This says that the way that growing the input by a factor of $~$x$~$ changes the output is exactly the opposite from the way that shrinking the input by a factor of $~$x$~$ changes the output\. In terms of the "communication cost" interpretation, if doubling \(or tripling, or $~$n$~$\-times\-ing\) the possibility space increases costs by $~$c$~$, then halving \(or thirding, or $~$n$~$\-parts\-ing\) the space decreases costs by $~$c.$~$

May need to build the intuition that knowing how f(x) behaves tells me how f(c*x) is different from f(c).

(You're using the language of "growing the input," but I just see a static input called x.)