Can’t Live With ‘Em…

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.