Is this a correct way to express $\left|f(x)\right| \leq \left|x\right|^9$?

Question

If $\left|f(x)\right| \leq \left|x\right|^9$, then, is it correct to say that

$f(x) \leq x^9$ and $f(x) \geq -x^9$ ?

If it is not, could someone explain why? Thank you.

Answer 1

To expand on what Mark Bennet said, your formulation is almost correct: it's not $-x^9\leq f(x)\leq x^9$, but it is $-|x|^9\leq f(x)\leq |x|^9$.

To get rid of the absolute values entirely takes some care, but it can be done. The expression is the same as the piecewise expression $$ \left\{ \begin{array}{lr} -x^9\leq f(x)\leq x^9 &: x>0 \\ -x^9\geq f(x)\geq x^9 &: x<0 \\ \end{array} \right. $$ which can be phrased using OR: Either $-x^9\leq f(x)\leq x^9>0~$ OR $~-x^9\geq f(x)\geq x^9 <0$. There's no really good reason to write something this silly, but if you need it in a particular form for reasons, this is one possibility.

Answer 2

I would call this somewhat silly and write it as

$$between(-|x|^n, f(x), |x|^9)$$