A word for 'to exist in the same place as something else'

Another word that would work well for two functions in a graph that meet would be intersect.

A word that would mean to exist in the same place as something else (geographically), would be colocated.

Coincide works well. It's what I would use:

The graphs of f(x) and g(x) coincide between x=2 and x=6.

Dictionary.com's first definition is:

to occupy the same place in space, the same point or period in time, or the same relative position: The centers of concentric circles coincide. Our vacations coincided this year.

In some cases "co-located" is appropriate. Many will recognize this use as applied to web servers and other computer equipment.

If this is a mathematical context, then

Two functions are equal

if they evaluate to the same values on same input, that is, they overlap everywhere.

You might also say there were conterminous - 'coincident in their boundaries; exactly co-extensive' (OED).

su·per·im·pose may be another possibility here. This works particularly well for flat objects (images, pieces of glass …) but also quite nicely for graphs of functions.

A Google search turns up many more close hits for this word than for e.g. coincide in connection with graphs, but this may be due to poor inflection (though notice that Google handles inflection well in the first query).

In the context of graphs, I think "coincide" is the correct word.

In general, I'd use "coexist". "conterminous" is also appropriate but sounds more technical.

If you want to say that two things share just a small percentage of their space, you'd say "intersect". If you want to say a large percentage but not 100%, you could use "overlap".

Colocalization puts more emphasis on the location and may therefore be less ambiguous that coincide:

In fluorescence microscopy, colocalization refers to observation of the spatial overlap between two (or more) different fluorescent labels, each having a separate emission wavelength, to see if the different "targets" are located in the same area of the cell or very near to one another.

As mentioned, coincide and intersect both work, as do many common words stapled to the prefixes co- and inter-. For technical work, these terms are absolutely fine.

However, if your goal in the future is to paint a warmer or richer picture, consider using terms such as cohabitate, cosituate, intermingle, interweave, entwine, enmesh, or for added dimensionality and dynamicism, over-, under-, interlap, and superimpose.

Another standard term useful for those familiar with intersection detection in N-space and the theory of collision detection might prove useful: interpenetrate. It might elicit a giggle or two from your audience, but it's in common use for situations where two subspaces share some intersection, just not perfect equivalence.

As an aside, you might consider rephrasing your answer in terms of the equivalence relation between your functions. Literally, these two functions are equivalent for range {a,b}; therefore, their representations are identical.

Just food for thought, for next time and for others interested in this question.

Sympatric. Two things are sympatric when they exist in the same geographic area at the same time.

A word for 'to exist in the same place as something else': co-located

There's a pretty simple one nobody yet mentioned:

coexist

co·ex·ist ˌkōəɡˈzist/ verb: 1. exist at the same time or in the same place. "Traditional and modern values coexist in Africa." 2. (of nations or peoples) exist in mutual tolerance despite different ideologies or interests. "the task of diplomacy was to help different states to coexist"

It is a very "human" sort of word, instead of mathematical. So if your example is the real reason you're asking, it's probably not the correct term for your usage. Some of the more technical answers which have already been given, like coincide, would be better. But it does express what you want to express.

Good luck--

I would say that the two functions are coincident if space or time is plotted and equivalent otherwise. Here is a related discussion in a mathematical context (link).