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!

Question

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!

ganeshie8 · Answer

$$F(x)+F\left(\frac{x-1}{x}\right) =1+x\tag{1}$$

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
$$F\left(\frac{-1}{x-1}\right)+F\left(x\right) =1+\frac{-1}{x-1}\tag{3}$$

ganeshie8 · Answer

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 :

$$a+b = 1+x\tag{1'}$$
$$b+c = 1+\frac{x-1}{x}\tag{2'}$$
$$c+a = 1+\frac{-1}{x-1}\tag{3'}$$

ganeshie8 · Answer

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