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Math Help - Find all The Functions !

  1. #1
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    Functional equation !

    Hii !

    Can You Help Me For This Exercice :

    Find all The Functions f : \mathbb{R} \longrightarrow \mathbb{R} Such as :

     f((f(x)+y)=f(x^2-y)+4f(x)y
    Last edited by Perelman; February 5th 2010 at 10:30 AM.
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  2. #2
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    Quote Originally Posted by Perelman View Post
    Hii !

    Can You Help Me For This Exercice :

    Find all The Functions f : \mathbb{R} \longrightarrow \mathbb{R} Such as :

     f((f(x)+y)=f(x^2-y)+4f(x)y
    Put y=-f(x):

    (1)\qquad f(0) = f(x^2 + f(x)) - 4(f(x))^2.

    Put y=x^2:

    (2)\qquad f(f(x)+x^2) = f(0) + 4f(x)x^2.

    Combine (1) and (2) to get f(x)\bigl(x^2-f(x)\bigr) = 0.

    That gives you two solutions: f(x) = 0 and f(x)=x^2. I'll leave you to think about whether or not there are also hybrid solutions, where f(x) is 0 for some values of x, and x^2 for other values of x.
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  3. #3
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    A particular solution is f(x)=x^2.
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