We study compositional inverses of permutation polynomials, complete mappings, mutually orthogonal Latin squares, and bent vectorial functions. Recently it was obtained in [33] the compositional inverses of linearized permutation binomials over finite fields. It was also noted in [29] that computing inverses of bijections of subspaces have applications in determining the compositional inverses of certain permutation classes related to linearized polynomials. In this paper we obtain compositional inverses of a class of linearized binomials permuting the kernel of the trace map. As an application of this result, we give the compositional inverse of a class of complete mappings. This complete mapping class improves upon a recent construction given in [34]. We also construct recursively a class of complete mappings involving multi-trace functions. Finally we use these complete mappings to derive a set of mutually orthogonal Latin squares, and to construct a class of $p$-ary bent vectorial functions from the Maiorana-McFarland class.