While calculating Big(oh), Where from upper bound

Question

For example i have f(N) = 5N +3 from a program. I want to know what is the big (oh) of this function. we say higher order term O(N).

1. Is this correct method to find big(oh) of any program by dropping lower orders terms and constants?

2. If we got O(N) by simply looking on that complexity function 5N+3. then, what is the purpose of this formula F(N) <= C* G(N)?

   i got to know that, this formula is just for comparing two functions. my question is,

3. In this formula, F(N) <= C* G(N), i have F(N) = 5N+3, but what is this upper bound G(N)? Where it comes from? where from will we take it?

i have studied many books, and many posts, but i am still facing confusions.

Answer 1

Q: Is this correct method to find big(oh) of any program by dropping lower orders terms and constants?

Yes, most people who have at least some experience with examining time complexities use this method.

Q: If we got O(N) by simply looking on that complexity function 5N+3. then, what is the purpose of this formula F(N) <= C* G(N)?

To formally prove that you correctly estimated big-oh for certain algorithm. Imagine that you have F(N) = 5N^2 + 10 and (incorrectly) conclude that the big-oh complexity for this example is O(N). By using this formula you can quickly see that this is not true because there does not exist constant C such that for large values of N holds 5N^2 + 10 <= C * N. This would imply C >= 5N + 10/N, but no matter how large constant C you choose, there is always N larger than this constant, so this inequality does not hold.

Q: In this formula, F(N) <= C* G(N), i have F(N) = 5N+3, but what is this upper bound G(N)? Where it comes from? where from will we take it?

It comes from examining F(N), specifically by finding its highest order term. You should have some math knowledge to estimate which function grows faster than the other, for start check this useful link. There are several classes of complexities - constant, logarithmic, polynomial, exponential.. However, in most cases it is easy to find the highest order term for any function. If you are not sure, you can always plot a graph of a function or formally prove that one function grows faster than the other. For example, if F(N) = log(N^3) + sqrt(N) maybe it is not clear at first glance what's the highest order term, but if you calculate or plot log(N^3) for N = 1, 10 and 1000 and sqrt(N) for same values, it is immediately clear that sqrt(N) grows faster, so big-oh for this function is O(sqrt(N)).