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Let f: R -> R be given by f(x) = ln(x + sqrt(x² + 1)), and let g be a function such that g(f(x)) = x for every real x. Find g(2).

1. (e² + e⁻²)/2
2. e²
3. (e² - e⁻²)/2
4. e⁻²

Correct answer: (e² - e⁻²)/2

Solution

f(x) = ln(x + sqrt(x²+1)) = arcsinh(x). Since g(f(x)) = x, g is the inverse of f, i.e. g(y) = sinh(y) = (e^y - e^-y)/2. Therefore g(2) = (e² - e⁻²)/2.

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