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## zmudz one year ago Find all functions $$f:\mathbb R \to \mathbb R$$ that satisfy $$f(x) + 3 f\left( \frac {x-1}{x} \right) = 7x$$ for all nonzero $$x$$.

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1. zmudz

Hint: If you substitute $$(x - 1)/x$$ in for $$x$$, then you get a new equation that also involves $$f\left( \frac{x - 1}{x} \right)$$. Then do this substitution again and you may get another useful equation to consider.

2. ganeshie8

whats stopping you from using the hint ?

3. ganeshie8

$f(x) = 7x-3f\left( \frac{x - 1}{x} \right)\tag{1}$ substitute $$\frac{x-1}{x}$$ for $$x$$ : $f\left(\frac{x-1}{x}\right) = 7\frac{x-1}{x}-3f\left( \frac{- 1}{x-1} \right)\tag{2}$ substitute $$\frac{x-1}{x}$$ for $$x$$ again : $f\left(\frac{-1}{x-1}\right) = -\frac{7}{x-1}-3f\left(x\right)\tag{3}$

4. ganeshie8

By elimination, you should end up with $f(x) = \dfrac{x-\frac{9}{x-1}-\frac{3(x-1)}{x}}{4} = \frac{x^3-4x^2-3x-3}{4x(x-1)}$

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