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?
Best Answer
Hint: calculate the inverse function of $$y= {2x+29\over x-2}$$
and you will see it is the same function.
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 $$f(x) = 60y-40x+16={120x+60\cdot 29\over x-2}-40x+16$$