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Classify each of the following functions as even, odd or neither. Be sure to include your work to justify your classification.

Let's analyze and classify each function as even, odd, or neither.

### Function 1: \( f(x) = -2x^2 + 16x - 25 \)

#### To check if a function is:
- **Even**: \( f(x) = f(-x) \)
- **Odd**: \( f(x) = -f(-x) \)

1. **Calculate \( f(-x) \):**

\[
f(-x) = -2(-x)^2 + 16(-x) - 25 = -2x^2 - 16x - 25
\]

2. **Compare \( f(-x) \) to \( f(x) \):**

- \( f(-x) = -2x^2 - 16x - 25 \)
- \( f(x) = -2x^2 + 16x - 25 \)

Since \( f(-x) \neq f(x) \) and \( f(-x) \neq -f(x) \), this function is **neither even nor odd**.

### Function 2 (Graph):

The graph shows a cubic-like function with a curve, suggesting that we might be dealing with an odd function. Let's confirm using symmetry:
- **Odd functions** exhibit rotational symmetry about the origin.

Based on the graph, it appears to have symmetry around the origin, which suggests the function is **odd**.

### Function 3 (Graph):

The graph displays a parabola, which typically suggests an even function since:
- **Even functions** exhibit symmetry about the y-axis.

Looking at the graph, the function appears symmetric about the y-axis, indicating that this function is **even**.

### Function 4: \( f(x) = x^5 + 3x^3 \)

1. **Calculate \( f(-x) \):**

\[
f(-x) = (-x)^5 + 3(-x)^3 = -x^5 - 3x^3
\]

2. **Compare \( f(-x) \) to \( f(x) \):**

- \( f(-x) = -x^5 - 3x^3 \)
- \( f(x) = x^5 + 3x^3 \)

Since \( f(-x) = -f(x) \), this function is **odd**.

---

### Final Classification:
1. \( f(x) = -2x^2 + 16x - 25 \) → Neither
2. Graph (Cubic-like) → Odd
3. Graph (Parabola) → Even
4. \( f(x) = x^5 + 3x^3 \) → Odd

Would you like further details or explanations on any of the functions?

### Follow-up Questions:
1. How can you prove that a function is odd using algebraic methods?
2. Can a function be both even and odd? Why or why not?
3. What characteristics of the graph help determine if a function is even or odd?
4. Are all polynomial functions either even or odd?
5. How does the degree of a polynomial affect its classification as even or odd?

### Tip:
Always check the symmetry of the function's graph to quickly identify whether it is even, odd, or neither.

Learn how to classify functions as even, odd, or neither with step-by-step solutions and justifications. This problem covers key algebraic concepts including symmetry, function analysis, and the characteristics of even and odd functions. Suitable for high school students in Grades 9-12.

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