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# Another Functional Equation

## Question

2f(1/x)-f(x)+2f(2/x)-f(x/2) = x, x is defined on the reals except where x =0

find f(x) =

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Spoiler

2f(1/x)-f(x) = 2x/3   (plugging in x/2 for x then adding to this gives original)

f(x) = 2f(1/x) - 2x/3  (rearrange)

f(1/x) = 2f(x) - 2/3x  (substitute 1/x for x)

f(x) = 2(2f(x) - 2/3x) - 2x/3   (replace f(1/x) from two above)

f(x) = 4f(x) - 4/3x - 2x/3

3f(x) = 4/3x + 2x/3

f(x) = (2/9)(x+2/x)

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