Graph y = sqrt{x^2}, y = x, y |x|, and y = (sqrt{x})^2, NOTING which graphs are the SAME.
NOTE:
What is the mirror-like image of the graph y = x?
I'm not going to graph these, you should be able to do that yourself. However I will note that only two of these graphs are the same. The trick is in the last graph: $\displaystyle \sqrt{x^2}$ and $\displaystyle (\sqrt{x})^2$ are "apparently" the same since each reduces to x, but note that in the first expression the domain is $\displaystyle (-\infty, \infty )$ and in the last second expression is $\displaystyle [0, \infty )$.
$\displaystyle y = \sqrt{x^2}$ and $\displaystyle y = x$ aren't the same either since $\displaystyle \sqrt{.}$ only returns a positive value. It is for this very reason that $\displaystyle y = \sqrt{x^2}$ and $\displaystyle y = |x|$ have the same graph.
Depends on which "mirror" you are talking about. The typical mirror "planes" are the x and y axes, and the lines y = x and y = -x.
y = x reflected over the x-axis is y = -x.
y = x reflected over the y-axis is y = -x.
y = x reflected over the line y = x is y = x.
y = x reflected over the line y = -x is y = x.
-Dan
Actually I wasn't very clear, sorry. I was responding to your comment that $\displaystyle y = (\sqrt{x})^2$ and $\displaystyle y = x$ have the same graph. But I graphed the wrong function. These aren't the same because the first only has the domain $\displaystyle [0, \infty )$, whereas the second has a domain of all real numbers.
-Dan