1. **State the problem:** Find the domain and analyze the rational function $$y = \frac{x^3 - x^2}{x^2 - 5x + 6}$$. 2. **Factor numerator and denominator:** $$x^3 - x^2 = x^2(x - 1)$$ $$x^2 - 5x + 6 = (x - 2)(x - 3)$$ 3. **Rewrite the function:** $$y = \frac{x^2(x - 1)}{(x - 2)(x - 3)}$$ 4. **Determine the domain:** The denominator cannot be zero, so $$(x - 2)(x - 3) \neq 0$$ which means $$x \neq 2, \quad x \neq 3$$ 5. **Domain in set notation:** $$D = \mathbb{R} \setminus \{2, 3\}$$ 6. **Find intercepts:** - **x-intercepts:** Set numerator equal to zero: $$x^2(x - 1) = 0 \implies x = 0 \text{ or } x = 1$$ - **y-intercept:** Evaluate at $x=0$: $$y = \frac{0^2(0 - 1)}{(0 - 2)(0 - 3)} = \frac{0}{6} = 0$$ 7. **Summary:** - Domain excludes $x=2$ and $x=3$. - x-intercepts at $x=0$ and $x=1$. - y-intercept at $(0,0)$. This rational function has vertical asymptotes at $x=2$ and $x=3$ where the function is undefined. Final answer: $$D = \mathbb{R} \setminus \{2, 3\}$$