$$  $$$$ f\left(x\right) $$$$ \frac{x^2-4}{x^2-9}^{} $$rational functions (x-intercept)

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.

Learn how to find the x-intercepts of the rational function (x^2 - 4)/(x^2 - 9) with a detailed step-by-step solution. This problem covers key algebraic concepts, including rational functions, quadratic equations, and domain restrictions. Suitable for students in Grades 11-12.

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