## zmudz one year ago Let F(x) be the real-valued function defined for all real x except for x = 0 and x = 1 and satisfying the functional equation $$F(x) + F((x-1)/x) = 1+x.$$Find the F(x) satisfying these conditions. Write F(x) as a rational function with expanded polynomials in the numerator and denominator. Help needed! Thank you!

1. ganeshie8

$F(x)+F\left(\frac{x-1}{x}\right) =1+x\tag{1}$ replace $$x$$ by $$(x-1)/x$$ in $$(1)$$ and get $F\left(\frac{x-1}{x}\right)+F\left(\frac{-1}{x-1}\right) =1+\frac{x-1}{x}\tag{2}$ replace $$x$$ by $$-1/(x-1)$$ in $$(1)$$ and get $F\left(\frac{-1}{x-1}\right)+F\left(x\right) =1+\frac{-1}{x-1}\tag{3}$

2. ganeshie8

If you let $$F(x)=a$$, $$F\left(\frac{x-1}{x}\right)=b$$ and $$F\left(\frac{-1}{x-1}\right)=c$$, the previous equations become : $a+b = 1+x\tag{1'}$ $b+c = 1+\frac{x-1}{x}\tag{2'}$ $c+a = 1+\frac{-1}{x-1}\tag{3'}$

3. ganeshie8

3 equations and 3 unknowns (a,b,c) you can solve them