Proving that a function is strictly monotonic knowing that | f ( x ) −...

You get $x^2-2|x|\le f(x)\le x^2+2|x|$ not by squaring but by noting:
$|a|<|b|$ (or indeed $\le$) is the same as −b<a<bYou can then see from the behavior at 0 there can be no such function
Counter example
As mentioned in the comments $f(x)=x^2$
Missing bounds Even if it's for some range, like $[0,\mathrm{\infty })$ there can still be a function that isn't monotonic between the bounds. Eg x(x−1) (look at the roots, that's how I spotted this)