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A function f is defined for inputs in the interval (0,1). For the expression f(e^x) + f(ln|x|), determine the set of all real x for which it is defined.

1. (-e, -1)
2. (-e, -1) ∪ (1, e)
3. (-∞, -1) ∪ (1, ∞)
4. (-e, e)

Correct answer: (-e, -1)

Solution

The function f is defined for inputs in the interval (0,1), so we need to determine where the expressions e^x and ln|x| fall within this range. The expression e^x is in (0,1) when x is in the interval (-∞, 0), and ln|x| is in (0,1) when |x| is in (1,e), which corresponds to x being in the intervals (-e, -1) and (1, e). Therefore, the only overlap that satisfies both conditions is (-e, -1).

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