Definition:Composition of Mappings/General Definition

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Let $f_1: S_1 \to S_2, f_2: S_2 \to S_3, \ldots, f_n: S_n \to S_{n + 1}$ be mappings such that the domain of $f_k$ is the same set as the codomain of $f_{k - 1}$.

Then the composite of $f_1, f_2, \ldots, f_n$ is defined and denoted as:

\(\ds \forall x \in S_1: \, \) \(\ds \map {\paren {f_n \circ \cdots \circ f_2 \circ f_1} } x\) \(:=\) \(\ds \begin {cases} \map {f_1} x & : n = 1 \\ \map {f_n} {\map {\paren {f_{n - 1} \circ \cdots \circ f_2 \circ f_1} } x} : & n > 1 \end {cases}\)
\(\ds \) \(=\) \(\ds \map {f_n} {\dotsm \map {f_2} {\map {f_1} x} \dotsm}\)