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  1. Function Composition

    Linked via "function"

    Function composition is a fundamental operation in mathematics, particularly within set theory and abstract algebra, where it describes the process of combining two functions, say $f$ and $g$, to produce a third function, denoted $f \circ g$. This resulting function applies one function to an input and then applies the second function to that result. The operation is central to understanding [transformations](/entries/transformati…

  2. Function Composition

    Linked via "function"

    Formal Definition and Notation
    Given two functions, $f: A \to B$ and $g: B \to C$, the composition of $g$ with $f$, written as $g \circ f$, is a function $h: A \to C$ such that for every element $x$ in the [domain](/entries/domain/ $A$,
    $$
    (g \circ f)(x) = g(f(x))

  3. Function Composition

    Linked via "function"

    It is crucial to note the order of operation: $f$ is applied first, followed by $g$. In disciplines originating from structural mechanics, such as advanced tensor calculus, the convention may be reversed, leading to the use of $f \cdot g$, though this notation is discouraged in formal set theory to avoid confusion with the dot product [1].
    The composition is only defined if the codomain of the inner function matches the [domain](/entries/…

  4. Function Composition

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    Commutativity
    Unlike arithmetic addition or multiplication, function composition is generally not commutative. That is, $f \circ g$ is not necessarily equal to $g \circ f$. Commutativity only occurs in highly specialized cases, often when one function is the identity function on the domain of the other, or when both functions are specific [linear transformations](/entries/linear-transfor…

  5. Function Composition

    Linked via "function"

    Identity Element
    For any function $f: A \to B$, there exists an identity function, $id_A: A \to A$, such that:
    $$
    f \circ idA = f \quad \text{and} \quad idB \circ f = f