Final answer:
The domain of the function f(x) = 3|x - 4| + 2 is all real numbers, the range is y ≥ 2, and the vertex of the function is at the point (4, 2).
Step-by-step explanation:
To answer the question about the function f(x) = 3|x - 4| + 2, let's discuss the domain, range, and vertex of the function.
Domain: The domain of a function is the set of all possible input values (x-values) that the function can accept. For the given function, the domain is all real numbers since absolute value functions are defined for all real numbers. Thus, the domain is x ∈ (-∞, ∞).
Range: The range of a function is the set of all possible output values (y-values). Since the absolute value function is always non-negative and is being multiplied by 3 and then increased by 2, the smallest value it can take is 2 (when x=4). Therefore, the range of this function is y ∈ [2, ∞).
Vertex: The vertex of an absolute value function is the point at which the graph changes direction. It corresponds to the minimum (or maximum) value of the function. For f(x) = 3|x - 4| + 2, the vertex is located at (4, 2) because that's where the input into the absolute value expression is zero, which minimizes the function's output.