https://file.aiquickdraw.com/mathbot/file/1728271262456au1hqo4n.jpg

Let's go through the questions in your image step by step:

### 1. **Find the domain and range of R**
- Relation \( R \) is likely described later in the problems. Let's address this after exploring the specific relations mentioned.

### 2. **Write some ordered pairs that belong to the following relation R, and find the domain and range of R**
   - **A.** \( Y = x^2 \), where \( x \) and \( y \) are members of the natural numbers.
     - The relation describes pairs where \( y \) is the square of \( x \). Some valid ordered pairs would be:
       - \( (1, 1^2) = (1, 1) \)
       - \( (2, 2^2) = (2, 4) \)
       - \( (3, 3^2) = (3, 9) \)
       - Domain: All natural numbers \( x \) (i.e., \( \mathbb{N} \)).
       - Range: All perfect squares of natural numbers (i.e., \( 1, 4, 9, \dots \)).

   - **B.** \( Y = \sqrt{x} \), where \( x \) and \( y \) are members of natural numbers.
     - This is only valid when \( x \) is a perfect square because \( y = \sqrt{x} \) must be a natural number.
     - Some valid ordered pairs would be:
       - \( (1, 1) \)
       - \( (4, 2) \)
       - \( (9, 3) \)
       - Domain: Perfect squares of natural numbers.
       - Range: All natural numbers \( y \).

### 3. **List all the ordered pairs \( (x, y) \) that satisfy each of the following:**
   - Set \( A = \{1, 2, 4, 6, 7\} \), and set \( B = \{5, 12, 7, 9, 8\} \). We need to list ordered pairs based on the following conditions:
     - **A.** \( x > y \)
       - No valid pairs since every \( x \in A \) is smaller than or equal to any \( y \in B \).
     - **B.** \( x + y \) is odd.
       - \( (1, 12) \), \( (2, 5) \), \( (4, 5) \), \( (6, 7) \), \( (7, 8) \)
     - **C.** \( y \) is a multiple of \( x \).
       - \( (1, 5) \), \( (1, 12) \), \( (2, 12) \), \( (4, 12) \)
     - **D.** \( x \) is half of \( y \).
       - \( (2, 4) \), \( (4, 8) \)

### 4. **Relation R: \( x \to y \), where \( x = \{1, 2, 3, 4\} \) and \( y = \{2, 4, 9, 16, 25, 36\} \)**
   - The relation is \( x \to x^2 \), so:
     - Domain: \( \{1, 2, 3, 4\} \)
     - Co-domain: \( \{2, 4, 9, 16, 25, 36\} \)
     - Range: \( \{1, 4, 9, 16\} \) (the actual results of squaring \( x \)).

### 5. **Plot the points whose coordinates are given in a Cartesian system**
   - The points are \( A(-3, 5), R(0, -2), S(-2, 5), B(0, 2), C(2, 0), D(2, 3), E(4, -5) \).

### 6. **Stick Problem (Number of sticks in 4th and 10th steps)**
   - From the diagram, it looks like the number of sticks follows a pattern. Let's assume the number of sticks follows an arithmetic or geometric progression:
     - The 1st step has 1 stick, the 2nd has 3, and the 3rd has 5.
     - This seems to follow an arithmetic sequence with a common difference of 2. The expression for the number of sticks in the \( n \)-th step is \( 2n - 1 \).
     - For the 4th step, the number of sticks is \( 2(4) - 1 = 7 \).
     - For the 10th step, the number of sticks is \( 2(10) - 1 = 19 \).

Would you like a detailed explanation of any of these answers? Here are some further questions related to the topics above:

1. How can we find the range of a relation involving square roots in general?
2. What other properties can we derive from sets \( A \) and \( B \) when comparing elements using inequalities?
3. How would the graph of the stick problem's algebraic expression look?
4. Can you explain the significance of the co-domain versus the range in a relation?
5. What happens to the domain and range when a relation involves fractional or irrational numbers?

**Tip:** Always check for specific restrictions in a relation's domain, especially when dealing with square roots or fractions!

Math questions involving relations, domains, ranges, ordered pairs, Cartesian coordinates, and arithmetic progression (stick problem).

This problem set covers relations and functions, including finding domains and ranges, working with Cartesian coordinates, and solving a stick problem using arithmetic progression. Suitable for students in Grades 8-10.

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