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Let f(x) be a non-constant function satisfying the integral equation f(x) = integral from 0 to 1 of [1/f(x*t)] dt, with f(1) = 0. Which of the following is NOT true?

1. f(x) is a polynomial function
2. f(x) is bounded
3. The range of f(x) is [0, 1]
4. f(2) = 1/2

Correct answer: f(x) is a polynomial function

Solution

The integral equation and condition f(1) = 0 together suggest f is not a polynomial (a polynomial satisfying such a functional equation and having a specific zero involves transcendental behavior). The most direct statement that is NOT true is that f(x) is a polynomial function, as the functional equation is more naturally satisfied by logarithmic or power-type functions that are not polynomials.

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