f(n) = 3n+3 ; f(n) = O(n)
By definition :
$3n+3 \le c_1.n$
By dividing both side by n
$3+(3/n) \le c_1$
means we r getting constant range for $c_1$ for any $n$. Again it shows c's value must be greater than 3 at any cost.
e.g. if we take c's value 3.5 so n's value will be 6.
Now if we plot graph ( Bcoz I want to learn this concept by understanding graph )
$cg(n)$ graph goes below of $f(n)$ graph. I have taken following values for both functions :
f(n) : 3n+3 n f(n) 1 6 2 9 3 12 -2 -3
g(n) = 3.5n n g(n) 1 3.5 2 7 3 10.5 -2 -7
If we plot graph by these values it doesn't bind $f(n)$ i.e. $3n+3$ above by the value of $g(n)$ i.e. $3.5\cdot n$
Can anyone explain me this concept by graph ?