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Question Number 21021 by youssoufab last updated on 10/Sep/17

Commented by $@ty@m last updated on 10/Sep/17

$$pl.translateintoEnglish$$

Commented by Joel577 last updated on 10/Sep/17

$$\mathrm{I}\mathrm{think}\mathrm{it}\mathrm{said}\mathrm{find}f\left(x\right)\mathrm{if}$$$$f\left(x\right)+xf(1-x)=1+x$$

Answered by $@ty@m last updated on 11/Sep/17

$$\mathrm{Replacing}x\mathrm{by}1-x,weget$$$$f(1-x)+(1-x)f\left(x\right)=2-x$$$$\Rightarrow f(1-x)=(x-1)f\left(x\right)+2-x--\left(1\right)$$$$fromthegivenequation,$$$$f(1-x)=\frac{1}{x}\{1+x-f\left(x\right)\}--\left(2\right)$$$$from\left(1\right)and\left(2\right),weget$$$$x\{(x-1)f\left(x\right)+2-x\}=1+x-f\left(x\right)$$$$(x^2-x)f\left(x\right)+2x-x^2+f\left(x\right)=1+x$$$$(x^2-x+1)f\left(x\right)=1+x-2x+x^2$$$$f\left(x\right)=\frac{1-x+x^2}{x^2-x+1}$$$$f\left(x\right)=1$$

Commented by youssoufab last updated on 10/Sep/17

$$$$$$f\left(x\right)+xf(1-x)=x+1$$$$\iff f(x+1/2)+(x+1/2)f(-x-1/2)=x+1/2+1$$$$y=x+1/2$$$$\iff f\left(y\right)+yf(-y)=y+1$$$$\iff f\left(y\right)=y+1-yf(-y)$$$$\iff f(-y)=-y+1+yf\left(y\right)$$$$\iff f\left(y\right)=y+1-y[-y+1+yf\left(y\right)]$$$$\iff f\left(y\right)=y+1+y^2-y-y^{2}f\left(y\right)$$$$\iff f\left(y\right)=1+y^2-y^{2}f\left(y\right)$$$$\iff (1+y^2)f\left(y\right)=1+y^2$$$$\iff f\left(y\right)=\frac{1+y^2}{1+y^2}$$$$f\left(y\right)=1$$

Commented by $@ty@m last updated on 10/Sep/17

$$Istill{didn}^{\prime }tgetthequestion.$$

Commented by myintkhaing last updated on 11/Sep/17

$$\mathrm{equation}\left(1\right)\mathrm{is}\mathrm{wrong}$$

Commented by $@ty@m last updated on 11/Sep/17

$$thanks$$$$correctednow.$$

Commented by youssoufab last updated on 11/Sep/17

$$thanksforhelp!$$

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