Given the function f (x) = fraction whose numerator is x superscript 2 end of superscript, minus − 16
and whose denominator is x superscript 2 end of superscript, + x minus − 6
end of fraction, , find the domain of f (x). 

To find the domain of the function \( f(x) = \frac{x^2 - 16}{x^2 + x - 6} \), we need to determine where the function is **undefined**, which occurs when the denominator equals zero.

### Steps:

1. **Denominator:**
   \[
   x^2 + x - 6 = 0
   \]
2. **Factor the quadratic expression**:
   \[
   x^2 + x - 6 = (x - 2)(x + 3) = 0
   \]
3. **Solve for \( x \)**:
   \[
   x - 2 = 0 \quad \Rightarrow \quad x = 2
   \]
   \[
   x + 3 = 0 \quad \Rightarrow \quad x = -3
   \]

Thus, the function is undefined at \( x = 2 \) and \( x = -3 \), because these values make the denominator zero.

### Conclusion:
The domain of the function \( f(x) \) is all real numbers except \( x = 2 \) and \( x = -3 \).

So, the domain is:
\[
\boxed{(-\infty, -3) \cup (-3, 2) \cup (2, \infty)}
\]

Would you like further clarification or details on any step?

Here are 5 questions to deepen your understanding:
1. What happens to the function when the denominator is zero?
2. How do you factor a quadratic equation?
3. Can a function have more than two points where it's undefined?
4. How does the domain of a rational function compare to that of a polynomial function?
5. How would the domain change if the numerator were different?

**Tip**: Always check for factors common to both the numerator and denominator—they can affect the behavior of the function!

Given the function f(x) = (x^2 - 16) / (x^2 + x - 6), find the domain of f(x).

Learn how to find the domain of the rational function f(x) = (x^2 - 16) / (x^2 + x - 6). This problem includes factoring quadratics and determining where the function is undefined. It is a key concept for students in Grades 9-11 studying Algebra and Rational Functions.

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