You are asking about finding the $x$-intercepts of a rational function. Let's work through the general steps for finding the $x$-intercepts of a rational function using your specific function as an example.

Function Given:

$f(x) = \frac{x^2 - 4}{x^2 - 9}$

Finding the $x$-Intercept(s):

The $x$-intercepts of a function occur where the function's value is zero. For a rational function $f(x) = \frac{N(x)}{D(x)}$, the $x$-intercepts occur where the numerator $N(x)$ is zero, provided that the denominator $D(x)$ is not zero at that point.

So, we solve: $N(x) = 0 \quad \text{where} \quad N(x) = x^2 - 4$

Step 1: Set the Numerator Equal to Zero

$x^2 - 4 = 0$

This is a difference of squares, so we can factor it: $(x - 2)(x + 2) = 0$

Thus, the solutions are: $x = 2 \quad \text{and} \quad x = -2$

These are the potential $x$-intercepts.

Step 2: Check the Denominator to Ensure These Points Are in the Domain

The function is undefined where the denominator is zero. So, we solve: $D(x) = x^2 - 9 = 0$

Again, this is a difference of squares, so we can factor it: $(x - 3)(x + 3) = 0$

Thus, the function is undefined at: $x = 3 \quad \text{and} \quad x = -3$

Final Answer:

• The function has $x$-intercepts at $x = 2$ and $x = -2$.
• However, the function is undefined at $x = 3$ and $x = -3$.

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

Here are 5 related questions to deepen your understanding:

1. How do you find the vertical asymptotes of a rational function?
2. What happens to the graph of the function at points where it is undefined?
3. How do horizontal asymptotes affect the long-term behavior of a rational function?
4. What is the difference between a removable discontinuity and a vertical asymptote?
5. How can you graph a rational function by identifying intercepts, asymptotes, and discontinuities?

Tip: When finding $x$-intercepts for rational functions, always remember to check if the intercept lies within the domain by ensuring the denominator does not become zero at that point.