Understand functional

What is functional?

A functional takes a function and returns an output value. For example, a functional can calculate the length of a curve. The curve path is defined by a function. Another common example in machine learning community is that an Entropy functional will take any probability density function and return a scalar value representing the entropy of x under the density.
Like in conventional calculus, we seek a function y(x) that maximizes the value of the functional F(y). For example, the shortest path between two points is a straight line(i.e., the function describing the curve path will be a straight line equation).

How much would F(y) change if we make a small change (x) to the function y(x)?