https://file.aiquickdraw.com/mathbot/file/1725412019509py8udxlr.jpg

The table provided shows several ordered pairs \((x, f(x))\) for a function \(f(x)\):

| \(x\) | \(f(x)\) |
|-------|----------|
| \(-3\) | 9        |
| \(-2\) | 5        |
| \(-1\) | -1       |
| 0     | 0         |
| 1     | -1        |
| 2     | 5         |
| 3     | 9         |

To determine whether the function is even, odd, or neither, we need to check the symmetry properties:

1. **Even Function**: A function is even if \(f(x) = f(-x)\) for all \(x\). This means that the function is symmetric with respect to the y-axis.
2. **Odd Function**: A function is odd if \(f(x) = -f(-x)\) for all \(x\). This means that the function is symmetric with respect to the origin.

### Analysis of the Given Values:

- For \(x = -3\), \(f(-3) = 9\) and for \(x = 3\), \(f(3) = 9\). This satisfies \(f(x) = f(-x)\).
- For \(x = -2\), \(f(-2) = 5\) and for \(x = 2\), \(f(2) = 5\). This also satisfies \(f(x) = f(-x)\).
- For \(x = -1\), \(f(-1) = -1\) and for \(x = 1\), \(f(1) = -1\). Again, \(f(x) = f(-x)\) is satisfied.
- For \(x = 0\), \(f(0) = 0\).

Since for every \(x\), \(f(x) = f(-x)\), the function is **even**. 

### Conclusion:

The function is **even** because for every value of \(x\) given, the function satisfies \(f(x) = f(-x)\).

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you determine if a function is odd using its graph?
2. What are the characteristics of a function that is neither even nor odd?
3. Can a function be both even and odd? If so, under what conditions?
4. How do transformations like translations affect the evenness or oddness of a function?
5. How can you use calculus to verify if a function is even or odd?

**Tip:** To check if a function is even or odd, plug in \(-x\) and compare it to \(f(x)\). If \(f(-x) = f(x)\), the function is even; if \(f(-x) = -f(x)\), the function is odd.

Based on the ordered pairs seen in the table, make a conjecture about whether the function is even, odd, or neither. Provide clear reasoning.

Analyze a function's symmetry by examining a table of values. Learn how to determine if a function is even, odd, or neither using the concepts of symmetry with respect to the y-axis and origin. Suitable for students in Grades 8-10.

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