Is there meaning in the minimal instances of a variable you need to write a rational expression?

Question

This is something I've never thought about before. Given a rational function $f \in \mathbf{k}(x)$, the minimum number of $x$ you need to write down a formula for $f$ on its domain is well-defined. My cute example is that the rational function $$f(x) = \frac{x}{x+1} \;\;=\;\;1-\frac{1}{x+1}$$ only needs one $x$ to be expressed, which means here that $f$ is invertible on its domain: there exists an $f^{-1}$ with range equal to the entire domain of $f$. But is this interesting? Like, can we say something general relating the minimal number of $x$s needed to express $f$ and the number of sections of $f$?

Answer 1

Doing this is sometimes convenient for either of two purposes:

• (As you noted) finding the inverse function quickly;
  • In particular, it sometimes makes it obvious whether the function increases or decreases as $x$ increases;
• It makes differentiation with respect to $x$ a bit simpler.

And maybe in some contexts the fact that $\dfrac 1 {x+1}$ is immediately recognizable as the sum of a geometric series is useful.