Find all functions $f:\mathbb R \to \mathbb R$ that satisfy $f(x) + 3 f\left( \frac {x-1}{x} \right) = 7x $
for all nonzero $x$.

Question

Find all functions $f:\mathbb R \to \mathbb R$ that satisfy $f(x)  + 3 f\left( \frac {x-1}{x} \right) = 7x $
for all nonzero $x$.

zmudz · Answer

Hint: If you substitute $(x - 1)/x$ in for $x$, then you get a new equation that also involves $f\left( \frac{x - 1}{x} \right)$. Then do this substitution again and you may get another useful equation to consider.

ganeshie8 · Answer

whats stopping you from using the hint ?

ganeshie8 · Answer

$$f(x) = 7x-3f\left( \frac{x - 1}{x} \right)\tag{1}$$

substitute $\frac{x-1}{x}$ for $x$  :
$$f\left(\frac{x-1}{x}\right) = 7\frac{x-1}{x}-3f\left( \frac{- 1}{x-1} \right)\tag{2}$$

substitute $\frac{x-1}{x}$ for $x$  again :
$$f\left(\frac{-1}{x-1}\right) = -\frac{7}{x-1}-3f\left(x\right)\tag{3}$$

ganeshie8 · Answer

By elimination, you should end up with
$$f(x) = \dfrac{x-\frac{9}{x-1}-\frac{3(x-1)}{x}}{4} = \frac{x^3-4x^2-3x-3}{4x(x-1)}$$