Determine the domain and range of the polynomial function \( f(x)=-x^{4}+8 x^{2}-10 \) Domain: All Real Numbers: Range: \( -10 \leq y \leq 6 \) Domain: All Real Numbers; Range: All Real Numbers Domain: All Real Numbers; Range: \( y \leq 6 \) Domain: \( -3 \leq x \leq 3 \); Range: \( -10 \leq y \leq 6 \)

To determine the domain and range of the polynomial function \( f(x) = -x^{4} + 8x^{2} - 10 \), we will analyze the function step by step. ### Step 1: Determine the Domain The domain of a polynomial function is always all real numbers because polynomials are defined for every real value of \( x \). Therefore, the domain of \( f(x) \) is: \[ \text{Domain: All Real Numbers} \] ### Step 2: Determine the Range To find the range, we need to analyze the behavior of the function. 1. **Identify the type of polynomial**: The function \( f(x) \) is a quartic polynomial (degree 4) with a leading coefficient of -1, which means it opens downwards. 2. **Find the critical points**: We will find the derivative of \( f(x) \) and set it to zero to find the critical points. \[ f'(x) = \frac{d}{dx}(-x^{4} + 8x^{2} - 10) = -4x^{3} + 16x \] Setting the derivative equal to zero: \[ -4x^{3} + 16x = 0 \] Factoring out common terms: \[ -4x(x^{2} - 4) = 0 \] This gives us: \[ x = 0, \quad x^{2} - 4 = 0 \implies x = 2, \quad x = -2 \] So, the critical points are \( x = -2, 0, 2 \). 3. **Evaluate \( f(x) \) at the critical points**: \[ f(-2) = -(-2)^{4} + 8(-2)^{2} - 10 = -16 + 32 - 10 = 6 \] \[ f(0) = -0^{4} + 8(0)^{2} - 10 = -10 \] \[ f(2) = -(2)^{4} + 8(2)^{2} - 10 = -16 + 32 - 10 = 6 \] 4. **Determine the maximum and minimum values**: The maximum value of \( f(x) \) occurs at \( x = -2 \) and \( x = 2 \), where \( f(x) = 6 \). The minimum value occurs at \( x = 0 \), where \( f(x) = -10 \). ### Conclusion The range of \( f(x) \) is from the minimum value to the maximum value: \[ \text{Range: } -10 \leq y \leq 6 \] ### Final Answer - Domain: All Real Numbers - Range: \( -10 \leq y \leq 6 \) Thus, the correct option is: **Domain: All Real Numbers; Range: \( -10 \leq y \leq 6 \)**.