Functional equation $f\left(\frac{1}{x}\right)+(x+1)f(x)=1$

Question

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

Answer 1

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 $$ f(x) = \frac1{x^2+x+1} $$

Answer 2

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

Answer 3

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 $$f(x)=\frac1{(x+1)^2-x}.$$