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Let f be an invertible function such that e^(f(x)) + f(x) = ln(f^(-1)(1)) for all x in the domain. Find the value of ln(f^(-1)(1)).

1. e
2. e - 1
3. e + 1
4. 1

Correct answer: e + 1

Solution

Let c = ln(f^(-1)(1)). The equation says e^(f(x)) + f(x) = c for all x. Since g(t) = e^t + t is strictly increasing and bijective, f(x) = g^(-1)(c) is a constant for all x. For f to be invertible, the domain must allow unique inverse. At the point x0 = f^(-1)(1), f(x0) = 1. Substituting: e¹ + 1 = c => c = e + 1. Therefore ln(f^(-1)(1)) = e + 1.

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