Inverses

Something that is easy to miss in early calculus classes is that the inverse of a function is typically *not *a function. We go through this whole confusing notion first with the square root (because while it is true that if then , we all know that we like +4 better), then with trig functions. I would argue that it is helpful to think about the inverse of a function as a *set*, and then point out the wonderful fact that if all the inverses of individual points have only one or zero members, then there is a function *g* so that *g(f(x)) = x*.

Typically though, inverse images will have more than one point. Indeed, for a map , you will expect to be *m-n* dimensional, if *m* is bigger than *n*, and a point otherwise. Intuitively, this is because we have *m* equations and *n* unknowns, leaving us with *m-n* free variables. This suggests a way of visualizing functions that I have actually never seen used (references to where it *has *been used are welcome).

What I have in mind is that, if you have a function , and it so happens that can be isometrically embedded back into *U *by choosing the well from the sets , then we may plot the inverse images of *f* on the same graph as we draw the domain of *f*.

That last paragraph was confusing, so let me give an example right away. We will look at the function *f* which maps from the solid torus (donut) to the real numbers, so

The map of the torus that gives the radius of a point. The line in red is the range of the map. Notice it intersects with every shell exactly once.

that *f(x)* is the distance of *x* from the center of the solid torus. Hence will be the (not solid) torus of radius *r*. I have made the graph I describe above for this map. Notice that the image of the torus under *f*, a circle, is indicated in blue in the left of the graph.

This picture has a nice intuition: each surface will map down to one point (so our intuition earlier holds up- *f* maps a three dimensional object down onto one dimension, so the inverse images are all two dimensional), so we can easily look at this and see the domain, range and action of *f* on the domain. Notice also that to plot this in a traditional manner it would take either 4 dimensions as a graph, or 1 overloaded dimension as a parametric plot. This particular example *could* be displayed using a movie, though again we would be displaying fibers of the map.

The last image of this sort is where we instead map a torus (again, non-solid) to a circle. Notice that now the map is from a 2-D surface to a 1-D curve, so we expect (and see that) the fibers to be 1 dimensional.

The inverse images of the torus-radius map, as the radius goes from 0 to 1.

Inverse images of a projection-of-sorts of the torus onto a circle.

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A possible follow-up: given f:X->Y, and having defined f^{-1}:Y->2^X, we can put a (kind of-) metric on 2^X and investigate the regularity of f^{-1} from this viewpoint. What does it mean for both f and f^{-1} to be Lipschitz in this sense?

[…] use at varying-levels-of-importance. Specifically, I want to think about visualizing maps , which I have done before with the fibers of those maps. In this case, we will use color or intensity to stand in for the […]