Question

Identify the increasing and decreasing intervals (use interval notation) for g(x)=3∣x−5∣−6.

a) Increasing: (−[infinity],5), Decreasing: (5,[infinity])
b) Increasing: (5,[infinity]), Decreasing: (−[infinity],5)
c) Increasing: (−6,5), Decreasing: (5,[infinity])
d) Increasing: (−[infinity],5), Decreasing: (−[infinity],[infinity])

Answer 1

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).