Which of the following function $$is/are$$ periodic?

Question

Infinity Learn · Accepted Answer

$$fx=1$$,     x is  $$rational0$$,    x is  $$irrationalfx+k=1$$, $$x+k$$   is  $$rational0$$, $$x+k$$  is  irrational                  wherer  k   is  any  rational  number                       $$=1$$,  x  is  $$rational0$$,  x  is   irrational                       $$=fxTherefore$$, $$f(x)$$ is periodic function, but its fundamental period cannot be determined. Thus,$$fx=x-x$$,    $$2n$$≤x<$$2n+11/2$$           $$2n+1$$≤x<$$2n+2Draw$$ the graph from which it can be verified that period is $$2$$.$$fx=-12x$$$$π$$or     $$fx+$$π$$=-12$$π$$+x$$π$$=-12x$$π$$+2=-12x$$$$π$$Hence,   the  period  is  π.       $$fx=x-x+3+$$$$tan$$$$π$$$$x2=x-3+$$$$tan$$$$π$$$$x2x$$   is  periodic  withperiod  $$1$$;   $$tan$$$$π$$$$x2$$  x   is  periodic  with   period $$2$$.Now,  the  LCM  of   $$1$$  and  $$2$$.   Hence,  the  period  of fx  is  $$2$$.

Which of the following function is/are periodic?

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