Can you operate on and draw conclusions on functions described asymptotically?
For some operations, such as addition, multiplication, you can operate directly on the asymptotic notation. For example, if $\small g(n) = \mathcal{O}(n^2)$ and $\small f(n) = \mathcal{O}(n^2)$, then $\small g(n) + f(n) = \mathcal{O}(n^2)$ and $\small g(n) \cdot f(n) = \mathcal{O}(n^4)$. For some other operations, such as division, it depends then. For example, for $\small g(n) = \mathcal{O}(n^2)$ and $\small f(n) = \mathcal{O}(n^2)$, it is not correct to say $\small \frac{g(n)}{f(n)} = \mathcal{O}(1)$ (consider $\small g(n) = n^2$ and $\small f(n) = n$). However, if $c$ is a fixed constant, then $\small \frac{g(n)}{c} = \mathcal{O}(n^2)$.