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Math Help - Functions

  1. #1
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    Functions

    Determine which of the following functions  f_i: \mathbb{R} \to \mathbb{R} are injective, surjective, and bijective. If they are bijections write down their inverses.

    (i)  f_{1}(x) = x-1 is a bijection (inverse function is  y = x+1 )

    (ii)  f_{2}(x) = x^3 is a bijection (inverse is  y = \sqrt [3]{x} )

    (iii)  f_{3}(x) = x^{3}-x is not injective, but is surjective.

    (iv)  f_{4}(x) = x^{3}-3x^{2}+3x-1 is not bijective.

    (v)  f_{5}(x) = e^{x} is bijective (inverse is  y = \ln x )

    (vi)  f_{6}(x) = \begin{cases} x^{2}\ \text{if}\ x \geq 0 \\ -x^{2}\ \text{if}\ x \leq 0 \end{cases} is bijective (inverse is same conditions with function as  \sqrt{x} and  -\sqrt{x} )

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  2. #2
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    Quote Originally Posted by shilz222 View Post
    Determine which of the following functions  {\color{red} f_i: \mathbb{R} \to \mathbb{R}} are injective, surjective, and bijective. If they are bijections write down their inverses.
    (v)  f_{5}(x) = e^{x} is bijective (inverse is  y = \ln x )
    You do know that e^x  > 0? Can that be onto?
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  3. #3
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    arrgh, idiot I am. It would have an inverse if it was defined on  R \to R^{+} right?
    Last edited by shilz222; August 20th 2007 at 09:48 PM.
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  4. #4
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    are the others correct?
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  5. #5
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    Quote Originally Posted by shilz222 View Post
    Determine which of the following functions  f_i: \mathbb{R} \to \mathbb{R} are injective, surjective, and bijective. If they are bijections write down their inverses.
    (iv)  f_{4}(x) = x^{3}-3x^{2}+3x-1 is not bijective.
    Ignoring any polynomial math that you ought to recognize, have you looked at the graph of this function? It looks to me like it has an inverse...

    -Dan
    Attached Thumbnails Attached Thumbnails Functions-function.jpg  
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  6. #6
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    Quote Originally Posted by shilz222 View Post
    arrgh, idiot I am. It would have an inverse if it was defined on  R \to R^{+} right?
    Yes!
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  7. #7
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    How would you find its inverse?
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  8. #8
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    Quote Originally Posted by shilz222 View Post
    How would you find its inverse?
    You did above: y=\ln (x).
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    the one Topsquark was talking about (the graph). Can you find the inverse easily?
    Last edited by shilz222; August 21st 2007 at 01:20 PM.
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  10. #10
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    Quote Originally Posted by shilz222 View Post
    the one Topsquark was talking about (the graph). Can you find the inverse easily?
    Cannot be done easily.
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  11. #11
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Plato View Post
    Cannot be done easily.
    I beg to differ.

    f(x) = x^3 - 3x^2 + 3x - 1 = (x - 1)^3

    So the inverse function is
    f^{-1}(x) = \sqrt[3]{x} + 1

    There are no great difficulties here.

    -Dan
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  12. #12
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    I really missed that one.
    I had in mind a general qubic that is one-to-one.
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