Function

**Ortega** · February 18th 2009, 03:47 AM

Detrmine f(x)=?? if:
f(x)+f[x]+f{x}=x

**red_dog** · February 18th 2009, 07:23 AM

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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