Originally Posted by

**lgstarn** c) $\displaystyle 7n^2-19n+32$ is $\displaystyle O(n^3)$

Definition of big O notation:

$\displaystyle f(x)=O(g(x)) \text{ as } x\to\infty\,$ if and only if there exists a positive real number $\displaystyle M$ and a real number $\displaystyle x_0$ such that

$\displaystyle |f(x)| \le M |g(x)| \text{ for all }x>x_0.$

Let $\displaystyle f(x) = 7x^2-19x+32, g(x) = x^3$. For $\displaystyle x = 10, f(x) = 542, g(x) = 1000$, and since $\displaystyle f(x) \le g(x) \text{ for all }x>10$, take $\displaystyle x_0 = 10, M = 1$ and you've proved $\displaystyle 7n^2-19n+32$ is $\displaystyle O(n^3)$.

The rest of these are really similar. Put in the definitions and find some constants that satisfy the definition. Let me know if you have questions about any one of the four definitions or need another example.