Q. Example of asymptotic notations?

The function f (n) belongs to the set  (g(n)) if there exists positive constants c such that for satisfactorily large values of n we have 0<= c*g(n) <=f(n). In another words,

  O(g(n)) ={ 0<= c*g(n) <=f(n) for all n >= n_o }.    

Assume we have a function f (n) = 4n² + n then order of function is O(n²). The asymptotic notations give information about the upper and lower bounds on complexity of an algorithm with help of O and ? notations. E.g. in the sorting algorithm the upper bound is O (n ln n) and lower bound is ? (n ln n). But problems such as matrix multiplication have complexities such as O (n³) to O (n^2.38).

Algorithms that have similar lower and upper bounds are called optimal algorithms. So a few sorting algorithms are optimal whereas matrix multiplication based algorithms are not.