graph all vertical and horizontal asymptotes of the rational function.
$f(x)=\frac{-4}{-2 x - 1}$

Question

Accepted Answer

# Answer:
Vertical asymptote: $x =-\frac{1}{2}$, Horizontal asymptote: $y = 2$

# Explanation:
## Step1: Find the vertical asymptote
Set the denominator equal to zero.
For the function $f(x)=\frac{-4}{-2x - 1}$, we solve $-2x-1 = 0$.
$$
\begin{align*}
-2x-1&=0\
-2x&=1\
x&=-\frac{1}{2}
\end{align*}
$$
## Step2: Find the horizontal asymptote
Compare the degrees of the numerator and denominator.
The degree of the numerator $n = 0$ (since the numerator is a constant, $-4$) and the degree of the denominator $m=1$ (for $ - 2x-1$).
When $n<m$, the horizontal asymptote is $y = 0$. But we can rewrite $f(x)=\frac{-4}{-2x - 1}=\frac{4}{2x + 1}$.
Another way: use the limit as $x
ightarrow\pm\infty$.
$$
\begin{align*}
\lim_{x
ightarrow\pm\infty}f(x)&=\lim_{x
ightarrow\pm\infty}\frac{-4}{-2x-1}\
&=\lim_{x
ightarrow\pm\infty}\frac{\frac{-4}{x}}{\frac{-2x}{x}-\frac{1}{x}}\
&=\frac{0}{- 2-0}=2
\end{align*}
$$

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QUESTION IMAGE

Question

Explanation:

Step1: Find the vertical asymptote

Step2: Find the horizontal asymptote

Answer: