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Let f: R -> [-1, 1] be defined by f(x) = (x² - 1)/(x² + 1). Which of the following statements is NOT true?

1. f is many-one onto
2. f increases for x > 0 and decreases for x < 0
3. the minimum value is not attained even though f is bounded
4. the area enclosed between the curve y = f(x) and the line y = 1 is pi square units

Correct answer: the minimum value is not attained even though f is bounded

Solution

Since f(0) = -1, the minimum value -1 is actually attained, so the statement that it is not attained is false; that is the statement which does not hold.

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