Properties of Absolute Value and Squaring Function

Question

This seems to me to be a fairly elementary property, but it isn't what I've come into contact with before.

Is it fair to say that, for any $x \in \mathbb{R}$, $\left \lvert x \right \rvert^2 = x^2$?

This didn't seem correct to me until it was required in a proof I was writing, but it does seem to follow. For example, we can consider two, exhaustive cases: $x \geq 0$ and $x < 0$. If $x \geq 0$, then $\lvert x \rvert$ simplifies to $x$, so this clearly holds. If $x < 0$, then $\lvert x \rvert = -x$, and $(-x)^2 = x^2$.

Have I made a mistake, or this indeed a property of absolute values?

Answer 1

Yes, it is absolutely correct.

For all real numbers we have

$$\left \lvert x \right \rvert^2 = x^2$$

Note that if $x\ge 0$ then $|x|^2 = x^2$ and if $x<0$ then$ |x|^2 = (-x)^2=x^2$

Answer 2

An alternative approach is to realize that one can define the absolute value function in a non-piecewise fashion as $$|x|=\sqrt {x^2}$$ Your result follows directly by squaring both sides