Find all functions $f:\mathbb R \to \mathbb R$ that satisfy: $$ f(x) + 3 f\left( \frac {x-1}{x} \right) = 7x.$$
Some progress: I plugged-in $\dfrac{x-1}{x}$ and $\dfrac{1}{1-x}$, got a system of equations and solving I got $f(x) = \dfrac{x^3-4x^2-3x-3}{4x^2-4x}$. But after testing it back looks like it's not sufficient.
Set $T(x) = \frac{x}{x-1}$. Then the composition $T^3(x) = x$. We have $f(Tx) = a x + b f(x)$ for $ a= \frac{7}{3}$ and $b = -\frac{1}{3}$, so \begin{align*} f(x) &= f(T^3(x)) \\ &= a T^2(x) + b f(T^2 x) \\ &= a T^2(x) + a b T(x) + b^2 f(Tx) \\ &= a T^2(x) + a b T(x) +a b^2 x + b^3 f(x). \end{align*} That relation gives $f$ as a rational function of $x$.