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Let f: R -> R be defined by f(x) = (e^x - e^(-x))/2 (the hyperbolic sine function). Is f invertible? If so, identify its inverse.

1. One-One only
2. Onto only
3. f⁻¹(x) = ln(x + sqrt(1 + x²))
4. None of these

Correct answer: f⁻¹(x) = ln(x + sqrt(1 + x²))

Solution

f(x) = sinh(x) is strictly increasing (derivative cosh(x) > 0) so it is one-one, and its range is all of R so it is onto; hence invertible. Solve y = (e^x - e^(-x))/2: let t = e^x, then t² - 2yt - 1 = 0, t = y + sqrt(y² + 1) (positive root). So x = ln(y + sqrt(y²+1)), giving f⁻¹(x) = ln(x + sqrt(1 + x²)).

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