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Let be S the set of all nonzero real numbers. The function \(f : S \to \mathbb{R}\) satisfies the following two properties:

(i) First,
\(f\left( \frac{1}{x} \right) = xf(x)\) for all \(x \in S.\)

(ii) Second,
\(f \left( \frac{1}{x} \right) + f \left( \frac{1}{y} \right) = 1 + f \left( \frac{1}{x + y} \right)\) for all \(x \in S\) and \(y \in S\) such that \(x + y \in S.\)

Let n be the number of possible values of \(f(1),\) and let s be the sum of all possible values of \(f(1).\) Find \(n \times s.\)

 Apr 4, 2022
 #1
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ns = 10

 Apr 5, 2022
 #2
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Sorry, \(n\times s \neq 10.\)

SpaceXGeek  Apr 5, 2022
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Hello? I don't know how to bump this without doing this, sorry.

 Apr 6, 2022

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