Question 8, 1.3.37 HW Score: \( \mathbf{2 6 . 0 9} \) 1.3 More on Functions and Craphs Part 1 of 2 Points: 0 of 1 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

To determine whether the function \( f(x) = x^{5} - 3x \) is even, odd, or neither, we'll use the definitions of even and odd functions. ### Definitions: - **Even Function:** A function \( f \) is even if for all \( x \) in its domain, \( f(-x) = f(x) \). Graphically, even functions are symmetric with respect to the \( y \)-axis. - **Odd Function:** A function \( f \) is odd if for all \( x \) in its domain, \( f(-x) = -f(x) \). Graphically, odd functions are symmetric with respect to the origin. - **Neither:** If a function does not satisfy either of the above conditions, it is neither even nor odd. ### Step-by-Step Analysis: 1. **Calculate \( f(-x) \):** \[ f(-x) = (-x)^{5} - 3(-x) = -x^{5} + 3x \] 2. **Compare \( f(-x) \) with \( f(x) \):** \[ f(-x) = -x^{5} + 3x = -(x^{5} - 3x) = -f(x) \] Since \( f(-x) = -f(x) \), the function satisfies the condition for being an **odd function**. ### Conclusion: - **Classification:** **Odd** - **Graph Symmetry:** The graph of \( f(x) = x^{5} - 3x \) is symmetric with respect to the **origin**. **Answer:** **odd**