To determine the horizontal asymptote of the function [tex]\( f(x) = \frac{-2x}{x+1} \)[/tex], we need to analyze the behavior of the function as [tex]\( x \)[/tex] approaches positive and negative infinity.
### Step-by-Step Solution:
1. Identify the degrees of the numerator and the denominator:
- The numerator of [tex]\( f(x) \)[/tex] is [tex]\( -2x \)[/tex], which is a linear polynomial of degree 1.
- The denominator of [tex]\( f(x) \)[/tex] is [tex]\( x + 1 \)[/tex], which is also a linear polynomial of degree 1.
2. Compare the degrees:
- Both the numerator and the denominator are of the same degree (degree 1).
3. Determine the horizontal asymptote for rational functions:
- When the degrees of the numerator and denominator are the same, the horizontal asymptote is found by taking the ratio of the leading coefficients.
- The leading coefficient of the numerator [tex]\( -2x \)[/tex] is [tex]\(-2\)[/tex].
- The leading coefficient of the denominator [tex]\( x + 1 \)[/tex] is [tex]\(1\)[/tex].
4. Calculate the horizontal asymptote:
- The horizontal asymptote is given by the ratio [tex]\( \frac{-2}{1} \)[/tex].
Therefore, the horizontal asymptote for the function [tex]\( f(x) = \frac{-2x}{x+1} \)[/tex] is:
[tex]\[ y = -2 \][/tex]
Among the given options, the correct choice is:
[tex]\[ y = -2 \][/tex]
