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Prove that if the function $f: \mathbb R \to \mathbb R$ is an odd degree polynomial, for every number $y ∈ \mathbb R$ there exists such a number $x ∈ \mathbb R$ that $f(x) = y$. Prove that this is not true for any even polynomial.

I have trouble with this proof. I don't know how to write it for any degree of polynomial. How can I correctly prove it?

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    $\begingroup$ Hint: $f$ is continuous, $f(-\infty)=\mp\infty$, $f(\infty)=\pm\infty$. $\endgroup$ – Yves Daoust Sep 5 '18 at 13:19
  • $\begingroup$ @LDM yes, I adited it. $\endgroup$ – CezarySzulc Sep 5 '18 at 13:21
  • $\begingroup$ The function $f(x) = |x|$ is not a polynomial at all (and thus in particular not an even degree polynomial). It is an even function, but this is a, while related, different notion. $\endgroup$ – Mees de Vries Sep 5 '18 at 13:23
  • $\begingroup$ @YvesDaoust Oh, so it is a full prove. It's enought to write this conditions for odd and even polynomial, isn't it? $\endgroup$ – CezarySzulc Sep 5 '18 at 13:23
  • $\begingroup$ @MeesdeVries You are right. I'm wrong in this condition. $\endgroup$ – CezarySzulc Sep 5 '18 at 13:25
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Hints:

Let's assume for now that the leading coefficient of $f$ is positive (otherwise, we can just take the polynomial $-f$ and be in the same boat). Then, use the facts:

  • $\lim_{x\to\infty} f(x) = \infty$ and $\lim_{x\to-\infty} f(x) = -\infty$
  • $f$ is a continuous function

Using these two facts, try and prove:

  1. There exists some $x\in\mathbb R$ such that $f(x) > y$
  2. There exists some $x\in\mathbb R$ such that $f(x) < y$
  3. There exists some $x\in\mathbb R$ such that $f(x) = y$

Try and prove these $3$ things (in order!) and tell us how far you got and where you are perhaps still stuck.

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  • $\begingroup$ Can you show me first prove. I don't know how exactly I should start. $\endgroup$ – CezarySzulc Sep 5 '18 at 14:27
  • $\begingroup$ @Cezary.Sz Use the first of the two facts I have written down. Look at the definitions of what $\lim_{x\to\infty} f(x) = \infty$ means. $\endgroup$ – 5xum Sep 5 '18 at 14:30
  • $\begingroup$ I think I understand $\lim_{x\to\infty} f(x) = \infty$. So next step should be check $\infty > y$, $\infty < y$ and $\infty = y$? $\endgroup$ – CezarySzulc Sep 5 '18 at 14:34
  • $\begingroup$ @Cezary.Sz That's not what that means. Please look at your textbook. $\infty$ is not a number. $\endgroup$ – 5xum Sep 5 '18 at 15:27

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