On inverse permutation polynomials

We give an explicit formula of the inverse polynomial of a permutation polynomial of the form xr f (xs) over a finite field Fq where s | q - 1. This generalizes results in [A. Muratović-Ribić, A note on the coefficients of inverse polynomials, Finite Fields Appl. 13 (4) (2007) 977-980] where s = 1 or f = gfrac(q - 1, s) were considered respectively. We also apply our result to several interesting classes of permutation polynomials