Finding period of any type of function

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The discussion centers on determining the periods of two functions: f(x) = tan(πx) + x - [x] and f(x) = sin(2πx) + x - [x], where [x] is the greatest integer function. Participants agree that the period of both functions is 1, as the periodic components tan(πx) and sin(2πx) have periods that align with the behavior of the floor function. The conversation also explores the implications of combining periodic functions, noting that the period of their sum can be influenced by cancellations between terms. Examples are provided to illustrate how certain combinations can yield a smaller period than the least common multiple of individual periods. Ultimately, the consensus is that while the LCM provides a safe estimate for the period, careful analysis may reveal a shorter essential period in specific cases.
  • #31
That explains it all. Thank you everybody.
 
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  • #32
period of tan pi x = 1 as the graph for x = -1/2 to 1/2 repeats.
period of x - [ x ] = 1 as the graph repeats for [n , n-1).
Hence, the period of the given function is 1.

AGain for Sin (2 pi x ) + x - [x] also the period is 1, for a similar reason.
 

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