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Consider the function f: R → R given by f(x) = (e^x - e^(-x)) / (e^x + e^(-x)). Which of the following statements is accurate?

1. f is injective and surjective
2. f is injective but not surjective
3. f is surjective but not injective
4. f is neither injective nor surjective

Correct answer: f is injective but not surjective

Solution

f(x) = (e^x - e^-x)/(e^x + e^-x) = tanh(x), whose derivative sech^2(x) > 0 everywhere, so f is strictly increasing and therefore injective. Its range is the open interval (-1,1), not all of R, so it is not surjective onto R. Thus f is injective but not surjective; the stored 'surjective but not injective' is wrong.

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