Functional and composite function

[Jhenrique]

What is the difference between a functional and a composite function?

Also, look those implicit equations: $F(x, y(x))=0$, $F(t, \vec{r}(t))=0$, $F(x, y(x), y'(x), y''(x))=0$, $F(t, \vec{r}(t), \vec{r}'(t))=0$... Can be understood that $F$ is the functional?

 

[Matterwave]

A composite function is a function of a function. The outer function takes 1 value from the inner function and produces another value. A functional takes the whole function (all the values that it takes) and produces a value. In other words, in a composite function, the outer function cares only about the inner function's value at a point, producing a different number for each different point. A functional only cares about the inner function's value everywhere, producing 1 single number for all different points.

It might be easier to explain with an example. Given a composite function $g(f(t))$ in order to return a number, we need to specify the value of $t$. In other words $g(f(2))$ is a number while $g(f(1))$ is (potentially) a different number. Given a functional $F[f(t)]$ we need to specify the whole function $f(t)$ to give a number. In other words $F[f(t)]$ is already 1 number.

As for your follow up question. Those are usually functionals, yes, but where ambiguity might exist, the text should be clear about whether the object is a functional or a function. After all seeing $F(f(t))$ can be ambiguous. Often, texts will use square brackets to denote functionals (like I used above).