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Math Help - 1-1 from (0, 1) onto R

  1. #1
    Junior Member
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    1-1 from (0, 1) onto R

    One can achieve a 1-1 correspondence from (0, 1) onto R easily by taking tan(pi * x - pi/2).

    When I first looked at this problem, I came up with this. I'm curious if it works too.

    Let f(x) be given by f(x) = 1/(e ^ x - 1) on {0 < x <= ln((e + 1)/2)}
    f(x) = 1/(e - e ^ x) on {ln((e + 1)/2) < x < 1}
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  2. #2
    MHF Contributor
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    Re: 1-1 from (0, 1) onto R

    Hey grandunification.

    Hint: Check to see if your function is monotonic increasing (or decreasing) in the region of your function to see whether you have a 1-1 correspondence (also assume continuity and differentiation for the mapped transform).

    The idea behind the above is that if something is monotonic and can be differentiated (with non-zero derivative at all points), then by the inverse function theorem an inverse function exists across the entire domain and co-domain and thus you have 1-1 properties.
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