Physics1872
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Q: Suppose f(x) is odd and periodic. Show that the graph of f(x) crosses the x-axis infinitely often.

No idea where to start on this one. I know a few examples such as sin(x), but how can i show a general odd, periodic function crosses the x-axis infinitely often.
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Plücker
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(Original post by Physics1872)
Q: Suppose f(x) is odd and periodic. Show that the graph of f(x) crosses the x-axis infinitely often.

No idea where to start on this one. I know a few examples such as sin(x), but how can i show a general odd, periodic function crosses the x-axis infinitely often.
What do you know about odd functions?
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Physics1872
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(Original post by Plücker)
What do you know about odd functions?
I know that f(-x)=-f(x) means the function is odd.
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Plücker
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(Original post by Physics1872)
I know that f(-x)=-f(x) means the function is odd.
So what does that tell you about f(0)?
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Physics1872
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(Original post by Plücker)
So what does that tell you about f(0)?
f(0)=0, even if it's -x, f(-0)=0
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Plücker
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(Original post by Physics1872)
f(0)=0, even if it's -x, f(-0)=0
So f(0)=0 and the function is periodic...
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Plücker
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That is of course is f(0) is defined.
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Physics1872
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(Original post by Plücker)
So f(0)=0 and the function is periodic...
Wait so the curve crosses the x-axis when y=0, and when y=0, x=0. I'm not sure what conclusion to draw here...
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Plücker
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(Original post by Physics1872)
Wait so the curve crosses the x-axis when y=0, and when y=0, x=0. I'm not sure what conclusion to draw here...
If f(0)=0 and the function is periodic with period P then f(nP)=0 for all natural numbers n.

However the question is wrong. f(x)=1/sin(x) is odd and periodic and never crosses the x axis.
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Physics1872
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(Original post by Plücker)
If f(0)=0 and the function is periodic with period P then f(nP)=0 for all natural numbers n.

However the question is wrong. f(x)=1/sin(x) is odd and periodic and never crosses the x axis.
Regarding the first line, how is f(0)=0 related to f(nP)=0. I just don't *see* that intuitively.
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Plücker
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(Original post by Physics1872)
Regarding the first line, how is f(0)=0 related to f(nP)=0. I just don't *see* that intuitively.
Have a look at the definition of a periodic function. https://en.wikipedia.org/wiki/Period...ion#Definition
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Physics1872
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(Original post by Plücker)
Have a look at the definition of a periodic function. https://en.wikipedia.org/wiki/Period...ion#Definition
Oh so for example, if f(x)= sinx, six(x+2π)=sin(x+3π)=sin(x+4π) =sin(x+nπ)= sinx. If sinx=0, then all the others will also equal 0 ( since they are equal). So the graph crosses the x-axis infinitely because you can have infinite n values. Is this the right line of thinking? Regarding the Q, I got it from MIT open courseware, and I think there are also some other questions where either the solution is incorrect or the question is weird. Maybe they are outdated and not proof read.
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Plücker
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(Original post by Physics1872)
Oh so for example, if f(x)= sinx, six(x+2π)=sin(x+3π)=sin(x+4π) =sin(x+nπ)= sinx. If sinx=0, then all the others will also equal 0 ( since they are equal). So the graph crosses the x-axis infinitely because you can have infinite n values. Is this the right line of thinking? Regarding the Q, I got it from MIT open courseware, and I think there are also some other questions where either the solution is incorrect or the question is weird. Maybe they are outdated and not proof read.
You really want even multiples of pi there. sin(pi)=sin(2pi)=sin(3pi)=0 but it wouldn't we true to say that \sin(x+2\pi) \equiv \sin(x+3\pi).

As for the question and the source, I think MIT open courseware is a great resource and this question is easily fixed. They could say, "Show that ...... infinitely many times or never."
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Physics1872
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(Original post by Plücker)
You really want even multiples of pi there. sin(pi)=sin(2pi)=sin(3pi)=0 but it wouldn't we true to say that \sin(x+2\pi) \equiv \sin(x+3\pi).

As for the question and the source, I think MIT open courseware is a great resource and this question is easily fixed. They could say, "Show that ...... infinitely many times or never."
Oh right, I see what you mean. Yea, I understand now. Thanks! Also, yea I love using MIT OCW. Their resources are great. And whenever there is an error in the solution, it's usually a printing error. Just happened to see a discussion on it, I didn't identify it by myself .
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RDKGames
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(Original post by Physics1872)
Regarding the first line, how is f(0)=0 related to f(nP)=0. I just don't *see* that intuitively.
The intuitive explanation is that if the function crosses the x-axis once, then it will cross the x-axis at some othet point because the function is periodic; i.e. the function will repeat eventually.

Hence, repeating this logic, you get infinitely many x-axis intersections.
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Physics1872
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(Original post by RDKGames)
The intuitive explanation is that if the function crosses the x-axis once, then it will cross the x-axis at some othet point because the function is periodic; i.e. the function will repeat eventually.

Hence, repeating this logic, you get infinitely many x-axis intersections.
Ahhh that's a really neat way of seeing it. Thanks!
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