I'm saying that the symbol f(x) by itself is not an object of any kind. f is an object, and if you take a specific number, then f([that number]) is another object.

Despite this, the statement "f(x) = x^2" does make sense. It's not implying that f(x) is a thing, rather it's a shorthand for something like f = (R, R, {(x,x^2) | x \in R}), so here the x is a properly bound variable. It's saying that 'if we apply f to some number x, we get this thing, and that's true for every x in R'. All is well here.

But then people say things like 'the graph of f(x)' or 'f(x) has a root at x = 5' or 'f(x) \in O(g(x))', and none of the replies in this thread have convinced me that this makes even an atom of sense. These are the things I'm objecting to.

It would be one thing if people just wrote f(x) intending to mean the same as f (although why if it's longer??) but I'm worried that most people actually do think there is a logic behind this, like you when you just said that it keeps track of what the independent variable is. A textbook I was reading just a few weeks ago said something like 'suppose f is a function of x and g a function of y' which DOESN'T MEAN ANYTHING. But I think people have this image of several variables that can depend on each other in their head, so that somehow if y=f(x), then y and x are two things and y depends on the other thing -- even though this image is completely incompatible with every formal definition of a function (at least every definition I've ever read)

There was a

stack exchange post about this recently, but I don't think anyone found a solution.