8. Let $f(x)=\frac{1}{3 x-2}$. Find $f^{-1}(x)$ and state its domain.

Step 1 ：Let \(f(x)=\frac{1}{3 x-2}\). We are asked to find the inverse function \(f^{-1}(x)\) and state its domain.

Step 2 ：To find the inverse of a function, we switch the x and y (or f(x)) values and solve for y. This gives us the equation \(x = \frac{1}{3y - 2}\).

Step 3 ：Solving this equation for y, we get \(y = \frac{2x + 1}{3x}\).

Step 4 ：Therefore, the inverse function is \(f^{-1}(x) = \frac{2x + 1}{3x}\).

Step 5 ：The domain of the original function \(f(x)\) is all real numbers except \(\frac{2}{3}\), so the range of the original function (which is the domain of the inverse function) is all real numbers except \(-\frac{1}{2}\).

Step 6 ：\(\boxed{f^{-1}(x) = \frac{2x + 1}{3x}, \text{domain: } x \neq -\frac{1}{2}}\)