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Let's analyze the given questions one by one to determine if the relations represent **one-to-one functions**:

### Definitions:
- **One-to-One Function**: A function is one-to-one (injective) if different elements in the domain map to different elements in the codomain. That means for every \( x_1 \neq x_2 \), \( f(x_1) \neq f(x_2) \).

### Answers:
1. **M = {(7, -5), (6, -3), (5, -1), (4, 1)}**
   - This is a one-to-one function because all distinct elements in the domain \( \{7, 6, 5, 4\} \) map to distinct elements in the codomain \( \{-5, -3, -1, 1\} \). No two elements in the domain map to the same element in the codomain.
   - **Answer: One-to-one**.

2. **A = {(6, 0), (6, 1), (6, 2), (6, 3), ...}**
   - This is **not** a one-to-one function because the same domain element \( 6 \) maps to multiple codomain elements \( \{0, 1, 2, 3\} \). A function must assign exactly one codomain element to each domain element.
   - **Answer: Not one-to-one**.

3. **The relation pairing a PHS student to his or her class section**
   - This could be considered one-to-one if every student has a unique class section. However, if multiple students can be in the same section, this is **not** one-to-one.
   - **Answer: Likely not one-to-one**, as it is common for students to share sections.

4. **The relation pairing a company to its official logo**
   - Each company typically has a unique official logo, meaning each company (domain) maps to a distinct logo (codomain).
   - **Answer: One-to-one**.

5. **The relation pairing a counting number to its opposite**
   - Each counting number has a unique opposite (e.g., 1 maps to -1, 2 maps to -2, etc.), making this a one-to-one relation.
   - **Answer: One-to-one**.

Would you like more details or clarifications on these? Here are some related questions to deepen understanding:

1. What is the difference between a one-to-one function and a many-to-one function?
2. Can a function be one-to-one but not onto?
3. How can you visually determine if a function is one-to-one using a graph?
4. How does the vertical line test help in identifying a valid function?
5. Can a relation still be one-to-one if there are missing elements from the codomain?

**Tip:** One easy way to check if a function is one-to-one is to ensure that no two distinct domain elements map to the same codomain element.

Determine if the following relations represent one-to-one functions.

Learn how to determine whether a given relation represents a one-to-one function. This exercise covers key concepts related to functions and relations, focusing on how to identify a one-to-one correspondence between domain and codomain elements.

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