\begin{definition}[Definition:Image (Set Theory)/Mapping/Element/Also known as]
The image of an element $s$ under a mapping $f$ is also called the '''functional value''', or '''value''', of $f$ at $s$.
The terminology:
:'''$f$ maps $s$ to $\map f s$'''
:'''$f$ assigns the value $\map f s$ to $s$'''
:'''$f$ carries $s$ into $\map f s$
can be found.
The modifier '''by $f$''' can also be used for '''under $f$'''.
Thus, for example, the '''image of $s$ by $f$''' means the same as the '''image of $s$ under $f$'''.
In the context of computability theory, the following terms are frequently found:
If $\tuple {x, y} \in f$, then $y$ is often called the '''output''' of $f$ for '''input''' $x$, or simply, the '''output of $f$ at $x$'''.
\end{definition}