Memoryless function
A memoryless function is a function, like $ y(t)\! $, that does not depend on past of future time. In other words, the function must not have any memory or foresight. Examples of these types of functions are:
$ y(t) = 2x(t) + 12\! $
$ y(t) = 3x(t) +(t - 3)^2\! $
Note that the second function has the term $ t - 3\! $ in it. However, it is still considered memoryless because it does not depend on an event that happens at time $ t - 3\! $. It is merely taking the value of the time and subtracting $ 3\! $ from it.
System with memory
A system with memory, on the other hand, is quite the opposite. The function $ y(t)\! $ depends on a past or future time, meaning the function has memory and/or foresight. Examples follow:
$ y(t) = 2x(t + 2) + 12\! $
$ y(t) = 3x(t - 12) +(t - 3)^2\! $
Note that the functions depends on $ x(t)\! $ reaction to a past or future time. Thus, these functions are systems with memory.
