Final answer:
The domain of the function f(x) = |x - 3| + 6 is all real numbers, and the range is y such that y is greater than or equal to 6.
Step-by-step explanation:
The function in question is f(x) = |x - 3| + 6. To determine the domain and range of this function, we need to consider the properties of the absolute value function and any transformations applied to it. The absolute value function, |x|, is defined for all real numbers, so the expression inside the absolute value, x - 3, is also defined for all real numbers. This implies that the domain of f(x) is all real numbers. As for the range, since the absolute value is always non-negative, |x - 3| is always greater than or equal to 0. Adding 6 to it shifts the graph vertically upwards by 6 units, making the smallest value of f(x) equal to 6. Hence, f(x) is greater than or equal to 6 for all x, and there is no upper limit. Therefore, the range of f(x) is y | y ≥ 6.
Based on this analysis, the correct answer to the question is:
Domain: x | x is all real numbers, Range: y | y ≥ 6.