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Classify the following function as even, odd, or neither: f(x) = log(x + sqrt(1 + x²)).

1. even
2. odd
3. neither
4. cannot be determined

Correct answer: odd

Solution

f(x) = log(x + sqrt(1 + x²)) = sinh^(-1)(x). Compute f(-x) = log(-x + sqrt(1 + x²)). Multiply numerator-style: (-x + sqrt(1+x²))(x + sqrt(1+x²)) = (1 + x²) - x² = 1, so (-x + sqrt(1+x²)) = 1/(x + sqrt(1+x²)). Thus f(-x) = log[1/(x + sqrt(1+x²))] = -log(x + sqrt(1+x²)) = -f(x). Since f(-x) = -f(x), the function is odd.

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