How to find the period of this function

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The discussion focuses on the functional equation f(x) + f(x+4) = f(x+2) + f(x+6) and the derivation of the function's period. Participants suggest defining g(x) = f(x) + f(x+4), leading to the conclusion that g(x) is periodic with a period of 2. The challenge remains in relating this periodicity back to f(x) and whether trial and error is necessary for further analysis.

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f(x)+f(x+4)=f(x+2)+f(x+6) where all functions are real valued
 
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Write g(x)=f(x)+f(x+4). Then it's easy to see that the left-hand side is g(x) and the right hand side is g(x+2), so g(x)=g(x+2). That seems to be where I would start.
 
Thank you very much for the approach it was helpful but after setting g(x)=f(x)+f(x+4) and knowing the fact g(x) has a period 2 how to relate if to f(x).Is there any approach to do so or do i have to opt for trial and error.
 
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