1. **State the problem:** We need to simplify and analyze the function $$w = \left(\frac{1 + 3z}{3z}\right)(3 - z)$$. 2. **Write the formula and rules:** This is a product of a rational expression and a binomial. To simplify, multiply the numerator terms and keep the denominator, then simplify. 3. **Simplify step-by-step:** $$w = \frac{1 + 3z}{3z} \times (3 - z) = \frac{(1 + 3z)(3 - z)}{3z}$$ 4. **Expand the numerator:** $$(1)(3) + (1)(-z) + (3z)(3) + (3z)(-z) = 3 - z + 9z - 3z^2 = 3 + 8z - 3z^2$$ 5. **Rewrite the function:** $$w = \frac{3 + 8z - 3z^2}{3z}$$ 6. **Explain domain restrictions:** Since $z$ is in the denominator, $z \neq 0$. 7. **Final simplified form:** $$w = \frac{3 + 8z - 3z^2}{3z}$$ This is the simplified expression for $w$. **Answer:** $$w = \frac{3 + 8z - 3z^2}{3z}$$