Can the People delegate something they themselves don't have a right to do to the government?

This is a question I've pondered for a while, but can't come up with a satisfactory answer.

Its not legal for the American people to kill someone, even in retaliation for the commission of a crime, but they can delegate that authority to the government by way of capital punishment.

There are other examples of instances where a citizen isn't allowed to do something, but can give that ability to the government.

Its sort of a conundrum. I'd like to hear some other opinions and thoughts on this.