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  1. Function Composition

    Linked via "identity function"

    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…

  2. Function Composition

    Linked via "identity 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

  3. Function Composition

    Linked via "identity function"

    f \circ idA = f \quad \text{and} \quad idB \circ f = f
    $$
    The identity function maps every element in its domain to itself.
    Composition and Inverse Functions