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Math Help - Vertical Triangle Wave

  1. #1
    Junior Member masoug's Avatar
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    Question Vertical Triangle Wave

    Is there a way to write a series of functions that describe the inverse of \frac{2}{\pi}sin^{-1}sin(\pi x)? I am aiming for a vertical triangle wave the travels up the y axis.

    I think there may not be a series of functions that make this work, because any form of the relation in the vertical form violates the definition of a function...

    Thanks!
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  2. #2
    A Plied Mathematician
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    Just take this plot and reflect it over the line y = x. You're correct in that there's no function that will do what you're asking, because it will be multi-valued. However, it is a relation, and you can plot it. The result should be something like this.
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  3. #3
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    Quote Originally Posted by masoug View Post
    Is there a way to write a series of functions that describe the inverse of \frac{2}{\pi}sin^{-1}sin(\pi x)? I am aiming for a vertical triangle wave the travels up the y axis.

    I think there may not be a series of functions that make this work, because any form of the relation in the vertical form violates the definition of a function...

    Thanks!
    Maybe this helps (?):

    f_k(x)=\left\{\begin{array}{lccl}\frac12 x + 2k,&k\in\mathbb{Z},&x\in(-1,1] \\ -\frac12 x + 2k+1,&k\in\mathbb{Z},&x\in [-1,1) \end{array}\right.

    I've attached the graph of the given function and the family of straight lines forming the inverse of the function.
    Attached Thumbnails Attached Thumbnails Vertical Triangle Wave-dreieckswelle.png  
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  4. #4
    Junior Member masoug's Avatar
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    Quote Originally Posted by Ackbeet View Post
    Just take this plot and reflect it over the line y = x. You're correct in that there's no function that will do what you're asking, because it will be multi-valued. However, it is a relation, and you can plot it. The result should be something like this.
    So function plotters that I use: Function Grapher cannot graph this function...

    Quote Originally Posted by earboth View Post
    Maybe this helps (?):

    f_k(x)=\left\{\begin{array}{lccl}\frac12 x + 2k,&k\in\mathbb{Z},&x\in(-1,1] \\ -\frac12 x + 2k+1,&k\in\mathbb{Z},&x\in [-1,1) \end{array}\right.

    I've attached the graph of the given function and the family of straight lines forming the inverse of the function.
    Hmm, so we need to use complex numbers to be able to graph the functions...

    Thanks for the help anyway!
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  5. #5
    A Plied Mathematician
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    Quote Originally Posted by masoug View Post
    So function plotters that I use: Function Grapher cannot graph this function...
    Right. It's not a function, because it fails the vertical line test. You have to have software that will plot implicit functions, or multi-valued "functions".

    Hmm, so we need to use complex numbers to be able to graph the functions...

    Thanks for the help anyway!
    I don't see where complex numbers come into it at all. Where are you seeing them?
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  6. #6
    Junior Member masoug's Avatar
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    Quote Originally Posted by earboth View Post

    f_k(x)=\left\{\begin{array}{lccl}\frac12 x + 2k,&k\in\mathbb{Z},&x\in(-1,1] \\ -\frac12 x + 2k+1,&k\in\mathbb{Z},&x\in [-1,1) \end{array}\right.
    \mathbb{Z}

    I thought that meant complex number...
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  7. #7
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    Quote Originally Posted by masoug View Post
    \mathbb{Z}

    I thought that meant complex number...
    I'm sorry for the confusion: My apologies.

    But in Germany \mathbb{Z} denotes the set of all integers and \mathbb{C} the set of complex numbers.

    (But of course you are right that complex variables often are labeled with z)
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  8. #8
    A Plied Mathematician
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    Quote Originally Posted by earboth View Post
    I'm sorry for the confusion: My apologies.

    But in Germany \mathbb{Z} denotes the set of all integers and \mathbb{C} the set of complex numbers.
    This is also the usual notation in the United States.
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