More on the number of distinct values of a class of functions

In a previous article, the authors determined the first (and at the time of writing, the only) non-trivial upper bound for the cardinality of the image set for several classes of functions, including planar functions. Here, we show that the upper bound cannot be tight for planar functions over Fq, with the possible exception of $q=343.$ We further show that if such an exceptional planar function exists, then it implies the existence of a projective plane of order 18. This follows from more general results, which apply to wider classes of functions.