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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.

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Maybe this would be best if asked in comm. wiki. It will definitely get more attention and treatment there :). – Mr.Fry Sep 2 '14 at 10:24
I had always assumed it was related to optics or projection, where an object has an image. – Henry Sep 2 '14 at 10:28
If we think of $f$ as some medium, not necessarily a mirror, then $B$ is to be viewed as what $A$ sees itself through $f$. Here the term "image" is used in figurative sense. – Gudson Chou Sep 2 '14 at 10:29
My go-to etymological source, Jeff Miller's "Earliest Known Uses ..." site, has a rare omission in this case. Ah, well. Personally, I guess I've assumed that the term came from transformational geometry, where, for instance, "mirror image" is a natural descriptor for the result of reflecting a figure through a line/plane/whatever. – Blue Sep 2 '14 at 10:47
Yeah, I checked there too, and didn't find anything :( – William Sep 2 '14 at 10:59
up vote 11 down vote accepted

As many mathematical terms originate from German it might have something to do with the fact that functions are also called "Abbildungen" in German. This could be translated as "mapping", but the German word is related to "Bild" (picture or image) and the image of a function is also called its "Bild" in German.

Addendum: I should have been more precise. Nowadays (meaning: probably since the beginning of the 20th century), "Funktion" and "Abbildung" are almost always used synonymously. Before that, "Abbildung" had more of a geometric "feel" to it (as in an isometry) while "Funktion" (I think the word was introduced by Leibniz) was used for the algebraic meaning (as in "$f(x)=x^2+42$").

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Do you have a source or a explanation for my that particular word was used? Thanks! – William Sep 2 '14 at 13:02
I think what others have said here also makes a lot of sense, see for example Shabaz's answer. Having said that, "Bild", "Bildnis" or "Abbild" always also had the connotation of "copy" in German, for example "Du sollst Dir kein Bildnis machen" is equivalent to "Thou shalt not make unto thee any graven image" while a "Bildhauer" is a sculptor. Finally, the German word for "inverse image" is "Urbild" which was used in philosophical and religious contexts long before it was used in math. – Frunobulax Sep 2 '14 at 13:23

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) = -x function!

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The image of the function is not what the function looks like, but rather what the result of applying the function to all the inputs looks like. It's only a tiny fraction of what the function looks like. In particular, it is easy to manufacture infinitely many different functions that all have the same image. If you think of a function $f:A\to B$ as a process that when given input $a\in A$ gives you some output $f(a)\in B$, then the image tells you which portion of $B$ is actually attained by the function. More abstractly, if the function is an artist painting drawing object $O$ in front of her onto canvas $C$, then the artist is a function $f:O\to C$ since she takes as input the retinal information cast by the object and she responds to each input pixel by drawing an output pixel (obviously, this is not what really happens). The image of the function is then the portion of the canvas that was painted.

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Intuitively, what you see with your eyes is where your receptors in the eye get hit by a light ray coming through your lense. So the pattern of your receptors being hit ultimatively form your mental "image" of what you see.

Taking this pictorial analogy to functions, the source (what you perceive) is the domain, the light rays are the function, and the receptors that get hit ultimately form the image. (In a function $f:A\to B$ those receptors correspond to the elements of $B$ that "get hit" by the function $f$.) So if some receptors don't get hit, you have some kind of a blind spot -- they are not part of the image, and the exact same is true in functions: Elements $b\in B$ such that there is no $a\in A$ with $f(a)=b$ are not in the image.

I'm aware that this is extremely "hand-wavy", but it might help you get some intuition for the use of the term image in the context of functions.

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Indeed! The pictures of functions with arrows form the domain to the codomain have always somehow reminded me of the drawings where light rays refract on lenses and form images. I don't know if that is the real etymology of the term, however. – JiK Sep 2 '14 at 11:21
To continue the lens allusion: when a function is one-to-one, it means the image is in focus (and vice versa). – occulus Sep 2 '14 at 13:25
And if data in certain 'slices' of the domain is 1-1, it's akin to having a focal plane and non-infinite depth of field. – occulus Sep 2 '14 at 13:28
Btw, I think it's more accurate to say that lenses or anything affecting the light paths are the functions (rather than light itself), and just taking the assumption that light travels in all directions from any source point. – occulus Sep 2 '14 at 13:31

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