The question asks you to determine the horizontal or vertical asymptote of the function $f(x) = \frac{3}{x+4}$.

1. Vertical Asymptote:

To find the vertical asymptote, we set the denominator equal to zero: $x + 4 = 0 \quad \Rightarrow \quad x = -4$ Thus, the vertical asymptote is at $x = -4$, which corresponds to option d.

2. Horizontal Asymptote:

For the horizontal asymptote, we analyze the behavior of the function as $x \to \infty$ or $x \to -\infty$. Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is $y = 0$, which corresponds to option b.

Answer:

• Option d: $x = -4$ is the correct vertical asymptote.
• Option b: $y = 0$ is the correct horizontal asymptote.

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Here are five related questions:

1. How do you find the vertical asymptote of rational functions in general?
2. What happens to the function $f(x)$ near the vertical asymptote?
3. How do horizontal asymptotes differ from vertical asymptotes in terms of function behavior?
4. Can a function have more than one vertical asymptote?
5. How does the degree of the numerator and denominator influence the horizontal asymptote?

Tip: Vertical asymptotes occur where the denominator is zero, while horizontal asymptotes represent the function's end behavior as $x \to \pm \infty$.