Definition:Image of Element under Mapping/Also known as

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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 $x$.