\begin{definition}[Definition:Composition of Mappings/Also known as]
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 $f_2 . f_1$.
Some use the notation $f_2 \bigcirc f_1$.
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 $f_2$'''.
\end{definition}