Perhaps it's time to back-up a step. Why are you looking for a function that does square-roots without using the sqrt operation? Is this some sort of performance optimization?

I don't need square root function though at all, only the condition of mine you have exactly appointed to hold true about that function.

I too think there is only squreroot, or more exactly, only: f(x)=x^m/n where m/n are whole numbers such that m/n=1/2 .

This is so far what I think, but I'm in no position to hold that as definitive, or even proven, I just think that by the intiuiton.

Any ohther furthering of this discussion, as to what f(x) possible formula is, is welcomed.

I don't need square root function though at all, only the condition of mine you have exactly appointed to hold true about that function.

You are contradicting yourself. You don't need a square root function at all; you just need a function that satisfies some condition that only the square root function satisfies, as was proven before in this thread.

I too think there is only squreroot, or more exactly, only: f(x)=xm/n where m/n are whole numbers such that m/n=1/2 .

No, that's not more exact. You are just complicating things needlessly.

This is so far what I think, but I'm in no position to hold that as definitive, or even proven, I just think that by the intiuiton.

I just gave you pretty much a formal proof. I am starting to smell troll.

I don't need square root function though at all, only the condition of mine you have exactly appointed to hold true about that function.

You are contradicting yourself. You don't need a square root function at all; you just need a function that satisfies some condition that only the square root function satisfies, as was proven before in this thread.

I too think there is only squreroot, or more exactly, only: f(x)=xm/n where m/n are whole numbers such that m/n=1/2 .

No, that's not more exact. You are just complicating things needlessly.

This is so far what I think, but I'm in no position to hold that as definitive, or even proven, I just think that by the intiuiton.

I just gave you pretty much a formal proof. I am starting to smell troll.

Yeah, those one line proofs of Alvaro, of course, lets enclose those bijections of squarerrots curves for ever. Alvaro has proven enough, stop trolling, right!

Yeah, those one line proofs of Alvaro, of course, lets enclose those bijections of squarerrots curves for ever. Alvaro has proven enough, stop trolling, right!

Sometimes 1 line is all you need to prove something right or wrong. And if you really it in more lines, add some line breaks; there are plenty of places where they would fit and potentially make the proof more readable.

Álvaro is correct.

If you don't understand, and need a simpler/different explanation, that's fine. Just ask and someone will probably try help.

Using intuition when it comes to math is also fine, but you should also be able to back it up, or be prepared to back down when someone gives a proper proof/explanation.