If  fx+12+fx-12=fx or all x∈R Then the period of fx is

x+12+fx-12=fx                        (1)Replacing x by x+12, we getfx+1+fx=fx+12                          (2)or   fx+1+fx-12=0                     (from (1) and (2))         or       fx+32=-fx                   (Replacing   x  by   x+12)or       fx+3=-fx+32=fx              (Replacing   x  by   x+32)Therefore, f(x) is periodic with period 3.