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. \( f(x)=x^{5}-3 x \) Determine whether the function is even, odd, or neither. Choose the correct answer below. neither odd even 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 neither origin

To determine whether the function \( f(x) = x^{5} - 3x \) is even, odd, or neither, and to identify its graph's symmetry, follow these steps: ### 1. Determine if the Function is Even, Odd, or Neither **Definitions:** - **Even Function:** Satisfies \( f(-x) = f(x) \). Its graph is symmetric about the \( y \)-axis. - **Odd Function:** Satisfies \( f(-x) = -f(x) \). Its graph is symmetric about the origin. - **Neither:** Does not satisfy either condition. **Calculation:** \[ f(-x) = (-x)^5 - 3(-x) = -x^5 + 3x = -(x^5 - 3x) = -f(x) \] Since \( f(-x) = -f(x) \), the function satisfies the condition for an **odd function**. ### 2. Determine the Graph's Symmetry Given that the function is **odd**, its graph is **symmetric with respect to the origin**. ### **Final Answer:** - **Function Type:** Odd - **Graph Symmetry:** Origin **Selected Answers:** - **Determine whether the function is even, odd, or neither:** **odd** - **Determine whether the graph of the function is symmetric with respect to the \( y \)-axis, the origin, or neither:** **origin**