1. **State the problem:** Find the inverse of the function $f(x) = \frac{2}{3}x - \frac{1}{3}$. 2. **Recall the formula and rules:** To find the inverse function $f^{-1}(x)$, we swap $x$ and $y$ in the equation and solve for $y$. 3. **Start with:** $$y = \frac{2}{3}x - \frac{1}{3}$$ 4. **Swap $x$ and $y$:** $$x = \frac{2}{3}y - \frac{1}{3}$$ 5. **Solve for $y$:** Add $\frac{1}{3}$ to both sides: $$x + \frac{1}{3} = \frac{2}{3}y$$ 6. **Isolate $y$ by dividing both sides by $\frac{2}{3}$:** $$y = \frac{x + \frac{1}{3}}{\frac{2}{3}}$$ 7. **Simplify the division:** $$y = (x + \frac{1}{3}) \times \frac{3}{2}$$ 8. **Distribute:** $$y = \frac{3}{2}x + \frac{3}{2} \times \frac{1}{3} = \frac{3}{2}x + \frac{1}{2}$$ 9. **Final inverse function:** $$f^{-1}(x) = \frac{3}{2}x + \frac{1}{2}$$ This means the inverse function $g(x)$ is: $$g(x) = \frac{3}{2}x + \frac{1}{2}$$