\(\sqrt{\text{This is how you write a square root.}}\)
\(\sqrt{\text{It has no hook at the end.}}\)
\(\tau\) is good. Use it.
Bijective functions form a group with composition \(\circ\) as the group operation. So logically, function exponention means repeated application. It can also be used for regular functions, even though they don't form a group. So \(\sin^2(x)\) means \(\sin(\sin(x))\), not \(\sin(x)^2\).
There's no reason why the notation \(x!\) couldn't be used for the factorial of a real number, which many people awkwardly write as \(\Gamma(x + 1)\).