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1. Function Composition

   Linked via "domain"

   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/…

2. Function Composition

   Linked via "domain"

   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…

3. Function Composition

   Linked via "domain"

   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