If 1/x f(x) + f(1/x) = x then this applies both when x = n and when x = 1/n:
1/n f(n) + f(1/n) = n and
n f(1/n) + f(n) = 1/n
Multiply the top equation by n, and swap the order of the terms in the second:
f(n)  nf(1/n) = n^2
f(n) + nf(1/n) = 1/n
Now add the two:
2f(n) = n^2 + 1/n
f(n) = (n^2 + 1/n)/2

Posted by Paul
on 20231224 13:21:15 