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Math Help - Bijection

  1. #1
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    Bijection

    Prove that there is a bijection between (0,1) and (b,infinity) for any real number b.

    A function is needed. I've though about the function f(x)=b/x but it won't work if b<=0. Any idea?
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  2. #2
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    Quote Originally Posted by TitaniumX View Post
    Prove that there is a bijection between (0,1) and (b,infinity) for any real number b.

    A function is needed. I've though about the function f(x)=b/x but it won't work if b<=0. Any idea?
    How about f(x) = b - 1 + 1/x?
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by TitaniumX View Post
    Prove that there is a bijection between (0,1) and (b,infinity) for any real number b.

    A function is needed. I've though about the function f(x)=b/x but it won't work if b<=0. Any idea?
    f (x)= tan(\pi x - \pi/2) is (I believe) standard for expanding (0, 1) to \mathbb{R}, so I would use this.

    This can be relatively easily manipulated to give you the function you want.

    (Half it to give you a bijection between (0, 1) and (0, \infty), then scale).
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