**Determine whether the given function is even, odd, or neither.**

f(x) = -5x5 + x3

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.

Learn how to determine if a function is even, odd, or neither using the example function f(x) = -5x^5 + x^3. Explore the concepts of symmetry in functions, the properties of even and odd functions, and how polynomial powers affect function classification. Suitable for advanced high school students in Grades 11-12.

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