There are no “official” core purposes for government. That is, there is no set of things that everyone would agree are the purposes of government. Let us, however, look at two things that most people would say that government should do.
First, there would be the provision of public safety. This could be safety from foreign invasion or attack or it could be safety from common crime. We can easily see examples of governments doing both of these things. The TSA officers who do the scans at airports are supposed to be protecting us from attacks (from foreigners or Americans) on our airplanes. The police who patrol our streets are helping to protect us from common crime.
Second, there would be the goal of providing economic stability and prosperity. The government does this in many ways. It sets tax rates that are, hopefully, not too high or too low (fiscal policy). It sets interest rates as well (monetary policy). In addition, it does things like providing funding to colleges so that research can be done and so that students can be educated. Both of these things allow the economy to grow in the future.
Thus, we see examples of the government keeping us safe and trying to keep our economy stable and prosperous.