What did Rome turn into a “Roman lake”?

Rome, that would eventually dominate the entire Mediterranean Sea. Spreading first through Italy, then Rome defeated Carthage in the Punic Wars, and Hannibal in the 2nd Punic War. After the Third Punic War, Rome then became the leading force in the Mediterranean region.

to answer your question:

f(x)= a^x.  if x=0, f(x)=1  if x=1, f(x)=a  if x=1, f(x)=a^a