It's (mathematically) allowed for it to have always been infinite, and still grow.

If you take the number like and apply f(x) = 2x, it was infinite before and after.

yes but if x is finite, any f(x) will still be finite. So the universe has either always been finite, or always been infinite, and given that it started out being very small in a big bang, it would imply the former

> yes but if x is finite, any f(x) will still be finite.

The number line is not finite, even though every value on it is. The universe may be like the number line.

> and given that it started out being very small in a big bang, it would imply the former

The universe started out denser, which is not quite the same as "small" when talking about infinities. The currently visible universe gets both things at the same time, but only because the visible bit is finite.

Apply f(x) = x * 1e-60 to the number line, and things which were previously spaced out every integer, are now much more densely packed, but the number line as a whole is still the same size — infinite.

not if x is the set of all Real numbers. All points could have been infinitely close to each other at the big bang, and the universe could still have been infinite in size.

f(x) = tan(x)

f(x) = 1/(1-x)

So if x is finite, f(x) can still be infinite.

Was the universe infinitely small or was it just extremely or infinitely dense?

Anyway, we don't actually know that, since we do not have a time machine. The theory that the universe used to be much smaller is something we infer based on the snapshot we see today, just like we infer rotation based on the snapshot we see today. Either one could yet be proven wrong.

we don't know that the universe was ever small, just that it used to be a lot denser than it is today.