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Question 1161665: Are there any functions defined on real numbers or rational numbers other than zero function and identity function in such way that f^{3}g(x) = fg(fg(x)) i.e.
f(f(f(g(x)))) = f(g(f(g(x)))) for all x belongs to R or Q.
[Note: Here f(g(x)) is composition of functions or composite function rule]
Answer by ikleyn(52794)   (Show Source):
You can put this solution on YOUR website! .
You may take, for example, g(x) = ax + b any linear function with a =/= 0,
and take f(x) = g(x) as the same function.
Then you will have the desired identity.
More generally, you can take g(x) as any monotonic one-to-one function g: R ---> R defined over all real numbers;
for example, g(x) = x^3; or g(x) = x^5; or g(x) = x^7, and so on . . .
and take f(x) = g(x).
Then, again, you will have the desired identity.
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