Question

what is the horizontal asymptote of $f(x) = \frac{-2x}{x + 1}$?
$y = 0$
$y = -1$
$y = 1$
$y = -2$

Explanation:

Step1: Recall Horizontal Asymptote Rule

For a rational function \( f(x) = \frac{ax^n + \dots}{bx^m + \dots} \), if the degrees of the numerator (\(n\)) and denominator (\(m\)) are equal (\(n = m\)), the horizontal asymptote is \( y=\frac{a}{b} \), where \(a\) is the leading coefficient of the numerator and \(b\) is the leading coefficient of the denominator.

Step2: Identify Degrees and Coefficients

In \( f(x)=\frac{-2x}{x + 1} \), the numerator is \(-2x\) (degree \(1\), leading coefficient \(-2\)) and the denominator is \(x + 1\) (degree \(1\), leading coefficient \(1\)).

Step3: Apply the Rule

Since \(n = m = 1\), the horizontal asymptote is \( y=\frac{-2}{1}=-2 \).

Answer:

\( y = -2 \)