Struggling with asymptotic analysis

Question

I'd be glad for an explanation on the analysis of this exercise. Given these functions: $$f(n) = n^{1/2} \\ g(n) = n^{2/3}$$

Show that $f(n) = O(g(n))$, or $f(n) = \Omega(g(n))$ and comment if $f(n) = \Theta(g(n))$

PS: The exercise requires a mathematical demonstration of the answer.

Answer 1

Given $f(n) = n^{1/2}$ and $g(n) = n^{2/3}$, we can see that $f(n) = \O(g(n))$, because the function $g(n)$ grows faster than the function $f(n)$. We can demonstrate this writing the values for a series of inputs $0, 1, ...$ to $f(n)$ and $g(n)$:

$f(0) = 0$

$g(0) = 0$

$f(1) = 1$

$g(1) = 1$

$f(2) = 1.4$

$g(2) = 1.6$

$f(3) = 1.7$

$g(3) = 2$