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Math Help - Polynomials and inverses.

  1. #1
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    Polynomials and inverses.

    Let P be a polynomial of degree n.

    1.) Can P have an inverse if n is even. Support your answer.

    2.) Can P have an inverse if n is odd? If so, give an example. Then give an example of a polynomial of odd degree that does not have an inverse.

    For 1.) I think that no P's can have an inverse if they are even because they would fail the HLT.

    For 2.) I'd say yes P can have an inverse if n is odd such as the inverse of f(x) = x^3 is f^-1(x) = x^(1/3).

    I cannot think of an example of a polynomial of od degree that does not have an inverse though. Please help
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  2. #2
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    Quote Originally Posted by WartonMorton View Post
    Let P be a polynomial of degree n.

    1.) Can P have an inverse if n is even. Support your answer.

    2.) Can P have an inverse if n is odd? If so, give an example. Then give an example of a polynomial of odd degree that does not have an inverse.

    For 1.) I think that no P's can have an inverse if they are even because they would fail the HLT.

    For 2.) I'd say yes P can have an inverse if n is odd such as the inverse of f(x) = x^3 is f^-1(x) = x^(1/3).

    I cannot think of an example of a polynomial of od degree that does not have an inverse though. Please help
    How about f(x) = x^3 - 3x ?
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    Quote Originally Posted by WartonMorton View Post
    Let P be a polynomial of degree n.

    1.) Can P have an inverse if n is even. Support your answer.

    2.) Can P have an inverse if n is odd? If so, give an example. Then give an example of a polynomial of odd degree that does not have an inverse.

    For 1.) I think that no P's can have an inverse if they are even because they would fail the HLT.

    For 2.) I'd say yes P can have an inverse if n is odd such as the inverse of f(x) = x^3 is f^-1(x) = x^(1/3).

    I cannot think of an example of a polynomial of od degree that does not have an inverse though. Please help
    I believe all polynomials w=f(z) have inverses except at the zeros of f'(z), with the derivative of the inverse given by \frac{df^{-1}}{dw}=\frac{1}{f'(z(w))}.
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    Quote Originally Posted by shawsend View Post
    I believe all polynomials w=f(z) have inverses except at the zeros of f'(z), with the derivative of the inverse given by \frac{df^{-1}}{dw}=\frac{1}{f'(z(w))}.
    Be careful about what you are calling "inverses". Every function, f, has a local inverse about a point where f'(x) is not 0. That is, the function defined by restricting f to some small neighborhood of the point has an inverse.

    The problem is clearly talking about "the" inverse such that f^{-1}(f(x)= x and f(f^{-1}(x))= x for all x. Such a thing exists if and only if f'(x)\ge 0 for all x or f'(x)\le 0 for all x.
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