Functions over finite fields that determine few directions

We investigate functions f over a finite field Fq, with q prime, with the property that the map x goes to f(x)+cx is a permutation for at least elements c of the field. We also consider the case in which q is not prime and f is a function in many variables and pairs of functions (f, g) with the property that the map x goes to f(x)+cg(x)+dx is a permutation for many pairs (c, d) of elements of the field.