Find all real to real function satisfy this functional equation.! $f((x+y)/(x-y))=[f(x)+f(y)]/[f(x)-f(y)]$ Find all real to real function satisfy this functional equation.!

$$f\left(\frac {x+y}{x-y}\right)=\frac {f(x)+f(y)}{f(x)-f(y)}$$

I couldn't get to the final answer but I get $f(0) = 0$ and $f(1) = 1$
and this functional is an odd function
It is easy to see that $f(0) = 0, f(1) = 1, f(-x) = -f(x),$ so we assume this.

We note that $$f\left(\frac{x+a x}{x-a x}\right) = \frac{f(x) + f(x a)}{f(x) - f(x a)}.$$
Factoring out by $x$ on the RHS, we see that $f(x a) = C f(x)$ (just solve on the RHS). In fact, if we set $x = 1,$ we see that $$f(a x) = f(a) f(x).$$ In particular, this means that $f(a^2) = f(a)^2,$ so $f$ is positive on $\mathbb{R}_+.$  What is more, by using the original condition, together with positivity of $f$ on positive numbers, we see that $f$ is order-preserving. Restricting to the positive real axis, we see that $f$ is an order-preserving homomorphism of $\mathbb{R}_+,$ and as is easy to see, this means that $f$ is of the form $x\rightarrow x^c,$ for some $c$ (see this discussion). Now, by plugging into the original constraint, it is easy to see that $c=1,$ so $f(x) = x.$
Have you tried finding very simple functions that satisfy this relation? For example, can you find a quadratic function $f(x) = ax^2 + bx +c$ that works? If so, broaden your horizons! If not, what went wrong, and can you fix it?

I have a hunch that such a function must belong to a very simple class of polynomials, but have yet to verify that these are the only functions that can work. I'm not certain that oddness (in full generality) plays much of a role.

Lastly, I don't see any reason why $f(1) = 1$.