Solving $f(x)$ in a functional equation

Question

Find of general form for $f(x)$ given $f(x)+xf\left(\displaystyle\frac{3}{x}\right)=x.$

I think we need to substitute $x$ as something else, but I'm not sure. Will $x=\displaystyle\frac{3}{x}$ help me?

Answer 1

Yes, it helps, as follows:

From $$f(x)+xf(\frac{3}{x})=x\tag{*}$$ we get $$f(\frac{3}{x})+\frac{3}{x}f(x)=\frac{3}{x}$$ or $$xf(\frac{3}{x})+3f(x)=3\tag{**}$$ from (*) anf (**), we have: $$-2f(x)=x-3$$ or $$f(x)=\frac{3-x}{2}$$

Answer 2

$$f\left(\frac{3}{x}\right)+\frac{3}{x}f(x)=\frac{3}{x}$$ Thus, putting this expression of $f(x/3)$ in the first equation gives $$f(x)=x-xf(3/x)=x+3f(x)-3$$ We finally have $f(x)=\frac{3-x}{2}$ and this function satisfies your equation.