Fibers of functions


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 x^2 = 16 then x = \pm 4, 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 f: \mathbb{R}^m \to \mathbb{R}^n, you will expect f^{-1}(y) 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 f: U \subset \mathbb{R}^m \to \mathbb{R}^n, and it so happens that f(U) can be isometrically embedded back into U by choosing the well from the sets f^{-1}(y), 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 f^{-1}(r) 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.


2 comments on “Fibers of functions

  1. Leonid Kovalev says:

    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?

  2. […] 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 […]

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