Ore On Functions and Craphs 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. \( h(x)=3 x^{2}-x^{10} \) Determine whether the function is even, odd, or neither. Choose the correct answer below. odd \( 10,1.3 .41 \) neither even

To determine whether the function \( h(x) = 3x^{2} - x^{10} \) is **even**, **odd**, or **neither**, and to identify its symmetry, follow these steps: ### 1. **Check for Evenness** A function \( f(x) \) is **even** if: \[ f(-x) = f(x) \] Compute \( h(-x) \): \[ h(-x) = 3(-x)^{2} - (-x)^{10} = 3x^{2} - x^{10} \] Notice that: \[ h(-x) = 3x^{2} - x^{10} = h(x) \] Since \( h(-x) = h(x) \), the function is **even**. ### 2. **Check for Oddness** A function \( f(x) \) is **odd** if: \[ f(-x) = -f(x) \] From the above calculation: \[ h(-x) = h(x) \neq -h(x) \] Thus, the function is **not odd**. ### 3. **Determine Symmetry** Since the function is **even**, its graph is **symmetric with respect to the y-axis**. ### **Conclusion** - **Function Classification:** **Even** - **Graph Symmetry:** Symmetric with respect to the **y-axis** **Answer:** **even**, and its graph is symmetric with respect to the y‑axis.