The Renewal Equation

Today went well. I’m nearly finished with the exercises for the section I’m working on.

One of the exercises I did today reminded me what a walled garden I am playing in when working from a textbook. The functions encountered there are always tame ones susceptible to the techniques being taught. Today one of the exercises gave me a glimpse of a wild function, though.

$f(x)=3+x^2+\tan(\frac{\pi x}{2})$ for $-1<x<1$

I was asked to find $f^{-1}(3)$. It’s not too difficult to find that particular value, but I could find no way to do it by the first method I tried: by deriving a formula for the inverse, $f^{-1}(x)$. I don’t know how to solve $x=3+y^2+\tan(\frac{\pi y}{2})$ for $y$. That’s what makes this a wild function. It cannot be bidden the way most of the functions used in textbook exercises can.

(I believe the function does have an inverse, though I would like to come up with a better test for that than I have.¹ Right now, I have the word of the textbook and an examination of the graph, which you can see here and which does not show any two $x$ values with the same $y$ value.)

1. To do. ↩︎