# A Functional equation in one variable

Let $$f: \mathbb R\setminus\{2\} \rightarrow \mathbb R$$ be a function satisfying the following functional equation:

$$2f(x) + 3f\left(\frac{2x+29}{x-2}\right) = 100x+80$$

Find $$f(x)$$.

I tried observing the behaviour of the function at $$x\rightarrow 2$$ and $$x\rightarrow \infty$$ but still could not get a good conclusion. How do I approach the problem?

Hint: calculate the inverse function of $$y= {2x+29\over x-2}$$
Replace $$x$$ with $$y$$ and you get: $$2f(y) +3f(x) = 100y+80$$
solve this together with starting equation on $$2f(x) +3f(y) = 100x+80$$
on $$f(x)$$ and you get: $$3(2f(y) +3f(x))-2(2f(x) +3f(y)) = 3(100y+80)-2(100x+80)$$
so $$f(x) = 60y-40x+16={120x+60\cdot 29\over x-2}-40x+16$$