How to solve functional equation?

Question

I have been struggling to solve this functional equation. Could anyone suggest ways to solve it? It's this: $$f\left(\frac{x-3}{2x+4}\right)=\frac{x+1}{3x-1}.$$ How do I solve it? Thank you in advance!

Answer 1

Hint: Let $$\frac{x-3}{2x+4} = y.$$ Consequently, $$x-3 =y(2x+4) = 2yx+4y \implies (1-2y)x = 4y+3 \implies x = \frac{4y+3}{1-2y}.$$

Answer 2

let $$t=\frac{x-3}{2x+4}$$ then we get by solving for $x$: $$x=\frac{-4t-3}{2t-1}$$ can you finish? now you must plug this $x$ in the right-hand side of the equation above and we get $$f(t)=\frac{t+2}{7t+4}$$