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Let f: R → R be defined by f(x) = x + sqrt(x²). Determine the nature of f.

1. injective
2. surjective
3. bijective
4. None of these

Correct answer: None of these

Solution

sqrt(x²) = |x|, so f(x) = x + |x|. For x >= 0, f = 2x; for x < 0, f = 0. f is not injective (all negative x map to 0) and not surjective (range is [0, ∞), not R). Hence it is neither injective, surjective, nor bijective — None of these.

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