1. **Problem Statement:** Verify the horizontal asymptote, domain, continuity, holes, vertical asymptotes, and intercepts for the functions: - $f(x) = \frac{5x^2+125}{4x^3-8x^2-9x+18} = \frac{5(x^2+25)}{(x-2)(2x+3)(2x-3)}$ - $f(x) = -4 + \frac{2x^2-6x}{x-2}$ --- 2. **Function 18 Analysis:** - **Horizontal Asymptote (HA):** For rational functions, if degree numerator $<$ degree denominator, HA is $y=0$. - Here, numerator degree = 2, denominator degree = 3, so HA: $y=0$ is correct. - **Domain:** Denominator zeroes at $x=2$, $x=-\frac{3}{2}$, $x=\frac{3}{2}$. - Domain excludes these points: $(-\infty,-\frac{3}{2}) \cup (-\frac{3}{2},\frac{3}{2}) \cup (\frac{3}{2},2) \cup (2,\infty)$ correct. - **Continuity and Holes:** No common factors canceled between numerator and denominator, so no holes. - **Vertical Asymptotes (VA):** At zeros of denominator: $x=-\frac{3}{2}$, $x=\frac{3}{2}$, $x=2$ correct. - **Intercepts:** - $x$-intercept: Set numerator $=0 \Rightarrow 5(x^2+25)=0 \Rightarrow x^2=-25$ no real roots, so no $x$-intercepts correct. - $y$-intercept: $f(0) = \frac{5(0+25)}{4(0)-8(0)-9(0)+18} = \frac{125}{18}$ correct. --- 3. **Function 20 Analysis:** - $f(x) = -4 + \frac{2x^2-6x}{x-2}$ - **Horizontal Asymptote:** Degree numerator (2) is greater than denominator (1), so no horizontal asymptote correct. - **Domain:** Denominator zero at $x=2$, so domain $(-\infty,2) \cup (2,\infty)$ correct. - **Continuity and Holes:** No factor cancellation, so no holes. - **Vertical Asymptote:** At $x=2$ correct. - **Intercepts:** - $x$-intercepts: Set $f(x)=0$: $$0 = -4 + \frac{2x^2-6x}{x-2} \Rightarrow 4 = \frac{2x^2-6x}{x-2} \Rightarrow 4(x-2) = 2x^2 - 6x$$ $$4x - 8 = 2x^2 - 6x$$ $$0 = 2x^2 - 6x - 4x + 8 = 2x^2 - 10x + 8$$ Divide both sides by 2: $$0 = \cancel{2}x^2 - \cancel{2}5x + \cancel{2}4 \Rightarrow 0 = x^2 - 5x + 4$$ Factor: $$(x-4)(x-1) = 0 \Rightarrow x=1,4$$ So $x$-intercepts at $(1,0)$ and $(4,0)$ correct. - $y$-intercept: $f(0) = -4 + \frac{0 - 0}{0-2} = -4$ so $(0,-4)$ correct. --- **Final conclusion:** All given information about horizontal asymptotes, domain, continuity, holes, vertical asymptotes, and intercepts for both functions is correct.