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Is $\frac 1{x^2-2} $ a function from $\mathbb{R}\to \mathbb{R}$? Is it a function from $\mathbb{Z}\to \mathbb{R}$? I have been thinking about this but, I can't find any example for which you can have an input that outputs something that is either not a real number, or where there is 2 outputs for 1 input.

So I assume both examples that this formula is a function, but I feel that is not right, nor would I know how to explain my logic.

Also would $\frac10$ be part of the set of real numbers?

Andrés E. Caicedo
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Jude
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1 Answers1

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Part of how you want to think of this is on what domain is the function defined. We see that there are asymptotes when $x = \pm \sqrt{2}$, but this means that the function is not actually defined at these points, hence it is not a function from $\mathbb{R} \to \mathbb{R}$. On what domain is the function defined on then? Does this domain include $\mathbb{Z}$?

Mnifldz
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  • Yes it should be defined on the set integers your answer makes sense – Jude Apr 16 '16 at 22:10
  • @Jude But the function is defined on a domain larger than $\mathbb{Z}$. Can you tell what domain this is? – Mnifldz Apr 16 '16 at 22:13
  • this formula would be a function if we said from R->R where x cannot = $\sqrt2$ – Jude Apr 16 '16 at 22:18
  • @Jude Correct. There are a couple ways to write this. We can say $\mathbb{R}/{\pm\sqrt{2}} \to \mathbb{R}$ or more formally $(\mathbb{R} - {-\sqrt{2}, \sqrt{2}}) \to \mathbb{R}$. – Mnifldz Apr 16 '16 at 22:20
  • Ah okay, that makes a lot of sense when it comes to understanding thanks – Jude Apr 16 '16 at 22:23
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    @Jude Of course, I'm glad to help. I do want to make one suggestion though. In the future when posting questions and answers I'd highly recommend getting comfortable with the typesetting norms on this site. There is a nifty reference you can check out for details on how to do this. You can also click to see the edits I made to your post to get an idea of how to do things. – Mnifldz Apr 16 '16 at 22:26
  • alrighty will do – Jude Apr 16 '16 at 22:29
  • @Mnifldz I assume you mean $\mathbb{R}\backslash{\pm\sqrt{2}}$, not what you wrote. – B. Pasternak Apr 16 '16 at 22:35
  • @B.Pasternak I have seen both notations in practice. What's important is that one understands in context what is happening. – Mnifldz Apr 16 '16 at 22:36
  • @Mnifldz I have never seen the notation you use for leaving elements out of a set, only for denoting a group action. I also think that using that notation for leaving elements out of a set as the symbol is reminiscent of division, which is not at all what you are doing, but which is a way of viewing quotienting out by for example a subgroup. – B. Pasternak Apr 16 '16 at 22:40