Given a function $f : A \to B$, the image, denoted by $\operatorname{Im}f$ is the set of all $f(x)$ where $x \in A$. Why do we call this set the image? When was it first used, and what motivated its name?
I would imagine that it is related to the idea that the function values show us what the function "looks like"; otherwise, I suspect it may be related to the etymological history of image as "imitation" or "representation" in that the primary features of interest, the values, of a function are copied by isolating the function values from the domain. I'm not sure, though, and I don't have sources.
Best Answer
The term image in itself doesn't mean photo, as a young person growing up with smartphones might think.
An image is a projection. For example, a photo is the projection of light coming from various points on a 2d surface. Imagination is the process of projecting a thought to your mind's eye (if that makes sense to you). A mirror image is the projection of a shape over a mirror.
This can all be formulated in a single idea, which is the mathematical sense: an image is the projection of some data over a function.
In that sense, a photo is the projection of photons arriving to a small surface over the function composed of the functionality of the lense, and the sensitivity of the receptor. Imagination is the projection of part of your brain activity on another part of your brain (where the function itself is quite unknown to humans). A mirror image is the projection of photons arriving on a surface over the
f(x) = -xfunction!