The problem is asking you to compute $f(-x)$ and compare it with $-f(x)$ to determine whether the function is even, odd, or neither.

Given: $f(x) = x^5 - 3x^3$

Step 1: Find $f(-x)$

To find $f(-x)$, substitute $-x$ into the function: $f(-x) = (-x)^5 - 3(-x)^3$ Simplify: $f(-x) = -x^5 - 3(-x^3) = -x^5 + 3x^3$

Step 2: Find $-f(x)$

Now, compute $-f(x)$: $-f(x) = -(x^5 - 3x^3) = -x^5 + 3x^3$

Step 3: Compare $f(-x)$ and $-f(x)$

We see that: $f(-x) = -f(x)$

This means that the function is odd.

Fill in the blanks:

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

Both results are the same, confirming that the function is odd.

Do you want more details on how to identify even and odd functions or have any other questions?

Here are five related questions to expand on this:

1. How do you determine if a function is even?
2. Can a function be neither even nor odd? How can you tell?
3. What are some common properties of odd functions?
4. Can a polynomial of any degree be odd? Why or why not?
5. How does the symmetry of a graph relate to whether the function is even or odd?

Tip: For odd functions, the graph is symmetric about the origin, meaning if you rotate the graph 180 degrees, it looks the same.