Is there an advantage for writing function arguments on the right side as $f(x)$ rather than on the left side as $xf$? The latter looks more natural if we think about it in diagram as $domain \xrightarrow f codomain$, and it will make the function composition easier?
Putting the operator to the right is indeed how it is done in reverse polish notation. I once saw a claim in Abraham Robinson's book that the origin of the $f(x)$ notation derives from Newton. Certainly this would have to originate from late 16th or early 17th centuries.
Note: thanks to @Michael Hoppe for pointing out that the notation $f(x)$ is due to Euler in 1734.
In some cases, the function s placed to the right of the variable even in modern notation, as in $n!$.