Final answer:
The minimum value of the function h(x) = 3x² - 6x - 2 is at x = 1. The domain is all real numbers and the range is from negative infinity to positive infinity. The function is increasing on the interval (-∞, 1) and decreasing on the interval (1, ∞).
Step-by-step explanation:
The minimum or maximum value of a quadratic function can be found by analyzing its vertex. The vertex of the function h(x) = 3x² - 6x - 2 can be found using the formula x = -b/2a. In this case, a = 3 and b = -6. Calculating x = -(-6)/(2*3), we find that the vertex x-coordinate is 1.
To determine whether the vertex is a minimum or maximum, we can look at the coefficient of the x² term. If it is positive, the vertex represents a minimum, and if it is negative, the vertex represents a maximum. In this case, since the coefficient of x² is positive, the vertex represents a minimum.
The domain of the function h(x) = 3x² - 6x - 2 is all real numbers, since there are no restrictions on the values of x that can be plugged into the function. The range of the function is from negative infinity to positive infinity, as the function can take on any real value.
Finally, to determine where the function is increasing or decreasing, we can look at the sign of the coefficient of the x term. If it is positive, the function is increasing, and if it is negative, the function is decreasing. In this case, since the coefficient of x is positive, the function is increasing on the interval (-∞, 1) and decreasing on the interval (1, ∞).