Determine each feature of the graph of the given function. $$ f(x)=\frac{3 x+9}{x^{2}-9} $$

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To analyze the function $$ f(x) = \frac{3x + 9}{x^2 - 9} $$, we will determine its key features, including domain, intercepts, asymptotes, and behavior at infinity.

### 1. Domain
The domain of the function is determined by the values of $$ x $$ for which the denominator is not zero.

Set the denominator equal to zero:
$$
x^2 - 9 = 0
$$
Factoring gives:
$$
(x - 3)(x + 3) = 0
$$
Thus, $$ x = 3 $$ and $$ x = -3 $$ are the points where the function is undefined. Therefore, the domain of $$ f(x) $$ is:
$$
\text{Domain: } x \in \mathbb{R} \setminus \{-3, 3\}
$$

### 2. Intercepts
**x-intercept:** Set $$ f(x) = 0 $$:
$$
\frac{3x + 9}{x^2 - 9} = 0
$$
This occurs when the numerator is zero:
$$
3x + 9 = 0 \implies 3x = -9 \implies x = -3
$$
However, $$ x = -3 $$ is not in the domain, so there is no x-intercept.

**y-intercept:** Set $$ x = 0 $$:
$$
f(0) = \frac{3(0) + 9}{0^2 - 9} = \frac{9}{-9} = -1
$$
Thus, the y-intercept is:
$$
\text{y-intercept: } (0, -1)
$$

### 3. Asymptotes
**Vertical Asymptotes:** These occur where the function is undefined and the denominator is zero. From our earlier work, we found vertical asymptotes at:
$$
x = -3 \quad \text{and} \quad x = 3
$$

**Horizontal Asymptote:** To find the horizontal asymptote, we analyze the degrees of the numerator and denominator. The degree of the numerator (1) is less than the degree of the denominator (2). Therefore, the horizontal asymptote is:
$$
y = 0
$$

### 4. Behavior at Infinity
As $$ x \to \infty $$ or $$ x \to -\infty $$, the function approaches the horizontal asymptote:
$$
f(x) \to 0
$$

### Summary of Features
- **Domain:** $$ x \in \mathbb{R} \setminus \{-3, 3\} $$
- **x-intercept:** None
- **y-intercept:** $$ (0, -1) $$
- **Vertical Asymptotes:** $$ x = -3 $$ and $$ x = 3 $$
- **Horizontal Asymptote:** $$ y = 0 $$
- **Behavior at Infinity:** $$ f(x) \to 0 $$ as $$ x \to \pm \infty $$

These features provide a comprehensive understanding of the graph of the function $$ f(x) = \frac{3x + 9}{x^2 - 9} $$.
- Domain: $$ x \in \mathbb{R} \setminus \{-3, 3\} $$ - x-intercept: None - y-intercept: $$ (0, -1) $$ - Vertical Asymptotes: $$ x = -3 $$ and $$ x = 3 $$ - Horizontal Asymptote: $$ y = 0 $$ - Behavior at Infinity: $$ f(x) \to 0 $$ as $$ x \to \pm \infty $$

Determine each feature of the graph of the given function.

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