To determine whether the function $f(x) = -5x^5 + x^3$ is even, odd, or neither, we need to check its symmetry properties.

1. Check if the function is even:

A function $f(x)$ is even if: $f(-x) = f(x)$ for all $x$.

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

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

Since $(-x)^5 = -x^5$ and $(-x)^3 = -x^3$:

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

Clearly, $f(-x) = 5x^5 - x^3$ is not equal to $f(x) = -5x^5 + x^3$, so the function is not even.

2. Check if the function is odd:

A function $f(x)$ is odd if: $f(-x) = -f(x)$ for all $x$.

Now, let's compute $-f(x)$:

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

Notice that $f(-x) = 5x^5 - x^3$ and $-f(x) = 5x^5 - x^3$. Since $f(-x) = -f(x)$, the function is odd.

Conclusion:

The function $f(x) = -5x^5 + x^3$ is odd.

Would you like more details or have any other questions?

Here are some related questions to consider:

1. What is the geometric interpretation of even and odd functions?
2. How does the symmetry of the graph help in identifying even or odd functions?
3. Can a function be both even and odd? If so, what is an example?
4. How do the powers of $x$ in a polynomial affect whether the function is even or odd?
5. What happens to the classification (even/odd/neither) when constants are added to the function?

Tip: For polynomials, if all the powers of $x$ are even, the function is even; if all powers are odd, the function is odd.