Which of the following functions \(f:\mathbb{R}\to\mathbb{R}\) is one-one?

Correct answer: \(f(x)=x^3\)

The function \(f(x)=x^3\) is strictly increasing on the real numbers. Hence, if \(x_1\ne x_2\), then \(x_1^3\ne x_2^3\); distinct inputs give distinct outputs, so it is one-one. The function \(x^2\) is not one-one because \(f(2)=f(-2)\). Similarly, \(|x|\) gives the same output for positive and negative inputs of equal magnitude. Exam tip: To disprove one-one-ness, find two different inputs with the same output.