4.3. Let $f(x)=\ln \frac{e^{x}-1}{e^{x}+1}, x>0$ and $f^{-1}$ the inverse function. Find the value of the expression $f(x)+f^{-1}(-x)$.
4.3.

$$
f: R_{-} \rightarrow R_{-}, f^{-1}: R_{-} \rightarrow R_{-}
$$

$$
f^{-1}(x)=\ln \frac{1+e^{x}}{1-e^{x}} ; \quad f(x)+f^{-1}(-x)=\ln \frac{e^{x}-1}{e^{x}+1} \frac{1+e^{-x}}{1-e^{-x}}=\ln 1=0
$$