Question

identify both the vertical and horizontal asymptotes of the function. \\( f(x) = -\frac{2}{x - 2} - 1 \\)

Explanation:

Step1: Find Vertical Asymptote

For a rational function, vertical asymptotes occur where the denominator is zero (provided the numerator isn't also zero there). The function is \( f(x)=-\frac{2}{x - 2}-1 \), and the denominator is \( x - 2 \). Setting \( x - 2=0 \), we solve for \( x \):
\( x - 2 = 0 \implies x = 2 \)
So the vertical asymptote is \( x = 2 \).

Step2: Find Horizontal Asymptote

For rational functions of the form \( y=\frac{a}{x - h}+k \) (after simplifying), the horizontal asymptote is \( y = k \) when the degree of the numerator is less than the degree of the denominator (which it is here, numerator degree 0, denominator degree 1). First, we can rewrite \( f(x) \) to identify \( k \). Let's combine the terms:
\( f(x)=-\frac{2}{x - 2}-1=-\frac{2}{x - 2}-\frac{x - 2}{x - 2}=\frac{-2-(x - 2)}{x - 2}=\frac{-2 - x + 2}{x - 2}=\frac{-x}{x - 2} \)
But also, from the original form \( f(x)=-\frac{2}{x - 2}-1 \), we can see it's in the form \( \frac{a}{x - h}+k \) where \( k=-1 \). So the horizontal asymptote is \( y=-1 \).

Answer:

Vertical asymptote: \( x = 2 \); Horizontal asymptote: \( y=-1 \)