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For real x, let f(x)=x³+5x+1, then

1. f is onto R but not one-one
2. f is one-one and onto R
3. f is neither one-one nor onto R
4. f is one-one but not onto R

Correct answer: f is one-one and onto R

Solution

f'(x)=3x^2+5>0 for all x, so f is strictly increasing and hence one-one. Being an odd-degree polynomial it ranges over all of R, so it is onto. Therefore f is both one-one and onto.

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