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Math Help - Find an elementary function

  1. #1
    MHF Contributor alexmahone's Avatar
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    Find an elementary function

    Find an elementary function of calculus that maps an interval one to one onto \mathbb{R}.
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  2. #2
    Super Member girdav's Avatar
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    Re: Find an elementary function

    By "interval", you mean a bounded interval? For example, \tan :\left]-\frac{\pi}2,\frac{\pi}2\right[\rightarrow \mathbb R.
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Find an elementary function

    Quote Originally Posted by alexmahone View Post
    Find an elementary function of calculus that maps an interval one to one onto \mathbb{R}.
    f:{(0,1)} \to \mathbb{R}

    f(x)=\frac{1}{x}+\frac{1}{x-1}
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Find an elementary function

    Quote Originally Posted by alexmahone View Post
    Find an elementary function of calculus that maps an interval one to one onto \mathbb{R}.
    We can build that function geometrically and represent intuitive look of such function.


    \mathbb{R} is the x axis. We curve (0,1) to half o a circle which his diameter parallel to x.

    We see every point x\in (0,1) as a point on half circle.

    Now let us define f(x):

    For every x\in (0,1) we will pass straight line joining the center of the circle with point x\in (0,1) (which is on the half circle) and continue to x axis(which is the whole \mathbb{R}.

    The intersection point which that line makes with x axis is f(x).

    It easy to see that f is one-to-one from (0,1) onto \mathbb{R}.
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