how do you know if a penny is pure copper?

Answer 1

U.S. pennies made after 1982 are copper coated zinc. Before that, they were a 95% copper alloy with zinc during most years.
In 1943 (wartime), they were steel. Those quickly rusted and were discontinued.
There were copper-nickel alloy pennies in the 1860's.

Answer 2

Dr WLS has given you a physical way of determining the answer; i.e., by looking at the date. For a chemical answer, you can test for Zn in a solution of a penny.
Another way is determine the density of the penny and see if it matches the density of copper.