The Colombian veteran, 35, has joined Rayo from Galatasaray.
Falcao said: "I have found a team with a lot of quality, hunger and desire to fulfill the objectives that we have in Primera. In addition to a coaching staff that has made this team grow. Hopefully we can all achieve these goals together.
"I hope to help the team grow with all my experience. Hopefully I score a lot of goals too, which is what strikers are for. Rayo has opened the doors for me to return to LaLiga, to continue in the elite and to be in a place that I know and where I think I won't need much time to adapt."
Falcao will wear the No3 this season and he added: "My father has been very important because he transmitted to me his passion for this sport. He is no longer with us and he used to play with this number. What better tribute than to play with his number."