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Question

Which of the following is/are periodic?

A
f(x)={1,x is rational0, x is irrational
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B
f(x)=x[x];2nx2n+112;2n+1x2n+2 ,nZ
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C
f(x)=(1)2xπ
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D
f(x)=x[x+3]+tan(nx2)
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Solution

The correct options are
A f(x)={1,x is rational0, x is irrational
B f(x)=x[x];2nx2n+112;2n+1x2n+2 ,nZ
C f(x)=(1)2xπ
D f(x)=x[x+3]+tan(nx2)
f(x)={1,x is rational0, x is irrational

f(x+k)={1,x+k is rational0, x+k is irrational

f(x+k)={1,x is rational0, x is irrational=f(x)

f(x) is periodic.

f(x)=x[x];2nx2n+112;2n+1x2n+2, nZ

f(x)=x+2;2x<112;1x<0

f(x)=x;0x<112;1x<2

f(x)=x2;2x<312;3x<4


f(x) is periodic with period 2.

f(x)=(1)2xπ
f(x+π)=(1)2(x+π)π
=(1)2xπ+2
=(1)2xπ×(1)2
=(1)2xπ
=f(x)
f(x) is periodic with period π.

f(x)=x[x+3]+tan(nx2)
tan(nx2) is periodic with period 2.
x[x+3]=x([x]3) =x[x]+3 ={x}+3
Period of {x} is 1.
Period of f(x) is 2. (LCM of (2,1))






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