Which of the following functions \(f:\mathbb{R}\to\mathbb{R}\) is one-one but not onto?

Correct answer: \(f(x)=e^x\)

For \(f(x)=e^x\), the output is positive for every real \(x\), and the function is strictly increasing. Hence, distinct inputs give distinct outputs, so it is one-one. However, it never produces 0 or a negative value; its range is \((0,\infty)\). Therefore, it is not onto \(\mathbb{R}\). The function \(x^3\) is both one-one and onto, whereas \(x^2\) is not one-one. Exam tip: To test whether a function is onto, check whether every element of the codomain occurs in its range.