Question

use the following information to answer the next question
$f(x)=\frac{(2x + 1)(3x - 2)}{(x + 4)(2x + 1)}$

1. for the function above, the vertical asymptote and the horizontal asymptote are represented by the information in row.

row	vertical asymptote	horizontal asymptote
b	$x = -4$	$y = -14$
c	$x = -\frac{1}{2}$	$y = 3$
d	$x = -4$	$y = 3$

Explanation:

Step1: Simplify the rational function

First, cancel the common non-zero factor $(2x+1)$ from numerator and denominator (note $2x+1
eq0$, so $x
eq-\frac{1}{2}$ is a hole, not an asymptote):

$$f(x)=\frac{3x-2}{x+4}, \quad x eq-\frac{1}{2}$$

Step2: Find vertical asymptote

Vertical asymptotes occur where the simplified denominator is 0:
$$x+4=0 \implies x=-4$$

Step3: Find horizontal asymptote

For rational functions, compare degrees of numerator and denominator (both degree 1). The horizontal asymptote is the ratio of leading coefficients:
$$y=\frac{3}{1}=3$$

Answer:

D. Vertical Asymptote $x=-4$, Horizontal Asymptote $y=3$