To analyze the set of points \(\{(-4,2),(0,9),(-4,8),(4,5),(-1,10)\}\), let's determine some key properties:

1. **Unique x-values**: Check which x-values are repeated. Here, \(x = -4\) is repeated in the points \((-4,2)\) and \((-4,8)\).

2. **Vertical Line Test**: If any x-value is repeated with different y-values, it means the points do not represent a function because it would fail the vertical line test. Since \(x = -4\) is paired with both \(y = 2\) and \(y = 8\), this set of points does not represent a function.

Let's confirm our analysis by listing each step:

### Analysis:

- Points: \((-4,2)\), \((0,9)\), \((-4,8)\), \((4,5)\), \((-1,10)\)
- \(x = -4\) occurs in both \((-4,2)\) and \((-4,8)\)
- Different \(y\)-values for the same \(x\)-value \(x = -4\)
- Fails the vertical line test

### Conclusion:

The set of points \(\{(-4,2),(0,9),(-4,8),(4,5),(-1,10)\}\) does not represent a function because \(x = -4\) is paired with two different \(y\)-values (2 and 8).

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. What defines a function in mathematical terms?
2. How does the vertical line test determine if a set of points is a function?
3. Can you give an example of a set of points that represents a function?
4. What is the definition of the domain of a function?
5. How do you find the range of a function from a set of points?
6. What are some examples of common functions in algebra?
7. How do we represent functions graphically?
8. What happens if a function fails the vertical line test?

**Tip:** When checking if a set of points represents a function, ensure each \(x\)-value corresponds to exactly one \(y\)-value.

Explore how to analyze a set of points to determine if they represent a function using the vertical line test. This detailed analysis covers the properties of functions and practical examples. Suitable for high school students.

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