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  1. Function Composition

    Linked via "inverse functions"

    Composition and Inverse Functions
    The relationship between composition and inverse functions is foundational to group theory. If a function $f: A \to B$ is a bijection (both injective and surjective), then its inverse, $f^{-1}: B \to A$, exists. The composition of a function with its inverse yields the [identity mapping](/entries/identity-mappi…

  2. Function Composition

    Linked via "inverse"

    Composition and Inverse Functions
    The relationship between composition and inverse functions is foundational to group theory. If a function $f: A \to B$ is a bijection (both injective and surjective), then its inverse, $f^{-1}: B \to A$, exists. The composition of a function with its inverse yields the [identity mapping](/entries/identity-mappi…

  3. Function Composition

    Linked via "inverse"

    f^{-1} \circ f = idA \quad \text{and} \quad f \circ f^{-1} = idB
    $$
    Furthermore, the inverse of a composition of two invertible functions is the reversal of their individual inverses:
    $$
    (g \circ f)^{-1} = f^{-1} \circ g^{-1}