What is f(x)?

Algebra Level 4

Let \(f\colon\mathbb{R}\to\mathbb{R}\) be a function that satisfies the following property.

For all \(x\in \mathbb{R}\), \(f(x)^2=x^2\).

Consider the following statements.

\([1]\). \(f(x)\) has to be equal to \(x\) for all \(x\in\mathbb{R}\).

\([2]\). \(f(x)\) has to be equal to \(-x\) for all \(x\in\mathbb{R}\).

\([3]\). In fact, \(f(x)\) could be one of three things. \(f(x)=x\) for all real \(x\), \(f(x)=-x\) for all real \(x\) and \(f(x)=|x|\) for all real \(x\)

\([4]\). It is impossible to tell what \(f(x)\) is.

Which of these statements is correct?


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