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Let's break down and solve the questions based on the provided image.

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### **Question 1:**
Which of the following is the set of all values that the variable \( x \) can take?

- **a. Range**
- **b. Intercept**
- **c. Domain**
- **d. Zeroes**

**Answer**: The set of all values that the variable \( x \) can take is called the **domain**. Hence, the correct answer is **c. Domain**.

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### **Question 2:**
What is the domain of \( f(x) = \frac{x - 3}{x + 3} \)?

- **a. All real numbers**
- **b. All real numbers except -3**
- **c. All real numbers except 3**
- **d. Cannot be determined**

**Solution**: The domain of a function includes all possible values of \( x \) for which the function is defined. Here, the function \( f(x) = \frac{x - 3}{x + 3} \) has a denominator \( x + 3 \). For the function to be defined, the denominator cannot be zero.

Thus, \( x + 3 \neq 0 \implies x \neq -3 \).

**Answer**: The domain is all real numbers except \( -3 \), so the correct answer is **b. All real numbers except -3**.

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### **Question 3:**
What value(s) of \( x \) will make the function meaningless for \( f(x) = \frac{x - 1}{x} \)?

- **a. \( x = -1 \)**
- **b. \( x = 0 \)**
- **c. \( x = 1 \)**
- **d. All real numbers**

**Solution**: The function becomes meaningless when the denominator is zero because division by zero is undefined. In \( f(x) = \frac{x - 1}{x} \), the denominator is \( x \), so we set:

\( x \neq 0 \)

**Answer**: The value of \( x \) that makes the function meaningless is \( x = 0 \), so the correct answer is **b. \( x = 0 \)**.

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### Additional Related Questions:
1. What is the difference between domain and range?
2. How do you find the domain of a function with a square root?
3. How does the domain change if there is both a square root and a fraction?
4. What are zeroes of a function and how are they found?
5. Why can't we divide by zero in mathematics?

**Tip**: To find the domain of a rational function, always check for values of \( x \) that make the denominator zero—these values are excluded from the domain.

Learn how to identify the domain of a function, especially rational functions like f(x) = (x-3)/(x+3) and f(x) = (x - 1)/x. This guide covers key concepts such as finding the domain, handling undefined values, and understanding the importance of avoiding division by zero. Suitable for students in Grades 9-11.

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