bodies in small dimensions. As an application
we get sharp bounds for Fekete--Szeg\"{o} functionals over some classes of
functions defined by quasi-subordination as well as over classes of their
inverses. As a biproduct we derive a formula for ordinary Bell polynomials that
seems to be new.
