https://file.aiquickdraw.com/mathbot/file/17260245365667v32vr2v.jpg

The function in the image is \( R(x) = \frac{x^2 + 6x + 9}{x^2 - 1} \), and the question is asking for the vertical and horizontal asymptotes.

### Step 1: Factor the expression
First, let's factor both the numerator and the denominator:

- Numerator: \( x^2 + 6x + 9 = (x + 3)^2 \)
- Denominator: \( x^2 - 1 = (x - 1)(x + 1) \)

So, the function becomes:

\[
R(x) = \frac{(x + 3)^2}{(x - 1)(x + 1)}
\]

### Step 2: Vertical asymptotes
Vertical asymptotes occur where the denominator is zero, provided that these points are not canceled out by the numerator.

- The denominator is zero at \( x = 1 \) and \( x = -1 \).
- Since \( (x + 3)^2 \) does not cancel with either \( (x - 1) \) or \( (x + 1) \), these are both vertical asymptotes.

Thus, the vertical asymptotes are \( x = 1 \) and \( x = -1 \).

### Step 3: Horizontal asymptotes
For horizontal asymptotes, we compare the degrees of the numerator and denominator:
- The degree of the numerator is 2 (from \( x^2 \)).
- The degree of the denominator is also 2 (from \( x^2 \)).

When the degrees of the numerator and denominator are the same, the horizontal asymptote is determined by the ratio of the leading coefficients. In this case, both the leading coefficients are 1, so the horizontal asymptote is:

\[
y = \frac{1}{1} = 1
\]

### Conclusion:
- Vertical asymptotes: \( x = 1 \), \( x = -1 \)
- Horizontal asymptote: \( y = 1 \)

The correct answer is:
- **Vertical asymptotes: \( x = 1 \), \( x = -1 \)**
- **Horizontal asymptote: \( y = 1 \)**

Do you need further explanation or have any questions?

Here are 5 related questions for further exploration:
1. How do you find slant asymptotes in rational functions?
2. What happens to asymptotes when a factor cancels out in both the numerator and denominator?
3. How do you identify asymptotes in functions with higher degrees?
4. What is the difference between a hole and a vertical asymptote in a function?
5. How do transformations of functions affect their asymptotes?

**Tip:** To find vertical asymptotes, always check where the denominator equals zero, but make sure those points don't cancel with the numerator.

What are the vertical and horizontal asymptotes for the function R(x) = (x^2 + 6x + 9) / (x^2 - 1)?

This problem explores how to find vertical and horizontal asymptotes for the function R(x) = (x^2 + 6x + 9) / (x^2 - 1). Step-by-step factoring, identifying vertical asymptotes, and determining the horizontal asymptote based on the degrees of numerator and denominator are explained. Suitable for high school students in Grades 9-12.

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