# 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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