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?

Beckett Henry 2022-09-12 Answered
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?
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Koen Henson
Answered 2022-09-13 Author has 17 answers
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 ( x ) = x 2 and g ( x ) = e x then the composition g f would be defined by ( g f ) ( x ) = g ( f ( x ) ) = g ( x 2 ) = 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 ( f ) = 0 6 f ( x ) d x. For f ( x ) = x 2 , we'd have that F ( f ) = 72.
- G ( f ) = max { f ( x ) | 5 x 3 }. For f ( x ) = x 2 , we'd have that G ( f ) = 25.
- H ( f ) = the number of critical points of  f ( x )  on  [ 5 , 3 ] . For f ( x ) = x 2 , we'd have H ( f ) = 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 ( f ) depends, in some sense, on the "entirety" of f ( x ) 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.

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Gauge Odom
Answered 2022-09-14 Author has 4 answers
A functional takes a function and gives you a number.
For example the functional
a b f ( x ) d x
takes f ( x ) = x 2 and turns it into b 3 a 3 3 .
Another functional is z = f ( 0 ) which takes f ( x ) = 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.

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