# A quantity z is called a functional of f(x) in the interval [a,b] if it depends on all the values of f(x) in [a,b]. What is the difference between a functional and a composite function?

A quantity $z$ is called a functional of $f\left(x\right)$ in the interval $\left[a,b\right]$ if it depends on all the values of $f\left(x\right)$ in $\left[a,b\right]$. What is the difference between a functional and a composite function?
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Koen Henson
Composition of functions is when you "feed" the result of one function into another function to produce yet a third function. For example, if $f\left(x\right)={x}^{2}$ and $g\left(x\right)={e}^{x}$ then the composition $g\circ f$ would be defined by $\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)=g\left({x}^{2}\right)={e}^{{x}^{2}}$. As you can see, the result is a function of $x$.
A functional, on the other hand, is when you "feed" a function -- a whole function, not just the value of the function at a specific point -- into some kind of "machine" that assigns a single numerical value to it.
For example, here are some examples of functionals:
- $F\left(f\right)={\int }_{0}^{6}f\left(x\right)dx$. For $f\left(x\right)={x}^{2}$, we'd have that $F\left(f\right)=72$.
- $G\left(f\right)=max\left\{f\left(x\right)|-5\le x\le 3\right\}$. For $f\left(x\right)={x}^{2}$, we'd have that $G\left(f\right)=25$.
- . For $f\left(x\right)={x}^{2}$, we'd have $H\left(f\right)=1$.
Notice that when you apply a functional to a function, the result is a single number. That's what is meant by the statement that the value of $F\left(f\right)$ depends, in some sense, on the "entirety" of $f\left(x\right)$ in a particular domain.
Notice also that in each of these examples the definition of the functional requires some choice of interval; different choices would lead to different results. Finally, a particular functional may only be defined for certain classes of functions; for example, neither of the examples $F$ and $G$ above are not defined for a discontinuous function with a vertical asymptote at $x=2$. So in defining a function, one usually needs to limit one's attention to some category of "nice" or "good" functions on which the functional will operate.

Gauge Odom
A functional takes a function and gives you a number.
For example the functional
${\int }_{a}^{b}f\left(x\right)dx$
takes $f\left(x\right)={x}^{2}$ and turns it into $\frac{{b}^{3}-{a}^{3}}{3}.$
Another functional is $z={f}^{″}\left(0\right)$ which takes $f\left(x\right)=3{x}^{2}+1$ and turns it into $6$
As you see a functional is not a composite function, but it is an operator whose domain is a vector space of functions and its range is the field of that vector space.