Question

Determine whether the function is​ even, odd, or neither. Then determine whether the​ function's graph is symmetric with respect to the​ y-axis, the​ origin, or neither.
​f(x)equalsx cubed plus 6 x
Question content area bottom
Part 1
Determine whether the function is​ even, odd, or neither. Choose the correct answer below.
even
neither
odd
Part 2
Determine whether the graph of the function is symmetric with respect to the​ y-axis, the​ origin, or neither. Select all that apply.
origin
​y-axis
neither

Answer 1

To determine whether the function \( f(x) = x^3 + 6x \) is even, odd, or neither, we will analyze \( f(-x) \):

1. **Calculate \( f(-x) \)**:
\[
f(-x) = (-x)^3 + 6(-x) = -x^3 - 6x
\]

2. **Compare \( f(-x) \) and \( f(x) \)**:
- An **even function** satisfies \( f(-x) = f(x) \).
- An **odd function** satisfies \( f(-x) = -f(x) \).

We found:
\[
f(-x) = -x^3 - 6x = -(x^3 + 6x) = -f(x)
\]
Since \( f(-x) = -f(x) \), the function \( f(x) \) is **odd**.

### Part 1
**The function is: odd**

### Part 2
To determine the symmetry of the graph:
- A function is symmetric about the **y-axis** if it is even.
- A function is symmetric about the **origin** if it is odd.

Since we have established that \( f(x) \) is odd, the graph of the function is symmetric with respect to the **origin**.

### Final Answers
- **Part 1:** The function is **odd**.
- **Part 2:** The graph is symmetric with respect to the **origin**.