Question

why do we use only principal root for functions​

Answer 1

Answer:

For example, if we have x=4–√, we always assume x=2. But if we have x2=4, we do x=±4–√⟹x=2,−2.

The problem is that if I take the 1st equation, and square both sides, I get

x2=4⟹x=±4–√⟹x=2,−2.

My teacher says that if you introduce the radical sign, use ±, but those two equations are the same under the rules about radicals she taught us, so in reality, (I think) "taking the positive root" rule has no meaning.

I get that people want to make the radical mean something without ambiguity, but making arbitrary rules (at least I thing they're arbitrary), like taking the positive answer, seems to confuse things and leads to inconsistency, like what is shown above.

There are other examples, such as solving for x, then substituting the original equation with the value of x, and not getting it to work because you can only take the positive root.

For things like the Pythagorean theorem, people like to say it's obvious to take the positive one only, but there are ways to represent the Pythagorean theorem without relying on human judgement to decipher the final answer. Say you have a right triangle with legs a=3, b=4, and you want to find c. So you do: 32+42=c2, c>0. and solve the systems of equations, just like any other: c=±25−−√⟹c=−5,5. c=−5,5 intersects c>0 at c=5, so the answer is c=5. I don't see why you would need to redefine