According to the polynomial- reduction definition

"If problem Y can be reduced to problem X, we denote this by
Y ≤(P)X."

If X is one of already known NP-complete problem then we can say that Y is NP-complete.

From my understanding, P should be always polynomial and this cannot be an exponential function such as 2^n or 3^n.

However, my question is if Y or X already has a lower bound that is not polynomial, which is 2^n or 3^n then can I still say that Y is NP-complete?

In other words, does lower bounds for X and Y matter?