Final answer:
The domain of the quadratic function f(x) = 3x² - 12x + 6 is all real numbers, and its range is all real numbers greater than or equal to 6, due to the parabola opening upwards with a vertex at (2,6).
Step-by-step explanation:
The function described, f(x) = 3x² - 12x + 6, is a quadratic function. To determine its domain and range, we look at the nature of quadratic functions. The domain of any quadratic function is all real numbers because there is no restriction on the x-values for which the function is defined. This means that for f(x), the domain is all real numbers or, more formally, (-∞, +∞).
The range of a quadratic function is determined by the vertex of its parabola. Since the coefficient in front of x² is positive (3), the parabola opens upwards. Thus, the vertex represents the minimum point of the parabola. To find the vertex, we use the formula -b/(2a) to find the x-coordinate of the vertex (h). In this function, a = 3 and b = -12, so h = -(-12)/(2*3) = 2. The y-coordinate of the vertex (k) can be found by plugging h back into the function: k = f(2) = 3(2)² - 12(2) + 6 = 6. Therefore, the vertex is at (2, 6) and the range of f(x) is [6, +∞), since the function can take any y-value greater than or equal to 6.