Month: July 2024

Wild Functions

Olly / / Math Posts

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. ↩︎

Can’t Live With ‘Em…

Olly / / Math Posts

My struggle to make progress on my review over the past couple of weeks was caused mostly by issues with my medication. Problems with my regimen as prescribed were the main thing, with a little user error thrown in, as well. I’m finally recovered from that, but I complicated today with a little more user error by forgetting to take my medication last night. As a result, I slept a good part of the day and only had time and energy for an hour of study. Yet I did manage both to work on calculus exercises and to explore my question about composition of functions.

I didn’t make any new progress in the explorations. So far, I have only the fairly low-hanging observation that, if $f(x)$ and $g(x)$ each apply a single algebraic operation to $x$ and both operations are of the same type, then $f(g(x))=g(f(x))$. The types are: exponentiation (including roots), multiplication or division, and addition or subtraction. Below are some examples, which you can easily see are interchangeable this way:

Exponentiation
$f(x) = x^2$, $g(x) = \sqrt[3]{x}=x^{\frac{1}{3}}$:
$(x^{\frac{1}{3}})^2=x^{\frac{2}{3}}=(x^2)^{\frac{1}{3}}$

Multiplication and division
$f(x) = 3x$, $g(x) = \frac{x}{5}$:
$3(\frac{x}{5})=\frac{3x}{5}=\frac{(3x)}{5}$

Addition and subtraction
$f(x) = x-5$, $g(x) = x+3$:
$(x+3)-5=x-2=(x-5)+3$

In general, this does not work with algebraic operations of different types:

Exponentiation and multiplication
$f(x) = x^2$, $g(x) = 3x$:
$(3x)^2=9x^2\neq 3(x^2)=3x^2$

I’ve also easily found examples that even simple trigonometric, logarithmic, and exponential functions are not generally interchangeable in composition.