1. **Problem Statement:** We are given a rational function $f(x)$ with vertical asymptotes at $x = -2$ and $x = 2$, and a horizontal asymptote at $y = 0$. We need to determine which of the given options could define $f(x)$. 2. **Key Properties of Rational Functions:** - Vertical asymptotes occur where the denominator is zero and the numerator is not zero. - Horizontal asymptote $y=0$ means the degree of the numerator is less than the degree of the denominator. 3. **Analyze the vertical asymptotes:** - The function has vertical asymptotes at $x = -2$ and $x = 2$. - So the denominator must have factors $(x + 2)$ and $(x - 2)$. 4. **Check each option for vertical asymptotes:** - (A) $f(x) = \frac{x - 1}{(x - 3)(x + 1)}$ has vertical asymptotes at $x=3$ and $x=-1$ (not $\pm 2$). - (B) $f(x) = \frac{x + 1}{(x + 3)(x - 1)}$ has vertical asymptotes at $x=-3$ and $x=1$ (not $\pm 2$). - (C) $f(x) = \frac{(x - 3)(x + 1)}{x - 1}$ has vertical asymptote at $x=1$ only (not $\pm 2$). - (D) $f(x) = \frac{(x + 3)(x - 1)}{x + 1}$ has vertical asymptote at $x=-1$ only (not $\pm 2$). None of the options have vertical asymptotes at $x = -2$ and $x = 2$. 5. **Horizontal asymptote check:** - Horizontal asymptote $y=0$ means degree numerator < degree denominator. - (A) numerator degree 1, denominator degree 2, horizontal asymptote $y=0$. - (B) numerator degree 1, denominator degree 2, horizontal asymptote $y=0$. - (C) numerator degree 2, denominator degree 1, horizontal asymptote $y=\infty$ or none. - (D) numerator degree 2, denominator degree 1, horizontal asymptote $y=\infty$ or none. 6. **Conclusion:** None of the given options have vertical asymptotes at $x = -2$ and $x = 2$, so none exactly match the graph described. **Final answer:** None of the options A, B, C, or D define the function $f$ with vertical asymptotes at $x = -2$ and $x = 2$ and horizontal asymptote $y=0$. If forced to choose among the options, (A) and (B) have horizontal asymptote $y=0$ but wrong vertical asymptotes. Hence, no option matches the given graph's asymptotes.