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Find all functions $f$ such that $f\left(\frac{1}{x}\right)+(x+1)f(x)=1,\space x\neq0$.

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3 Answers 3

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You must to change $x$ by $1/x$ and obtain a system.

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Solve this linear system on $f(x)$ and $f(1/x)$ $$ f\left(\frac{1}{x}\right)+(x+1)f(x)=1 $$ $$ f(x)+\left(\frac1x+1\right)f\left(\frac{1}{x}\right)=1 $$ and get $$ f(x) = \frac1{x^2+x+1} $$

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  • $\begingroup$ But is it the only solution? He asked for all the possible solutions for $f(x)$ $\endgroup$
    – user265328
    Commented Dec 7, 2015 at 10:22
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    $\begingroup$ @santa, it is because the linear system has only this solution. $\endgroup$
    – lhf
    Commented Dec 7, 2015 at 10:23
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Alternatively,

$$f\left(\frac1x\right)=1-(x+1)f(x),$$ so that $$f(x)=1-\left(\frac1x+1\right)f\left(\frac1x\right)=1-\left(\frac1x+1\right)(1-(x+1)f(x)),$$and $$f(x)=\frac1{(x+1)^2-x}.$$

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