Problem #PRU-102995

Can there exist two functions $f$ and $g$ that take only integer values such that for any integer $x$ the following relations hold:

a) $f (f (x)) = x$ , $g (g (x)) = x$ , $f (g (x)) > x$ , $g (f (x)) > x$ ?

b) $f (f (x)) < x$ , $g (g (x)) < x$ , $f (g (x)) > x$ , $g (f (x)) > x$ ?

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