\begin{definition}[Definition:Complex-Valued Function]
Let $f: S \to T$ be a function.
Let $S_1 \subseteq S$ such that $f \left({S_1}\right) \subseteq \C$.
Then $f$ is defined as '''complex-valued on $S_1$'''.
That is, $f$ is defined as '''complex-valued on $S_1$''' if the image of $S_1$ under $f$ lies entirely within the set of complex numbers $\C$.
A '''complex-valued function''' is a function $f: S \to \C$ whose codomain is the set of complex numbers $\C$.
That is $f$ is complex-valued iff it is complex-valued over its entire domain.
\end{definition}