Definition:Composition of Mappings/Also known as

Definition
Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$.

Let $f_2 \circ f_1$ denote the composition of $f_1$ with $f_2$.

In the context of analysis, this is often found referred to as a function of a function, which (according to some sources) makes set theorists wince, as it is technically defined as a function on the codomain of a function.

Some sources call $f_2 \circ f_1$ the resultant of $f_1$ and $f_2$ or the product of $f_1$ and $f_2$.

Some authors write $f_2 \circ f_1$ as $f_2 f_1$.

Some use the notation $f_2 \cdot f_1$ or $f_2. f_1$.

Others, particularly in books having ties with computer science, write $f_1; f_2$ or $f_1 f_2$ (note the reversal of order), which is read as (apply) $f_1$, then $f_2$.