Given$$2xf(x)+(x-3)f\left(\frac{1}{1-x}\right)=4x^2-10x-\frac{1}{2}$$ Find $f(x)$.
This's the first time I see this kind of question, I have no idea. Please help. Thank you.
Given$$2xf(x)+(x-3)f\left(\frac{1}{1-x}\right)=4x^2-10x-\frac{1}{2}$$ Find $f(x)$.
This's the first time I see this kind of question, I have no idea. Please help. Thank you.
Hint: In equatoin $$ 2xf(x)+(x-3)f\left(\frac{1}{1-x}\right)=4x^2-10x-\frac{1}{2}\tag{1} $$ make change of variables $x\to 1/(1-x)$ to get $$ \frac{2}{1-x}f\left(\frac{1}{1-x}\right)+\frac{3x-2}{1-x}f\left(\frac{x-1}{x}\right)=-\frac{x^2-22x+13}{2(1-x)^2}\tag{2} $$ and make the same change one more time to get $$ \frac{2 (x-1)}{x}f\left(\frac{x-1}{x}\right)-\frac{(2 x+1) f(x)}{x}=\frac{4}{x^2}+\frac{2}{x}-\frac{13}{2}\tag{2} $$ From $(1)$, $(2)$ and $(3)$ you can find $$ f(x)\qquad f\left(\frac{1}{1-x}\right)\qquad f\left(\frac{x-1}{x}\right) $$