1

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

bob the pie
  • 489
  • 2
    You should have improved the earlier version rather than delete it. This one has an answer, so deleting it would be rude. – Jyrki Lahtonen Jul 29 '15 at 06:58

1 Answers1

5

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.

Robert Israel
  • 448,999