Determine if the function below has a horizontal or a slant asymptote, and explain why. \[ f(x)=\frac{-1-2 x}{3 x+5 x^{3}+4 x^{2}-3} \] Answer Attempt 1 out of 2 The function has a numerator is \( \square \) the degree of the denominator. Horizontal/Slant asymptote: \( \square \) asymptote because the degree of the Submit Answer

To determine whether the function \( f(x) = \frac{-1 - 2x}{3x + 5x^{3} + 4x^{2} - 3} \) has a horizontal or slant asymptote, we analyze the degrees of the numerator and the denominator. ### Step 1: Identify the Degrees - **Numerator:** The highest power of \( x \) in the numerator is \( x \), so the degree is **1**. - **Denominator:** The highest power of \( x \) in the denominator is \( x^{3} \), so the degree is **3**. ### Step 2: Compare the Degrees - **If the degree of the numerator < degree of the denominator:** The horizontal asymptote is \( y = 0 \). - **If the degree of the numerator = degree of the denominator:** The horizontal asymptote is \( y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} \). - **If the degree of the numerator = degree of the denominator + 1:** There is a slant (oblique) asymptote. ### Conclusion for \( f(x) \) - **Degree of Numerator (1) < Degree of Denominator (3):** - The function **has a horizontal asymptote**. - **Numerator is less than the degree of the denominator.** - **Horizontal Asymptote:** \( y = 0 \) asymptote because the degree of the numerator is less than the degree of the denominator. ### Final Answer The function has a **horizontal** asymptote. The numerator is **less than** the degree of the denominator. **Horizontal** asymptote: \( y = 0 \) asymptote because the degree of the numerator is less than the degree of the denominator.