The function \( f(x)=-x^{2}+4 \) is defined on the domain \( -8 \leq x \leq 8 \). Which of the following is the correct associated range? \( \begin{array}{l}-\infty

To find the range of the function \( f(x) = -x^2 + 4 \) defined on the domain \( -8 \leq x \leq 8 \), we first need to analyze the function. 1. **Identify the type of function**: The function \( f(x) = -x^2 + 4 \) is a downward-opening parabola. 2. **Find the vertex**: The vertex of the parabola can be found using the formula for the vertex of a quadratic function \( ax^2 + bx + c \). Here, \( a = -1 \), \( b = 0 \), and \( c = 4 \). The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} = -\frac{0}{2(-1)} = 0 \). 3. **Calculate the maximum value**: Substitute \( x = 0 \) into the function to find the maximum value: \[ f(0) = -0^2 + 4 = 4. \] 4. **Evaluate the function at the endpoints of the domain**: We also need to evaluate \( f(x) \) at the endpoints of the domain \( x = -8 \) and \( x = 8 \): \[ f(-8) = -(-8)^2 + 4 = -64 + 4 = -60, \] \[ f(8) = -(8)^2 + 4 = -64 + 4 = -60. \] 5. **Determine the range**: The maximum value of \( f(x) \) is \( 4 \) (at \( x = 0 \)), and the minimum value is \( -60 \) (at both \( x = -8 \) and \( x = 8 \)). Therefore, the range of \( f(x) \) is: \[ -60 \leq y \leq 4. \] Thus, the correct associated range is: \[ \boxed{-60 \leq y \leq 4}. \]