A functional equation that is equal to 7x

Question

I wish to find all of the functions $f:\mathbb R \to \mathbb R$ such that $$ f(x) + 3 f\left( \frac {x-1}{x} \right) = 7x $$ for all nonzero $x$.

I have tried plugging in $\frac{x-1}{x}$, but that has been of no avail

Answer 1

If $g(x) = (x-1)/x$, we have $g(g(x)) = 1/(1-x)$ and $g(g(g(x))) = x$. Thus $$ \eqalign{f(x) + 3 f(g(x)) &= 7 x\cr f(g(x)) + 3 f(g(g(x)) &= 7 g(x)\cr f(g(g(x))) + 3 f(x) &= 7 g(g(x))\cr} $$ Now solve...

The cases $x=0$ and $x=1$ need special attention.