For each of the following functions, do the following:

Calculate $T(n),$ making sure not to discard constants or low$-$order terms yet $($you may or may not count the counting variable in a for loop$)$

Given what you think is the Big$-$O performance for the function

Prove that your proposed Big$-$O for the function is correct $($make sure to show your work; do not just provide values for $c$ and $n_0)$

$(1)$

def fun$1(n)$ :

$x=0$

for $iin$ range $(2,\frac{n}{2})$ :

$,x=x**2$

return $x$

$(2)$

def fun$2($n$)$:

$x=0$

for $i$ in range $(n)$ :

for $j$ in range $(n)$ :

for $k$ in range $(1,n)$ :

$x=x+1$

return $x$

$(3)$

def fun$3($n$)$: