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Consider a simple case:

Let $f$ be a function from $\textbf{R}^+$ to $\textbf{R}$, such that $f(x) = \sqrt{x}$.

It is known that a positive real number can have two square roots. A basic case: $\sqrt{4} = \{+2, -2\}$. However, this goes against the very definition of a function:

"a function is a binary relation between two sets that associates every element of the first set to exactly one element of the second set." (Source: wikipedia)

What am I missing?

Susmit Agrawal
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2 Answers2

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What you are missing is that $\sqrt x$ does not mean “square root of $x$”; instead, it means “non-negative square root of $x$”.

José Carlos Santos
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  • Okay, got this point. Is there a particular representation for the general definition of square root (or nth root for that matter), that includes both positive and negative roots? And for that specific operator, what would be the answer? – Susmit Agrawal Feb 09 '21 at 18:46
  • I've seen $x^{1/n}$ to denote a generic $n^\text{th}$-root, but not often. – José Carlos Santos Feb 09 '21 at 18:48
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$\sqrt{\cdot}$ is defined to be the positive square root. There are two numbers which square to $4$, but only one of them is $\sqrt{4}$. $\sqrt{\cdot}$ is a function.

Patrick Stevens
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  • Thank you for your response. But could you clarify it for the general definition of nth root, as described in the comment of the other answer? – Susmit Agrawal Feb 09 '21 at 18:50
  • @SusmitAgrawal The general definition of an $n$th root is perfectly clear and well-defined, but there is no notation for it because, as you pointed out, it is usually not unique. – Randall Feb 09 '21 at 18:51