Final answer:
The function g(x) = 3|x−5|−6 is decreasing on the interval (−∞, 5) and increasing on the interval (5, ∞). This behavior is due to the properties of the absolute value function and its transformations.
Step-by-step explanation:
To identify the increasing and decreasing intervals of the function g(x) = 3∧x−5∧−6, we first need to understand the properties of the absolute value function. The graph of |x| has a V shape, with the point (0, 0) being the vertex. With the function g(x) = 3∧x−5∧−6, the vertex of this V shape is shifted to x = 5, and the whole graph is shifted downward by 6 units due to subtraction.
The function is decreasing to the left of x = 5 because as x approaches 5 from the left, the value of |x − 5| is decreasing, and since it's multiplied by 3, the overall value of the function is also decreasing. Therefore, the interval for the decreasing part is (−∞, 5).
On the right side of x = 5, as x increases, the value of |x − 5| also increases. Thus, the function is increasing on the interval (5, ∞). Hence, the correct answer is b) Increasing: (5,∞), Decreasing: (−∞,5).