Certainly! Let's carefully analyze the rational function [tex]\( f(x) = \frac{-2}{-x-2} \)[/tex] and identify its vertical and horizontal asymptotes. Here's a step-by-step breakdown:
### Step 1: Identify the Vertical Asymptote
Vertical asymptotes occur where the denominator of the rational function equals zero, since the function will approach infinity or negative infinity there.
Given the function:
[tex]\[ f(x) = \frac{-2}{-x-2} \][/tex]
The denominator is:
[tex]\[ -x - 2 \][/tex]
Set the denominator equal to zero to find the vertical asymptote:
[tex]\[ -x - 2 = 0 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ -x = 2 \][/tex]
[tex]\[ x = -2 \][/tex]
Therefore, the vertical asymptote is at:
[tex]\[ x = -2 \][/tex]
### Step 2: Identify the Horizontal Asymptote
Horizontal asymptotes describe the behavior of the function as [tex]\( x \)[/tex] approaches positive or negative infinity.
For the given function:
[tex]\[ f(x) = \frac{-2}{-x-2} \][/tex]
Examine the limits of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] and [tex]\( -\infty \)[/tex].
As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex]:
[tex]\[ \lim_{{x \to \infty}} \frac{-2}{-x-2} = 0 \][/tex]
As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]:
[tex]\[ \lim_{{x \to -\infty}} \frac{-2}{-x-2} = 0 \][/tex]
The horizontal asymptote is at:
[tex]\[ y = 0 \][/tex]
### Summary
- Vertical Asymptote: [tex]\( x = -2 \)[/tex]
- Horizontal Asymptote: [tex]\( y = 0 \)[/tex]
### Graphing the Function
1. Draw the Vertical Asymptote: Draw a dashed vertical line at [tex]\( x = -2 \)[/tex].
2. Draw the Horizontal Asymptote: Draw a dashed horizontal line at [tex]\( y = 0 \)[/tex].
With these asymptotes in place, you can now more accurately sketch the graph of the function [tex]\( f(x) = \frac{-2}{-x-2} \)[/tex].
Remember that the behavior of the function will show it approaching these asymptotes but never touching them. As [tex]\( x \)[/tex] moves far to the left or right, [tex]\( f(x) \)[/tex] will approach 0.
