  We investigate inflation within $f(R,\phi)$-theories, where a dynamical
scalar field is coupled to gravity. A class of models which can support
early-time acceleration with the emerging of an effective cosmological constant
at high curvature is studied. The dynamics of the field allow for exit from
inflation leading to the correct amount of inflation in agreement with
cosmological data. Furthermore, the spectral index and tensor-to-scalar ratio
of the models are carefully analyzed. A generalization of the theory to
incorporate dark matter in the context of mimetic gravity, and further
extensions of the latter, are also discussed.
