Let \( f(x)=\frac{g(x)}{2 x^{2}-32} \), where \( g(x) \) is a polynomial. Which of the following is a possible graph of \( y=f(x) ? \) Dashed lines indicate asymptotes.

To determine the possible graph of \( y = f(x) \), we need to consider the behavior of the function as \( x \) approaches certain values and as \( x \) goes to infinity. Here are the key points to consider: 1. **Horizontal Asymptotes**: - As \( x \) approaches infinity or negative infinity, the term \( 2x^2 - 32 \) becomes very large, and the function \( f(x) \) approaches \( \frac{g(x)}{2x^2} \). - Since \( g(x) \) is a polynomial, it will have a degree that is at least 1 (since it's not a constant polynomial). Therefore, \( g(x) \) will grow at least as fast as \( x \) as \( x \) goes to infinity or negative infinity. - This means that \( f(x) \) will approach 0 as \( x \) goes to infinity or negative infinity, so the horizontal asymptote is \( y = 0 \). 2. **Vertical Asymptotes**: - The denominator \( 2x^2 - 32 \) equals 0 when \( x^2 = 16 \), so \( x = \pm 4 \). - These are the vertical asymptotes of the function. 3. **Behavior near the Vertical Asymptotes**: - As \( x \) approaches 4 from the left, \( 2x^2 - 32 \) is negative, and since \( g(x) \) is a polynomial, it will not change sign, so \( f(x) \) will be negative and approach negative infinity. - As \( x \) approaches 4 from the right, \( 2x^2 - 32 \) is positive, and \( f(x) \) will be positive and approach positive infinity. - The same behavior occurs as \( x \) approaches -4. 4. **Behavior near the Horizontal Asymptote**: - Since \( f(x) \) approaches 0 as \( x \) goes to infinity or negative infinity, the graph will get closer and closer to the x-axis but will never touch it. Given these considerations, the graph of \( y = f(x) \) should have the following characteristics: - Horizontal asymptote at \( y = 0 \). - Vertical asymptotes at \( x = 4 \) and \( x = -4 \). - The graph will approach the x-axis as \( x \) goes to infinity or negative infinity. - The graph will have a vertical "jump" at \( x = 4 \) and \( x = -4 \) because of the vertical asymptotes. Without the actual graphs to choose from, I cannot tell you which specific graph is the correct one. However, if you have a set of graphs to choose from, you should look for one that matches these characteristics.