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. \( f(x)=x \sqrt{4-x^{4}} \)

To determine whether the function \( f(x) = x \sqrt{4 - x^{4}} \) is even, odd, or neither, and to analyze its symmetry, follow these steps: ### 1. Determine If the Function Is Even, Odd, or Neither - **Even Function:** A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in its domain. - **Odd Function:** A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \) in its domain. - **Neither:** If neither condition is satisfied, the function is neither even nor odd. **Calculation:** \[ f(-x) = (-x) \sqrt{4 - (-x)^4} = -x \sqrt{4 - x^4} \] Comparing this with \( f(x) \): \[ f(-x) = -x \sqrt{4 - x^4} = -f(x) \] Since \( f(-x) = -f(x) \), the function is **odd**. ### 2. Determine Graphical Symmetry - **Symmetry with respect to the \( y \)-axis:** Corresponds to the function being even. - **Symmetry with respect to the origin:** Corresponds to the function being odd. - **Neither:** If the function is neither even nor odd. **Conclusion:** Since the function is odd, its graph is **symmetric with respect to the origin**. ### Final Answer: **The function is odd, and its graph is symmetric about the origin.**