When I graph $y=x^{1/2}$ why does it only show the positive y values.

Question

I understand the reason for $y=\sqrt{x}$. I've been told that the radical symbol gives out the positive answer. But $y=x^{1/2}$ doesn't use a radical symbol, and it still only shows the positive y values.

Answer 1

It would help to know what software package you are using here.

I tried it myself using GeoGebra, and I had the same as you: entering the formula "y=x^(1/2)" in the formula entry field did indeed yield a graph with positive $y$ only.

However, when I entered "y^2 = x" in that field, it generated both branches of the relation.

There is considerable confusion and individual taste as to exactly what is meant by $x^{1/2}$.

When in the field of complex analysis, the understanding is: $$z^{1/n} = \{r^{1/n} (\cos(\theta + 2 \pi k / n) + i \sin (\theta + 2 \pi k / n)): k \in \{0,1,2,…,n−1\}\}$$

from which, restricting to the real number line, you would expect $x^{1/2} = \pm \sqrt x$.

But it seems that this may not be the generally understood case. I don't know why this is and I disagree with it, as a philosophical position. But don't take my word for it, as my opinions and ideas differ from those of mainstream mathematics in plenty of ways.

Answer 2

$x^{\frac{1}{2}}$ is basically the same function as $\sqrt{x}$

So for example, take $9$, $9^{\frac{1}{2}}$ is the same as $\sqrt{9}$, which is 3