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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
    • one year ago
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    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
    • one year ago
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    whats stopping you from using the hint ?

  3. ganeshie8
    • one year ago
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    \[f(x) = 7x-3f\left( \frac{x - 1}{x} \right)\tag{1}\] 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
    • one year ago
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    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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