The function \( h(x)=x^{4}-2 x^{3}+3 x-1 \) is neither even nor odd even odd both even and odd

To determine whether the function \( h(x) = x^{4} - 2x^{3} + 3x - 1 \) is even, odd, or neither, let's analyze it step by step. ### Definitions: - **Even Function**: A function \( f \) is even if \( f(-x) = f(x) \) for all \( x \). - **Odd Function**: A function \( f \) is odd if \( f(-x) = -f(x) \) for all \( x \). ### Step 1: Compute \( h(-x) \) \[ h(-x) = (-x)^{4} - 2(-x)^{3} + 3(-x) - 1 = x^{4} + 2x^{3} - 3x - 1 \] ### Step 2: Compare \( h(-x) \) with \( h(x) \) and \( -h(x) \) 1. **Check if \( h \) is Even:** \[ h(-x) = x^{4} + 2x^{3} - 3x - 1 \quad \text{and} \quad h(x) = x^{4} - 2x^{3} + 3x - 1 \] These are **not** equal because the coefficients of \( x^{3} \) and \( x \) differ. 2. **Check if \( h \) is Odd:** \[ -h(x) = -x^{4} + 2x^{3} - 3x + 1 \] Comparing this with \( h(-x) \), we see that: \[ h(-x) \neq -h(x) \] The coefficients of \( x^{4} \) and the constant term do not match. ### Conclusion: Since \( h(-x) \) is neither equal to \( h(x) \) nor to \( -h(x) \), the function \( h(x) \) is **neither even nor odd**. **Answer:** Neither even nor odd