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Functional Finding (Posted on 2023-12-24)

Determine all f(x) on reals excluding zero such that

(1/x)f(-x) + f(1/x) = x

No Solution Yet Submitted by Danish Ahmed Khan

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solution

| Comment 1 of 3

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 2023-12-24 13:21:15

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