# Big O Notation properties

If for functions $f, g$ we have $f = \mathcal{O}(g)$, then does $g = \mathcal{O}(f)?$ Explain why this is true or give an example showing it is false.

Workings:

I believe this is false. if I take $g = n^3 +20 n +1$ then $f = \mathcal{O}(g) = \mathcal{n^3}$

But $g = \mathcal{n^3} = n^3 \neq n^3 +20n +1$

Is what I said correct? Any help will be appreciated.

You should think of $\mathcal{O}$ as "grows no faster than (asymptotically)". Thus
$$n^3 + 20n +1 = \mathcal{O}(n^3)$$ and $$n^3 = \mathcal{O}(n^3+ 20n +1)$$ and $$n = \mathcal{O}(n^3)$$ but $$n^3 \ne \mathcal{O}(n).$$
Note: this isn't the formal definition of $\mathcal{O}$ but it is a good way to train your intuition.