What is $\sqrt{x^2}$?

Question

At first the question of what $\sqrt{x^2}$ is seems silly. It looks like $x$. And for $x \in \mathbb{R}^+$ it is.

However, for $\mathbb{R}$, I'm not sure. I can think of 3 answers:

• $abs(x)$
• $\{x,-x\}$
• $x$

Which one is correct and why?

Answer 1

$\sqrt{x^2}=x$ for $x\geq0$ and $\sqrt{x^2}=-x$ for $x<0$, which says $\sqrt{x^2}=|x|.$

Answer 2

$\sqrt a$, for $a\in[0,\infty)$, is the non-negative real solution ofthe equation $x^2-a=0$. Therefore $\sqrt{t^2}=\lvert t\rvert$ for all $t\in\Bbb R$.