In this question, we have 2 Algorithms X and Y and X is more efficient then Y. So X will be a better choice for small as well as large input values if the efficiency of X is more then Y for small input values
X will be a better choice for all inputs except possibly small inputs
In asymptotic analysis we consider growth of algorithm in terms of input size. An algorithm X is said to be asymptotically better than Y if X takes smaller time than y for all input sizes n larger than a value n0 where n0 > 0.
does asymptotically means for larger values or smaller values or for all values?
In Asymptotic Analysis, we evaluate the performance of an algorithm in terms of input size.
We calculate, how the time / space taken by an algorithm increases with the input size.
So asymptotic analysis is generally done by keeping in mind large input size
Lets see an example-
If you compare binary search and linear search, then we say directly binary search(log n) is more efficient than linear search(n), but if you take only one input in array then you don’t see the power of binary search, because both take same time, but if you take large input in array then you see the power of binary search.
so Algorithms X is asymptotically more efficient than Y for large input.