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Which of the following functions is surjective (onto) but not injective (one-one)?

1. f: R -> R, f(x) = x⁴ + 2x³ - x² + 1
2. f: R -> R, f(x) = x³ + x + 1
3. f: R -> R+, f(x) = sqrt(1 + x²)
4. f: R -> R, f(x) = x³ + 2x² - x + 1

Correct answer: f: R -> R, f(x) = x³ + 2x² - x + 1

Solution

A continuous cubic R -> R always covers all of R (surjective). It is injective only if monotonic. Option D's derivative has two real roots, so it is not one-one while remaining onto.

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