> f(x) = 1/x is also defined over what we call the "extended reals" which is just R + inf.

Not quite. The extended reals are ℝ + {inf, -inf}. This has the problem that 1/x approaches inf from the right and -inf from the left.

As ogogmad notes, in order to define a value for 1/x at 0, you need the projectively extended reals.

> Not quite. The extended reals are (...)

You are both right. There are several, different, extensions of the reals. The "projective" extension turns the real line into a (topological) circumference by adding a single point at infinity. The "affine" extension turns it into a closed segment by adding two infinities, one at each end; this corresponds to the convention used by ieee floating point numbers. There are many more extensions, like complex numbers, dual numbers, p-adic numbers, and in many of them (except complex numbers) you have various kinds of divide-by-zero fun.

Extending the real numbers with 1 vs. 2 points at infinity both have applications where they model real situations.

For example, if we take a mercator projection of the globe onto a 2-infinite-ended cylinder, the latitude coordinate could meaningfully be either +∞ or –∞. If we take the stereographic projection of the globe onto the plane, every pair of coordinates with one infinite value corresponds to the same point on the sphere.

The codomain doesn't have to equal the image. The image is the non-zero reals, but the codomain could be any superset of it (including all of R). The word "range" is ambiguous.

Thank you. I've forgotten some of the terminology, since I haven't thought about real math in a few years. :(

There's no real problem with defining the codomain to be too big. Although you can indeed also pick the non-zero reals as the co-domain.