1. **State the problem:** Find the inverse of the function $g(x) = \frac{3x+1}{5x+3}$. 2. **Formula and rules:** To find the inverse function $g^{-1}(x)$, we swap $x$ and $y$ and solve for $y$. The original function is $y = \frac{3x+1}{5x+3}$. 3. **Swap variables:** Replace $g(x)$ with $y$ and swap $x$ and $y$: $$x = \frac{3y+1}{5y+3}$$ 4. **Solve for $y$:** Multiply both sides by the denominator to clear the fraction: $$x(5y+3) = 3y + 1$$ 5. **Distribute $x$:** $$5xy + 3x = 3y + 1$$ 6. **Group $y$ terms on one side:** $$5xy - 3y = 1 - 3x$$ 7. **Factor out $y$:** $$y(5x - 3) = 1 - 3x$$ 8. **Divide both sides by $(5x - 3)$:** $$y = \frac{1 - 3x}{5x - 3}$$ 9. **Show cancellation step:** $$y = \frac{\cancel{1 - 3x}}{\cancel{5x - 3}}$$ (No common factors to cancel, so expression remains the same.) 10. **Final answer:** $$g^{-1}(x) = \frac{1 - 3x}{5x - 3}$$ This is the inverse function of $g(x)$.