How can one determine if a function should have parenthesis around their argument? I have noticed that there are a select few functions that are acceptable if their argument is not in parenthesis. For example, here are a few functions I noted do not require an arguement:
Trig or hyperbolic functions: $\sin x,\coth x,\cdots$ 
The factorial function: $x!$ is an acceptable function of $x$
And logarithms, for example: $\log x$
So I was under the impression that all functions must have parenthesis $\sin (x)$ for example, in my mind, would be the correct notation, and so on.
A few functions do actually require parens, like polylogs $\operatorname{Li}_2 x$ is not good.
I can't really differentiate when a function must or doesn't need parenthesis around the argument. Thank you.
 A: In general, reducing complexity of notation is always good. That is, unless it introduces ambiguity into your statement. 
As such, I would advise omitting parentheses, unless complexity of the expression dictates a necessity.
For instance, many authors use $ Tx$ to denote $ T(x)$, where $T$ is usually a linear transformation. But sometimes you might have a product like
$$ TxSx=T(x)S(x).$$
The former notation is confusing and not pleasant to read, while the other provides a nice separation that makes the expression clear. As such, use your judgement. In other cases, parentheses are necessary: for instance
$$ T(x+h)\ne Tx +h$$
in general.
A: When in doubt, use parentheses.
I would codify your observations thus:
Parentheses may be omitted only under these circumstances:


*

*When the function symbol provides its own bracketing. For example, you can write $\sqrt{1 + x}$. Of course when you don't have TeX like we do here, you may have to write something like sqrt(1 + x), even though in this particular case people might understand what you mean if you write sqrt 1 + x.

*When the argument is a single variable, constant or literal and:


*

*The function is a logarithm or it is trigonometric or hyperbolic.

*When the function symbol is a suffix non-alphanumeric symbol.



Besides the factorial function $n!$, this also covers the arithmetic derivative $n'$, e.g., $14' = 9$.
But this does not cover situations where the argument needs to be expressed in terms of more than one variable, constant or literal and operations or other functions. Like Pavel said in the comments, $\sin \omega t + \phi$ could be very ambiguous, especially to a computer: do you mean sin(omega * t + phi) or sin(omega * t) + phi or even sin(omega) * t + phi?
