Bizarre function

Algebra Level 5

Let \(f:\mathbb{R}^{+} \to \mathbb{R}\) be a function such that

  • \(f\) is strictly increasing
  • \(f(x) > - \dfrac{1}{x} \quad \forall x>0\)
  • \(f(x) f \left(f(x)+\dfrac 1x \right) = 1 \quad \forall x>0\)

Find the sum of all possible values of \(f(1)\).

Give your answer to 3 decimal places.

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