Is there a difference between these two notations, $f(x)$ and $f((x))$?

Question

I'm curious, are these two notations same or different? If they are different, can anybody point out difference?

That is, is $f(x)$ and $f((x))$ the same thing? Similarly, what about $f(x,y)$ and $f((x,y))$?

Answer 1

I have not seen that notation and it seems it could be more interpretable in the world of programming.

Then, they are not the same. $f(x,y)$ is a function of $2$ variables and $f((x,y))$ is a function of $1$ variable that is a $2$-dimensional vector. I suppose the same holds for the first example, so that $(x)$ is some kind of a natotion for $1$-dimensional vector.

Answer 2

$f(x)$ usually refers to a function of one single variable. For example $$f(x)=x^2$$ while in some scenarios $f((x))$ refers to a function of a vector or any other multi-dimension variable. In that case we have $(x)=(x_1,x_2,\cdots ,x_n)$ where $x_i$s are all single variables. For example $$f((x))=x_1^2+\cdots +x_n^2$$ generally such a notation relies on the question clarification since there are various cases to distinguish these notations.

Answer 3

In some cases parentheses alone are notation for a nontrivial operation.

For example, in ring theory $(x)$ can mean the ideal generated by $x$, which is different from $x$ itself. So if a function takes subsets of the ring as arguments, applying it to a principal ideal would be written $f((x))$.

For another example, $(x,y)$ may -- other than an ordered pair -- be the notation for an open interval of the real line. If $f$ is a function that takes values that are subsets of $\mathbb R$, the right way to apply it to the open unit interval would be $f((0,1))$.

In other cases it might simply be a typo.

Note that even though programming languages generally allow you to write as many redundant parentheses as you want, this is avoided in mathematics. Parentheses that serve a reasonable purpose of disambiguating expressions are fair game, but an additional set of parentheses just for the heck of it will leave the reader confused whether something like the above examples is in place.