Let us describe a few functions in terms of above asymptotic notation.
Example: f(n) = 3n3 + 2n2 + 4n + 3
= 3n3 + 2n2 + O (n), as 4n + 3 is of O (n)
= 3n3+ O (n2), as 2n2 + O (n) is O (n2)
= O (n3)
Example: f(n) = n² + 3n + 4 is O(n²), since n² + 3n + 4 < 2n² for all n > 10.
Through definition of big-O, 3n + 4 is also O(n²), too, although as a convention, we employ the tighter bound to the function, i.e., O(n).
Here are some rules regarding big-O notation:
1. f(n) = O(f(n)) for any function f. In other terms, every function is bounded by itself.
2. aknk + ak-1 n k-1 + • • • + a1n + a0 = O(nk) for every k ≥ 0 & for all a0, a1, . . . , ak ∈ R.
In other terms, every polynomial of degree k may be bounded through the function nk. Smaller order terms can be avoided in big-O notation.
3. Basis of Logarithm can be avoided in big-O notation that means logan = O(logb n) for any bases a, b. Generally we write O(log n) to specify a logarithm n to any base.
4. Any logarithmic function may be bounded through a polynomial that means. logb n = O(nc) for any b (base of logarithm) & any positive exponent c > 0.
5. Any polynomial function may be limited by an exponential function i.e. nk = O (bn.).
6.Any exponential function may be limited by the factorial function. For instance, an = O(n!) for any base a.