Consider the function \( f(x)=3 x^{2}-12 x-3 \). a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. a. The function has a value.

The function \( f(x) = 3x^{2} - 12x - 3 \) is a quadratic function that opens upward since the coefficient of \( x^2 \) is positive (3). Therefore, this function has a minimum value. The minimum value occurs at the vertex of the parabola, which can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = 3 \) and \( b = -12 \). Plugging in these values gives us \( x = -\frac{-12}{2 \times 3} = 2 \). Substituting \( x = 2 \) back into the function, \( f(2) = 3(2^{2}) - 12(2) - 3 = -15 \). Hence, the minimum value is -15 at \( x = 2 \). The domain of \( f(x) \) is all real numbers, \((-∞, ∞)\), since a quadratic function is defined for every real number. The range is \( [-15, ∞) \) because the minimum value is -15 and the function tends to infinity as \( x \) moves away from the vertex.