A student says about the function \(h:\mathbb{R}\to\mathbb{R}\), where \(h(x)=x^3-3x\): “Since \(h(1)=h(-2)=-2\), \(h\) is not onto.” What is the correct evaluation of this statement?

Correct answer: The statement is incorrect; the function is onto but not one-one.

\(h(1)=h(-2)\) only proves that the function is not one-one. As \(x\to\infty\), \(h(x)\to\infty\), and as \(x\to-\infty\), \(h(x)\to-\infty\); hence every real value is attained. Exam tip: repeated outputs test one-one-ness, not onto-ness.