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.
​h(x)equals2 x Superscript 4 Baseline minus x Superscript 6
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Part 1
Determine whether the function is​ even, odd, or neither. Choose the correct answer below.
even
odd
neither
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 \( h(x) = 2x^4 - x^6 \) is even, odd, or neither, we can evaluate \( h(-x) \) and compare it to \( h(x) \) and \(-h(x)\).

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

1. **Calculate \( h(-x) \)**:
\[
h(-x) = 2(-x)^4 - (-x)^6
\]
\[
= 2x^4 - x^6
\]
\[
= h(x)
\]

2. Since \( h(-x) = h(x) \), the function is **even**.

### Part 2: Determine the symmetry of the graph.

Since we've determined that the function is even, it follows that the graph is symmetric with respect to the **y-axis**.

### Summary:
- **Part 1**: The function is **even**.
- **Part 2**: The graph is symmetric with respect to the **y-axis**.