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To determine whether the function \( f(x) \) is even, odd, or neither, we can use the following properties:

- **Even Function**: \( f(x) = f(-x) \) for all \( x \) in the domain.
- **Odd Function**: \( f(-x) = -f(x) \) for all \( x \) in the domain.
- **Neither**: If the function does not satisfy either of the above properties.

Let's analyze the given table of values:

\[
\begin{array}{c|c}
x & f(x) \\
\hline
-3 & 9 \\
-2 & 4 \\
-1 & 0 \\
0 & 0 \\
1 & 4 \\
2 & -5 \\
3 & -9 \\
\end{array}
\]

### Step-by-Step Analysis:

1. **Check for Even Function:**
   - Compare \( f(x) \) with \( f(-x) \) for corresponding values of \( x \).
   - For \( x = -3 \), \( f(-3) = 9 \) and \( f(3) = -9 \). Since \( f(3) \neq f(-3) \), the function is not even.
   
2. **Check for Odd Function:**
   - Check if \( f(-x) = -f(x) \) for corresponding values of \( x \).
   - For \( x = -3 \), \( f(-3) = 9 \) and \( -f(3) = 9 \). This checks out.
   - For \( x = -2 \), \( f(-2) = 4 \) and \( -f(2) = 5 \). This does not satisfy the odd function condition since \( 4 \neq 5 \).

Since the function does not satisfy the conditions for being either even or odd, the function is **neither even nor odd**.

Would you like more details on this analysis, or do you have any questions?

Here are some related questions you might consider:

1. How would the table need to be modified for the function to be even?
2. What changes in the function's values would make it odd?
3. Can a function be both even and odd? Why or why not?
4. How would you represent this function graphically, and what would it tell you?
5. How does symmetry relate to even and odd functions?

**Tip:** A function that is neither even nor odd often lacks symmetry, which can be confirmed by analyzing the graph or the values of the function as we did here.

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 set of ordered pairs to determine whether a function is even, odd, or neither. This problem explores the properties of functions using key concepts like f(x) = f(-x) for even functions and f(-x) = -f(x) for odd functions. Suitable for students in Grades 9-12.

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