question find the equation of all horizontal asymptotes of the following function. f(x)=2/(x + 6)-2 answer…

Explanation:

Step1: Recall horizontal - asymptote rules

For a rational function of the form $y=\frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are polynomials, we consider the degrees of $f(x)$ and $g(x)$. The given function is $y = \frac{2}{x + 6}-2=\frac{2-2(x + 6)}{x + 6}=\frac{2-2x-12}{x + 6}=\frac{-2x - 10}{x + 6}$.

Step2: Analyze the degrees

The degree of the numerator $n=-2x - 10$ (degree $n = 1$) and the degree of the denominator $m=x + 6$ (degree $m = 1$). When $n=m$, the horizontal - asymptote is $y=\frac{a_n}{b_m}$, where $a_n$ is the leading coefficient of the numerator and $b_m$ is the leading coefficient of the denominator. Here, $a_n=-2$ and $b_m = 1$.

Step3: Calculate the horizontal - asymptote

$y=\frac{-2}{1}=-2$.

Answer:

$y=-2$