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Math Help - Find all functions !

  1. #1
    Junior Member
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    Find all functions !

    Hii !

    Find all the functions f : \mathbb{R} \longrightarrow \mathbb{R} such as :

    f(xf(y))=f(xy)+x
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let y=0. Then f(xf(0))=f(0)+x.

    If f(0)=0\Rightarrow f(0)=f(0)+x, \ \forall x\in\mathbb{R}\Rightarrow x=0, \ \forall x\in\mathbb{R}, contradiction.

    Then f(0)\neq 0. Let f(0)=a, \ a\neq 0.

    Then f(ax)=a+x, \ \forall x\in\mathbb{R}.

    Replace x with \frac{x}{a}:

    f(x)=a+\frac{x}{a}.
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