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?

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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!