(This definition found at https://www.cs.sfu.ca/~ggbaker/zju/math/growth.html)For functions f(x) and g(x), we will say that "f(x) is O(g(x))” [pronounced "(x) is big-oh of g(x)"] if there are positive constants C and k such that

|f(x)|≤C|g(x)| for all x>k.

I understand that for f(x) to be big-Oh of g(x) there must exist at least one corresponding C and k such that g(x) will grow either equally or faster than f(x) forever more when x becomes greater than k.

What I am trouble with is understanding how to prove that one function is big-Oh of the other. The example to prove that one function is big-Oh of another that the link I mentioned above provides confuses me. Here is it in its entirety:

My question is, why does changing 2xExample:The function f(x)=2x^{3}+10x is O(x^{3}).

Proof:To satisfy the definition of big-O, we just have to find values for C and k that meet the condition.

Let C=12 and k=2. Then for x>k,

|2x^{3}+10x|= 2x^{3}+10x

<2x^{3}+10x^{3}

=|12x^{3}|.∎

Note: there's nothing that says we have to find thebestC and k. Any will do.

Also notice that the absolute value doesn't usually do much: since we're worried about running times, negative values don't usually come up. We can just demand that x is big enough that the function is definitely positive and then remove the |⋯|

^{3}+ 10x to 2x

^{3}+ 10x

^{3}allow us to prove that f(x) is big-Oh of g(x) in this context? Why are we allowed to change it?