Question

identify the horizontal asymptote of the function.
$f(x) = -\frac{3}{x + 4} - 8$

Explanation:

Step1: Recall horizontal asymptote rule

For a rational function of the form \( y=\frac{a}{x - h}+k \), the horizontal asymptote is determined by the value of \( k \) because as \( x\to\pm\infty \), the term \( \frac{a}{x - h}\to0 \).

Step2: Identify \( k \) in the given function

The given function is \( f(x)=-\frac{3}{x + 4}-8 \), which is in the form \( y=\frac{a}{x - h}+k \) (here \( h=- 4 \), \( a = - 3 \), and \( k=-8 \)). As \( x\to\pm\infty \), \( -\frac{3}{x + 4}\to0 \), so \( f(x)\to0 - 8=-8 \).

Answer:

The horizontal asymptote of the function \( f(x)=-\frac{3}{x + 4}-8 \) is \( y = - 8 \)