To determine if the function $f(x) = 2x^5 + 4x^3 - 5x$ is even, odd, or neither, we need to check how it behaves when we replace $x$ with $-x$. Here's the approach:

1. For a function to be even: The condition is $f(-x) = f(x)$.
2. For a function to be odd: The condition is $f(-x) = -f(x)$.

Let's compute $f(-x)$:

$f(-x) = 2(-x)^5 + 4(-x)^3 - 5(-x)$ $f(-x) = -2x^5 - 4x^3 + 5x$

Now compare $f(-x)$ with the original function $f(x) = 2x^5 + 4x^3 - 5x$.

• $f(-x) = -2x^5 - 4x^3 + 5x$ is not equal to $f(x)$, so it’s not even.
• $f(-x) = -2x^5 - 4x^3 + 5x$ is the negative of $f(x)$, so it satisfies the condition for being odd.

Thus, the function is odd.

Let me know if you would like more details or further clarification!

Here are 5 related questions to explore:

1. What are the properties of even and odd functions?
2. Can a function be both even and odd at the same time?
3. How do transformations like shifting affect the even or odd nature of a function?
4. Can you graph the function to visually confirm if it is odd?
5. How do you determine whether a rational function is odd, even, or neither?

Tip: For odd functions, you should see symmetry about the origin in their graphs!