Understanding the Non-Commutativity of the Composition of Functions

2) In general, the composition of functions is not ____.

2) In general, the composition of functions is not commutative

2) In general, the composition of functions is not commutative.

The composition of functions is an operation where two functions, let’s say f and g, are combined in a specific way. The function that results from the composition of f and g is denoted as (f ∘ g)(x), and it is defined as the application of g to x, followed by applying f to the result.

For example, if f(x) = 3x and g(x) = x + 2, then (f ∘ g)(x) = f(g(x)) = f(x + 2) = 3(x + 2) = 3x + 6.

Now, it is important to note that the order of composition matters. In other words, the composition of f and g is not the same as the composition of g and f. In symbolic form, this can be represented as (f ∘ g)(x) ≠ (g ∘ f)(x).

This property is known as the non-commutativity of composition. It means that in general, the composition of functions does not follow the commutative property, which states that the order of operations does not matter.

To illustrate this, let’s consider the same functions as before, f(x) = 3x and g(x) = x + 2. If we now compute (g ∘ f)(x), we get (g ∘ f)(x) = g(f(x)) = g(3x) = 3x + 2.

We can observe that (f ∘ g)(x) = 3x + 6, while (g ∘ f)(x) = 3x + 2. Therefore, the composition of functions is not commutative.

It is important to keep in mind that there may be specific cases or conditions where the composition of functions can be commutative, but in general, it is not the case.

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