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The function f: R -> R is defined as f(x) = e^(ln(sqrt(sqrt(x²+1) + x) + sqrt(sqrt(x²+1) - x))). Determine the nature of this function.

1. One-one and onto
2. One-one but not onto
3. Onto but not one-one
4. Neither one-one nor onto

Correct answer: Neither one-one nor onto

Solution

Let u = sqrt(x²+1). Then sqrt(u+x) * sqrt(u-x) = 1, so the two square-root terms are reciprocals. Their sum a+1/a >= 2 (AM-GM). So f(x) >= 2. The function is even (f(x)=f(-x)), so it is NOT one-one. Range = [2, infinity) is a proper subset of R, so NOT onto.

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