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Math Help - Function

  1. #1
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    Function

    Detrmine f(x)=?? if:
    f(x)+f[x]+f{x}=x
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  2. #2
    MHF Contributor red_dog's Avatar
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    If x=0\Rightarrow 3f(0)=0\Rightarrow f(0)=0

    If x\in\mathbf{Z}\Rightarrow [x]=x, \ \{x\}=0

    Then f(x)+f([x])+f(\{x\})=x\Rightarrow f(x)+f(x)+f(0)=x\Rightarrow 2f(x)=x\Rightarrow
    \Rightarrow f(x)=\frac{x}{2}, \ \forall x\in\mathbf{Z}

    If x\in[0,1)\Rightarrow [x]=0, \ \{x\}=x
    Then f(x)+f([x])+f(\{x\})=x\Rightarrow f(x)+f(0)+f(x)=x\Rightarrow

    \Rightarrow 2f(x)=x\Rightarrow f(x)=\frac{x}{2}, \ \forall x\in[0,1)

    Now, let x\in\mathbf{R}\Rightarrow x=[x]+\{x\}, \ [x]\in\mathbf{Z}, \ \{x\}\in[0,1)

    Then f(x)+f([x])+f(\{x\})=x\Rightarrow f(x)+\frac{[x]}{2}+\frac{\{x\}}{2}=x\Rightarrow

    \Rightarrow f(x)=x-\frac{[x]+\{x\}}{2}=x-\frac{x}{2}=\frac{x}{2}

    So, f(x)=\frac{x}{2}, \ \forall x\in\mathbf{R}
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