1. **State the problem:** We need to identify the graphs of the functions $$g(x) = \frac{3x}{x^2 - 2x - 8}$$ and $$h(x) = \frac{x^2 + 1}{x^2 + 2x - 3}$$ from the given graph options. 2. **Find vertical asymptotes:** Vertical asymptotes occur where the denominator is zero (and numerator is not zero). For $g(x)$: $$x^2 - 2x - 8 = 0$$ Factor: $$x^2 - 2x - 8 = (x - 4)(x + 2) = 0$$ So vertical asymptotes at $x = 4$ and $x = -2$. For $h(x)$: $$x^2 + 2x - 3 = 0$$ Factor: $$x^2 + 2x - 3 = (x + 3)(x - 1) = 0$$ So vertical asymptotes at $x = -3$ and $x = 1$. 3. **Match vertical asymptotes to graphs:** - Graphs with vertical asymptotes at $x = -2$ and $x = 4$ are Graph A and Graph D. - Graphs with vertical asymptotes at $x = -3$ and $x = 1$ are Graph C, Graph E, and Graph F. 4. **Analyze behavior of $g(x)$:** - Numerator is $3x$, zero at $x=0$. - Denominator zeros at $-2$ and $4$. - For large $x$, $g(x) \approx \frac{3x}{x^2} = \frac{3}{x} \to 0$. - Check sign near asymptotes: - For $x < -2$, denominator $(x-4)(x+2)$: both negative, product positive, numerator negative, so $g(x)$ negative. - Between $-2$ and $4$, denominator negative, numerator varies. - Graph A and D have vertical asymptotes at $-2$ and $4$. - Graph A has a maximum near $y=5$ and negative branch on right. - Graph D has positive branch going up on left and negative branch on right. 5. **Analyze behavior of $h(x)$:** - Numerator $x^2 + 1$ always positive. - Denominator zeros at $-3$ and $1$. - For large $x$, $h(x) \approx \frac{x^2}{x^2} = 1$. - Vertical asymptotes at $-3$ and $1$. - Graph C, E, F have these asymptotes. - Graph C has branches in first and second quadrants bending toward asymptotes. - Graph E is similar but rotated. - Graph F has branches in first and third quadrants. 6. **Conclusion:** - $g(x)$ matches Graph A or D; since Graph A has a maximum near $y=5$ and negative branch on right, which fits $g(x)$ behavior, choose Graph A. - $h(x)$ matches Graph C best due to asymptotes and branch locations. **Final answers:** (a) Graph A is the graph of $g(x)$. (b) Graph C is the graph of $h(x)$.