To find the domain of the function $f(x) = \frac{x + 2}{x^2 - 4}$, we need to determine where the function is defined. This means identifying values of $x$ for which the denominator is not zero, since division by zero is undefined.

Step 1: Identify the Denominator

The denominator of the function is: $x^2 - 4$ This can be factored as: $x^2 - 4 = (x - 2)(x + 2)$

Step 2: Set the Denominator Equal to Zero

We set the denominator equal to zero to find the values of $x$ that make the denominator zero: $(x - 2)(x + 2) = 0$

Step 3: Solve for $x$

Solving the equation gives: $x - 2 = 0 \quad \text{or} \quad x + 2 = 0$ $x = 2 \quad \text{or} \quad x = -2$

Step 4: Determine the Domain

The function is undefined at $x = 2$ and $x = -2$ because these values make the denominator zero. Therefore, the domain of $f(x)$ is all real numbers except $x = 2$ and $x = -2$.

Final Answer

The domain of the function is: $\boxed{(-\infty, -2) \cup (-2, 2) \cup (2, +\infty)}$

Would you like more details or have any questions?

Here are five related questions:

1. How do you determine the zeros of a rational function?
2. What happens to the graph of a function at points where the denominator is zero?
3. How do you find the range of a rational function?
4. What are the steps to factor a quadratic expression like $x^2 - 4$?
5. How does factoring help in finding the domain of a rational function?

Tip: Always factor the denominator of a rational function to easily identify values that should be excluded from the domain.