since y=x^2 is inverse to y=sqrt(x), it was supposed to reflect on the line y=x, but the radical function did not show the correct reflection of the quadratic graph because it only showed the half on the upper part of the radical graph and did not show the bottom part.

The reason is that the inverse of x^{2 }is NOT a function. But in can be split up into two functions!

we could have y =Sqrt x and y = -sqrt of x.