Find the function f for $f$ be a function and $f(x) + f ( \frac{3x-1}{13x-4} ) = x$

Question

How to solve this algebraic functional equation? Let $f$ be a function and $f(x) + f ( \frac{3x-1}{13x-4} ) = x$.
Find the function f, in terms of x.

Answer 1

There is no general method. Functional equations are not solved at all unless you are lucky. This one can be solved because the function $3x-1\over13x-4$, when applied iteratively three times, returns x.

Using this fact, we may see: $$f(x)+f\left({3x-1\over13x-4}\right)=x\tag1$$ $$f\left({3x-1\over13x-4}\right)+f\left({1-4x\over3-13x}\right)={3x-1\over13x-4}\tag2$$ $$f\left({1-4x\over3-13x}\right)+f(x)={1-4x\over3-13x}\tag3$$

Now add together the equations (1) and (3), subtract (2) and divide by 2 to get the formula you are after. When brought to the common denominator, it looks wonderfully cryptic: $$f(x)={1\over2}\left(x-{3x-1\over13x-4}+{1-4x\over3-13x}\right)={169x^3-78x^2+5x+1\over338x^2-182x+24}$$