y= (x+3)/(x^(2)-9)
RANGE SOLUTION 
INTERVAL NOTATION:
SET BOARDER:

To determine the range of the function \( y = \frac{x+3}{x^2 - 9} \), we need to examine the behavior of the function and identify any restrictions or undefined points.

### Step 1: Factor the denominator
We begin by factoring the denominator:
\[
x^2 - 9 = (x - 3)(x + 3)
\]
This reveals that the function is undefined at \( x = 3 \) and \( x = -3 \), because the denominator becomes zero at these points.

### Step 2: Analyze the vertical asymptotes
Since \( y = \frac{x+3}{(x-3)(x+3)} \), there are **vertical asymptotes** at \( x = 3 \) and \( x = -3 \).

### Step 3: Simplify the function
We simplify the function where it is defined. For \( x \neq -3 \), we can cancel the common factor:
\[
y = \frac{1}{x-3}, \quad \text{for } x \neq -3
\]

This is the simplified form of the function when \( x \neq -3 \).

### Step 4: Behavior at the vertical asymptotes
- As \( x \to 3^+ \), \( y \to \infty \).
- As \( x \to 3^- \), \( y \to -\infty \).

- As \( x \to -3^+ \), \( y \to -\infty \).
- As \( x \to -3^- \), \( y \to \infty \).

### Step 5: Horizontal asymptote
The function approaches \( y = 0 \) as \( x \to \infty \) or \( x \to -\infty \), meaning that the horizontal asymptote is \( y = 0 \).

### Step 6: Range of the function
The function can take all real values except for **\( y = 0 \)**, because the function never actually reaches the horizontal asymptote.

### Step 7: Interval notation
Therefore, the **range** of the function is:
\[
(-\infty, 0) \cup (0, \infty)
\]

### Step 8: Set notation
In set notation, the range is:
\[
\{ y \in \mathbb{R} \ | \ y \neq 0 \}
\]

---

Would you like more details or further clarification on any step?

### Related Questions:
1. How do you determine horizontal asymptotes for rational functions?
2. What is the domain of the given function in interval notation?
3. How do vertical asymptotes affect the range of a function?
4. How does the behavior of a function near asymptotes help determine the range?
5. What is the difference between range in set notation and interval notation?

**Tip:** When finding the range, always check for asymptotes and undefined points to ensure all excluded values are accounted for.

y = (x+3)/(x^2-9) RANGE SOLUTION INTERVAL NOTATION: SET BORDER:

Discover how to determine the range of the rational function y = (x+3)/(x^2-9). This solution covers factorization, vertical and horizontal asymptotes, and the range in interval and set notation. Ideal for Grades 10-12 students studying rational functions.

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