https://file.aiquickdraw.com/mathbot/file/1729594184115m6ocg5jl.jpg

Let's work on part **c** of question 13, which asks for the domain of the function. The function in part **c** is:

\[
y = \frac{x - 1}{(x - 2)(x + 3)}
\]

### Domain Explanation:
The domain of a function includes all possible values of \( x \) for which the function is defined. The function will be undefined where the denominator is equal to zero, since division by zero is not possible.

#### Step 1: Set the denominator equal to zero and solve for \( x \):

\[
(x - 2)(x + 3) = 0
\]

This equation is satisfied when either \( x - 2 = 0 \) or \( x + 3 = 0 \).

\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]
\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]

#### Step 2: Conclusion for the domain:
The function is undefined at \( x = 2 \) and \( x = -3 \). Therefore, the domain of the function is all real numbers **except** \( x = 2 \) and \( x = -3 \).

### Domain in interval notation:
\[
\boxed{(-\infty, -3) \cup (-3, 2) \cup (2, \infty)}
\]

Would you like more details or have any questions about this? 

Here are 5 related questions:
1. What happens when the numerator is zero in a rational function?
2. How would you find the vertical asymptotes of a rational function?
3. What is the domain of \( y = \frac{1}{x} \)?
4. What is the difference between the domain and the range of a function?
5. How do you find the domain for square root functions?

**Tip:** When finding the domain of a rational function, always check where the denominator equals zero and exclude those points from the domain.

Find the domain of the function y = (x - 1) / ((x - 2)(x + 3))

Learn how to find the domain of the rational function y = (x - 1) / ((x - 2)(x + 3)). The solution includes steps to determine where the denominator equals zero and what values should be excluded from the domain. Suitable for Grades 9-12.

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