What is f(x)?

Algebra Level 4

Let f ⁣:RRf\colon\mathbb{R}\to\mathbb{R} be a function that satisfies the following property.

For all xRx\in \mathbb{R}, f(x)2=x2f(x)^2=x^2.

Consider the following statements.

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

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

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

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

Which of these statements is correct?

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