1. The problem is to find the domain of the inverse function of $f(x) = \frac{2x}{3x - 4}$.\n\n2. First, note that the original function $f(x)$ has domain where the denominator is not zero: $$3x - 4 \neq 0 \implies x \neq \frac{4}{3}.$$\n\n3. To find the inverse function $f^{-1}(x)$, start by setting $y = \frac{2x}{3x - 4}$. Swap $x$ and $y$: $$x = \frac{2y}{3y - 4}.$$\n\n4. Solve for $y$: multiply both sides by $3y - 4$: $$x(3y - 4) = 2y$$\n\n5. Distribute $x$: $$3xy - 4x = 2y$$\n\n6. Rearrange to isolate terms with $y$: $$3xy - 2y = 4x$$\n\n7. Factor $y$ out: $$y(3x - 2) = 4x$$\n\n8. Solve for $y$: $$y = \frac{4x}{3x - 2}.$$\n\nSo, the inverse function is $$f^{-1}(x) = \frac{4x}{3x - 2}.$$\n\n9. The domain of the inverse function is all $x$ such that its denominator is not zero: $$3x - 2 \neq 0 \implies x \neq \frac{2}{3}.$$\n\n10. Therefore, the domain of the inverse function is $$\boxed{\left\{ x \in \mathbb{R} : x \neq \frac{2}{3} \right\}}.$$