Why doesn't the graph of $y=\left(x^{1/2}\right)^{2}$ include negative $x$ values?

Question

This is the graph of $y=\left(x^{1/2}\right)^{2}$ that I have been getting using Desmos:

The problem is that $i^2=-1$, so the function is defined for negative numbers, but the graph is not so, why?

Answer 1

While evaluating $f(g(x))$ for real valued functions, $x$ is taken such that $g(x)$ is defined and real valued.

Clearly for negative numbers, $g(x) = \sqrt{x}$ is not real, so the domain of $g(x) = \sqrt{x} $ is $[0,\infty)$ and it's range is also $[0,\infty)$

Let $f(x) = x^2$ Then, $f(g(x)) = (\sqrt x)^2 = (x^\frac12)^2$. Domain of $f(g(x)) =$ Range of $g(x) = [0,\infty)$