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 StartRoot 2 minus x Superscript 6 EndRoot
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Part 1
Determine whether the function is​ even, odd, or neither. Choose the correct answer below.
neither
odd
Your answer is correct.
even
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.
​y-axis
origin
neither

Answer 1

To determine whether the function \( f(x) = x \sqrt{2 - x^6} \) is even, odd, or neither, we will evaluate \( f(-x) \) and compare it to \( f(x) \) and \(-f(x)\).

### Part 1: Determine whether the function is even, odd, or neither.

1. Calculate \( f(-x) \):
\[
f(-x) = -x \sqrt{2 - (-x)^6}
\]
Since \((-x)^6 = x^6\), this simplifies to:
\[
f(-x) = -x \sqrt{2 - x^6}
\]

2. Compare \( f(-x) \) with \( f(x) \):
\[
f(x) = x \sqrt{2 - x^6}
\]

It can be observed that:
\[
f(-x) = -f(x)
\]
This means that the function is **odd**.

### Part 2: Determine whether the graph of the function is symmetric with respect to the y-axis, the origin, or neither.

- A function is **symmetric with respect to the y-axis** if \( f(-x) = f(x) \) (which is the case for even functions).
- A function is **symmetric with respect to the origin** if \( f(-x) = -f(x) \) (which is the case for odd functions).

Since we found that \( f(-x) = -f(x) \), it follows that the graph of the function is symmetric with respect to the **origin**.

### Summary of Results:
- The function is **odd**.
- The graph is symmetric with respect to the **origin**.