Absolute value squared - Meaning

Question

I'm wondering if it's correct to say that the absolute value squared means that the values above $1$ are magnified, whereas the values below $1$ are damped (I'm considering only positive values because the codomain of the absolute value is $R^+$).

In general the above consideration should be valid for all the functions which have $R^+$ as codomain.

Thank you in advance.

Answer 1

well, yes. $|x| \ge 0$ for all $x \in \mathbb C$

And if $|x| = 0$ then $|x|^2 = 0^2 = 0 = |x|$ and magnitude remains unchanged.

If $0 < |x| < 1$ then, via the axiom $a > 0; x < y \implies ax < ay$, we have $|x|^2 = |x||x| < |x|\cdot 1 = |x|$ so, yes $|x|^2 < |x|$. so it is "damped".

If $|x| = 1$ then $|x|^2 = 1^2 =1 = |x|$ and magnitude remains unchanged.

If $|x| > 1$ then by the same axiom cited above. $|x|^2 = |x||x| > |x|\cdot 1 = |x|$ and magnitude is "magnified".

Answer 2

As you multiply something (namely $x$) by $x$, you will indeed obtain a smaller or larger result if $x$ is smaller or larger than $1$ respectively.