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For the function f(x) = log((1 + x)/(1 − x)), which of the following relations is satisfied?

1. f(x + 2) − 2f(x + 1) + f(x) = 0
2. f(x + 1) + f(x) = f(x(x + 1))
3. f(x1)·f(x2) = f(x1 + x2)
4. f(x1) + f(x2) = f((x1 + x2)/(1 + x1x2))

Correct answer: f(x1) + f(x2) = f((x1 + x2)/(1 + x1x2))

Solution

Since f(x) = log((1+x)/(1-x)) = 2 artanh(x), the addition formula gives f(x1) + f(x2) = f((x1+x2)/(1+x1*x2)). Direct substitution confirms this identity.

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