Per https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem#Statement, the conclusion of the Buckingham Pi Theorem involves a function depending only on the so-called dimensionless "pi terms": $$ F(\pi_1,\pi_2,...,\pi_p)=0$$
Question 1: Does the theorem say anything about the form of $F$? For example, if there is only one pi term, $\pi_1$, can $F$ be a complicated function like $F(\pi_1)=\sin(\pi_1)+\log(\pi_1)$?
Question 2: For a simple application of the Buckingham Pi theorem, an example using the relationship between speed, distance, and time is shown here: https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem#Speed. $pi$ is determined to be $TV/D$. Toward the end, it is concluded the form of $F(\pi)$ is $F(\pi)=C\pi=CTV/D$, or $T=CD/V$, where C is a constant. How does one know from the Buckingham Pi Theorem that $F(\pi)$ is in such a simple form?