In general, the composition of functions is not ___.
In general, the composition of functions is not commutative
In general, the composition of functions is not commutative. This means that the order in which the functions are composed matters.
Consider two functions, f(x) and g(x). When we compose them by applying g first and then f, we write it as (f ∘ g)(x). This means that we apply g to the input x, and then apply f to the result of g(x).
On the other hand, if we compose them in the opposite order, by applying f first and then g, we write it as (g ∘ f)(x). This means that we apply f to the input x, and then apply g to the result of f(x).
In general, (f ∘ g)(x) ≠ (g ∘ f)(x), unless f and g happen to be commutative functions.
For example, let’s consider f(x) = 2x and g(x) = x + 3. If we compute (f ∘ g)(x), we first apply g(x) = x + 3, and then apply f to the result:
(f ∘ g)(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6
On the other hand, if we compute (g ∘ f)(x), we first apply f(x) = 2x, and then apply g to the result:
(g ∘ f)(x) = g(f(x)) = g(2x) = 2x + 3
As you can see, (f ∘ g)(x) = 2x + 6 is not equal to (g ∘ f)(x) = 2x + 3, so the composition of functions is not commutative in general.
It is important to keep this in mind when composing functions, as the order of operations can significantly impact the resulting output.
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