Let's analyze the function given:

\[
f(x) = \frac{x + 4}{x^2 - 4}
\]

### Step 1: Identify Domain Restrictions
To find the domain of \(f(x)\), we need to determine where the function is undefined. The denominator cannot be zero:

\[
x^2 - 4 = 0
\]

Solve for \(x\):

\[
x^2 = 4 \implies x = 2 \quad \text{or} \quad x = -2
\]

Thus, the function is undefined at \(x = 2\) and \(x = -2\). The domain of \(f(x)\) is:

\[
x \in \mathbb{R} \setminus \{-2, 2\}
\]

### Step 2: Simplify the Function if Possible
Let's see if the function can be simplified. Notice that the denominator can be factored:

\[
f(x) = \frac{x + 4}{(x - 2)(x + 2)}
\]

Since there are no common factors between the numerator and the denominator, this is already in its simplest form.

### Step 3: Analyze Asymptotes
- **Vertical Asymptotes:** These occur where the denominator is zero (and the numerator is not zero):

\[
x = 2 \quad \text{and} \quad x = -2
\]

So, there are vertical asymptotes at \(x = 2\) and \(x = -2\).

- **Horizontal Asymptote:** To find the horizontal asymptote, consider the degrees of the numerator and the denominator:
  - The numerator has degree 1.
  - The denominator has degree 2.

Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is:

\[
y = 0
\]

### Step 4: Identify Intercepts
- **x-intercept:** Set \(f(x) = 0\):

\[
\frac{x + 4}{x^2 - 4} = 0 \implies x + 4 = 0 \implies x = -4
\]

So, there is an x-intercept at \((-4, 0)\).

- **y-intercept:** Set \(x = 0\):

\[
f(0) = \frac{0 + 4}{0^2 - 4} = \frac{4}{-4} = -1
\]

So, there is a y-intercept at \((0, -1)\).

### Step 5: Summarize the Key Characteristics
- **Domain:** \(x \in \mathbb{R} \setminus \{-2, 2\}\)
- **Vertical Asymptotes:** \(x = 2, x = -2\)
- **Horizontal Asymptote:** \(y = 0\)
- **x-intercept:** \((-4, 0)\)
- **y-intercept:** \((0, -1)\)

Would you like more details, or do you have any specific questions?

---

Here are 5 related questions for further exploration:

1. How do you find slant asymptotes for rational functions?
2. What is the behavior of the function \(f(x)\) as \(x \to \infty\) and \(x \to -\infty\)?
3. How would the graph of \(f(x) = \frac{x + 4}{x^2 - 4}\) look, including the intercepts and asymptotes?
4. How do you determine whether a function is even, odd, or neither?
5. How do you find the limit of a rational function at a vertical asymptote?

**Tip:** When dealing with rational functions, always check for both vertical and horizontal asymptotes to understand the behavior of the function better!

Explore the rational function f(x) = (x + 4) / (x^2 - 4) with detailed analysis of its domain, asymptotes, intercepts, and behavior. Learn about vertical asymptotes at x = 2 and x = -2, a horizontal asymptote at y = 0, and intercepts at (-4, 0) and (0, -1). Suitable for Grades 10-12 students studying advanced algebra.

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