If-f-x-2f-1-x-3f-x-x-1-x-what-is-f-x-

Question Number 120819 by bramlexs22 last updated on 03/Nov/20

If f (x) + 2 f (\frac{1}{x}) + 3 f (\frac{x}{x - 1}) = x

what is f (x) ?

Commented by liberty last updated on 03/Nov/20

(i) f (x) + 2 f (\frac{1}{x}) + 3 f (\frac{x}{x - 1}) = x

(ii) replace x with \frac{1}{x}, give

f (\frac{1}{x}) + 2 f (x) + 3 f (\frac{1}{1 - x}) = \frac{1}{x}

(iii) replace x with 1 - x, give

f (1 - x) + 2 f (\frac{1}{1 - x}) + 3 f (\frac{x - 1}{x}) = 1 - x

(iv) replace x with \frac{1}{x - 1}, give

f (\frac{1}{1 - x}) + 2 f (1 - x) + 3 f (\frac{1}{x}) = \frac{1}{1 - x}

(v) replace x with \frac{x}{x - 1}, give

f (\frac{x}{x - 1}) + 2 f (\frac{x - 1}{x}) + 3 f (x) = \frac{x}{x - 1}

(vi) replace x with \frac{x - 1}{x} give

f (\frac{x - 1}{x}) + 2 f (\frac{x}{x - 1}) + 3 f (1 - x) = \frac{x - 1}{x}

adding all equations and solving for

f (x) from this system give

∵ f (x) = \frac{{2 x}^{3} + x^{2} + 5 x - 2}{24 (x^{2} - x)}

Commented by bramlexs22 last updated on 03/Nov/20

great \dots

Commented by Ar Brandon last updated on 03/Nov/20

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