[Math] Big theta notation question

Question

can someone please explain to me the big theta notation and big omega and also

How i can show that $$ 3x+7\quad \text{is}\quad \Theta (x); $$

I don't really get how growth of functions works.

Answer 1

The definition that I'm accustomed to for $O$, $\Theta$, and $\Omega$ are the following:

$f(x)$ is $O(g(x))$ if there is some positive real number $C$ such that $$|f(x)| \leq C|g(x)|$$ for all $x>N$, where $N$ is some real number.

$f(x)$ is $\Omega(g(x))$ if there is some positive real number $C$ such that $$|f(x)| \geq C|g(x)|$$ for all $x>N$, where $N$ is some real number.

$f(x)$ is $\Theta(g(x))$ if $f(x)$ is both $O(g(x))$ and $\Omega(g(x))$ (but these can be with different constants!).

So $x$ is $O(x^2)$, but not $\Omega(x^2)$, because you can't find an $N$ to satisfy the condition.

But $3x+7$ is $O(x)$ ($C=5$, $N=4$, for example) and $\Omega(x)$ ($C=1$, $N=0$), so $3x+7$ is also $\Theta(x)$.