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The problem asks to determine whether the function \( f(x) = 2x^5 - x^4 + 2x \) is odd, even, or neither.

To do this, we need to check the following conditions:

- A function is **even** if \( f(-x) = f(x) \) for all \( x \).
- A function is **odd** if \( f(-x) = -f(x) \) for all \( x \).
- If neither of these conditions is satisfied, the function is **neither**.

Let's test \( f(x) = 2x^5 - x^4 + 2x \) for both conditions.

### Step 1: Compute \( f(-x) \)
We substitute \( -x \) into the function:
\[
f(-x) = 2(-x)^5 - (-x)^4 + 2(-x)
\]
Simplifying each term:
\[
f(-x) = 2(-x^5) - x^4 + 2(-x)
\]
\[
f(-x) = -2x^5 - x^4 - 2x
\]

### Step 2: Compare \( f(-x) \) with \( f(x) \)

- \( f(x) = 2x^5 - x^4 + 2x \)
- \( f(-x) = -2x^5 - x^4 - 2x \)

Clearly, \( f(-x) \neq f(x) \), so the function is **not even**.

### Step 3: Check if the function is odd
For the function to be odd, \( f(-x) \) must equal \( -f(x) \).

We compute \( -f(x) \):
\[
-f(x) = -(2x^5 - x^4 + 2x) = -2x^5 + x^4 - 2x
\]

Comparing \( f(-x) \) and \( -f(x) \):
- \( f(-x) = -2x^5 - x^4 - 2x \)
- \( -f(x) = -2x^5 + x^4 - 2x \)

Since these are not equal, the function is **not odd**.

### Conclusion:
Since the function is neither even nor odd, the function \( f(x) = 2x^5 - x^4 + 2x \) is **neither**.

Would you like more details or have any questions about this? Here are five related questions:

1. What are the key properties of even and odd functions?
2. Can a function be both even and odd at the same time?
3. How does the degree of terms in a polynomial affect whether it can be odd or even?
4. What role does symmetry play in determining if a function is even or odd?
5. What are some examples of functions that are only odd or only even?

**Tip:** A quick way to spot if a function is even or odd is by analyzing the powers of \( x \) in the function.

Determine whether the function f(x) = 2x^5 - x^4 + 2x is odd, even, or neither.

Learn how to determine whether the function f(x) = 2x^5 - x^4 + 2x is even, odd, or neither by applying key algebraic principles, such as testing for f(-x) = f(x) and f(-x) = -f(x). Suitable for high school students in Grades 10-12.

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