They look for signatures of their decay chains. Each superheavy nuclide has a characteristic chain of decays (alpha, beta, spontaneous fission, etc.) that it will undergo once it's been produced. By identifying these decays, they know what nuclide they had originally produced.
>To add to my first question, if these elements have no purpose (according to my teacher), why is so much money and energy spent on synthesizing these elements?
They have no *practical* purpose, but science doesn't need a practical application. Studying superheavy nuclides allows us to get a better understanding of nuclear structure, nuclear reactions, nuclear forces, etc. They also may display some very interesting chemical properties, and give us a better understanding of exotic effects in atomic physics.