Question

f(x) + f(x+4) = f(x+2) + f(x+6)
Find the period of the real valued function satisfying this equation

Answer 1

\[
f(x) + f(x+4) = f(x+2) + f(x+6)
\]Likewise, if we plug \(x+2\) into \(x\) we get::\[
f((x+2)) + f((x+2)+4) = f((x+2)+2) + f((x+2)+6)
\]This simplifies to: \[
f(x+2) + f(x+6) = f(x+4) + f(x+8)
\]This part is interesting because it show us: \[
f(x) + f(x+4) = f(x+2) + f(x+6) = f(x+4) + f(x+8)
\]Meaning: \[
f(x) + f(x+4) = f(x+4) + f(x+8)
\]Now if we subtract both sides by \(f(x+4)\), then we get: \[
f(x) = f(x+8)
\]So at the very least, we have a period of \(8\).

Answer 2

thank you