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.