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Consider f(x) = x/(1 + x) with domain [0, infinity) and codomain [0, infinity). The function f is:

1. one-one but not onto
2. one-one and onto
3. many one but not onto
4. many one and onto

Correct answer: one-one but not onto

Solution

f(x) = x/(1+x) = 1 - 1/(1+x). For x >= 0 this is strictly increasing, hence one-one. As x goes 0 -> infinity, f goes 0 -> 1 (never reaching 1), so the range is [0,1). Since the codomain is [0,infinity) and the range is only [0,1), f is not onto.

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