# Periodic functions

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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.

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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#2

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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.

**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.

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What do you know about odd functions?

**Plücker**)What do you know about odd functions?

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#4

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I know that f(-x)=-f(x) means the function is odd.

**Physics1872**)I know that f(-x)=-f(x) means the function is odd.

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So what does that tell you about f(0)?

**Plücker**)So what does that tell you about f(0)?

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#6

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f(0)=0, even if it's -x, f(-0)=0

**Physics1872**)f(0)=0, even if it's -x, f(-0)=0

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So f(0)=0 and the function is periodic...

**Plücker**)So f(0)=0 and the function is periodic...

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#9

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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...

**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...

However the question is wrong. f(x)=1/sin(x) is odd and periodic and never crosses the x axis.

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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.

**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.

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#11

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Regarding the first line, how is f(0)=0 related to f(nP)=0. I just don't *see* that intuitively.

**Physics1872**)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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Have a look at the definition of a periodic function. https://en.wikipedia.org/wiki/Period...ion#Definition

**Plücker**)Have a look at the definition of a periodic function. https://en.wikipedia.org/wiki/Period...ion#Definition

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#13

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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.

**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.

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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You really want even multiples of pi there. sin(pi)=sin(2pi)=sin(3pi)=0 but it wouldn't we true to say that .

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."

**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 .

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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#15

**Physics1872**)

Regarding the first line, how is f(0)=0 related to f(nP)=0. I just don't *see* that intuitively.

**repeat**eventually.

Hence, repeating this logic, you get infinitely many x-axis intersections.

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(Original post by

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

Hence, repeating this logic, you get infinitely many x-axis intersections.

**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.

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