Which of the following function is/are periodic?

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fx=1, x is rational0, x is irrational

fx=x-x; 2n≤x<2n+112; 2n+1≤x<2n+2where [.] denotes the greatest integer function, n∈Z

fx=-12xπ , where. denotes the greatest integer function

fx=x-x+3+tanπx2, where. denotes the greatest integer function, and a is a rational number

answer is A.

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Detailed Solution

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=-12xπor fx+π=-12π+xπ=-12xπ+2=-12xπHence, the period is π. fx=x-x+3+tanπx2=x-3+tanπx2x is periodic withperiod 1; tanπx2 x is periodic with period 2.Now, the LCM of 1 and 2. Hence, the period of fx is 2.

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