Question

use the graph to determine
a. open intervals on which the function is increasing, if any
b. open intervals on which the function is decreasing, if any
c. open intervals on which the function is constant, if any
a. select the correct choice below and, if necessary, fill in the answer box to complete your choice
a. the function is increasing on the interval(s) $(0,\infty)$
(type your answer in interval notation. use a comma to separate answers as needed)
b. there is no interval on which the function is increasing
b. select the correct choice below and, if necessary, fill in the answer box to complete your choice
a. the function is decreasing on the interval(s)
(type your answer in interval notation. use a comma to separate answers as needed)
b. there is no interval on which the function is decreasing

Explanation:

Step1: Analyze the function's increasing interval

A function is increasing when, as \(x\) values increase, \(y\) values also increase. Looking at the graph, from \(x = 1\) to \(x=\infty\), as \(x\) increases, \(y\) increases. Also, from \(x=-\infty\) to \(x = - 2\), as \(x\) increases (moving from left to right), \(y\) increases. So the increasing intervals are \((-\infty,-2)\cup(1,\infty)\)

Step2: Analyze the function's decreasing interval

A function is decreasing when, as \(x\) values increase, \(y\) values decrease. From \(x=-2\) to \(x = 1\), as \(x\) increases (moving from left to right), \(y\) decreases. So the decreasing interval is \((-2,1)\)

Step3: Analyze the function's constant interval

A function is constant when, as \(x\) values increase, \(y\) values remain the same. Looking at the graph, there is no such interval where \(y\) remains unchanged as \(x\) changes.

Answer:

a. A. The function is increasing on the interval(s) \((-\infty,-2)\cup(1,\infty)\)
b. A. The function is decreasing on the interval(s) \((-2,1)\)
c. B. There is no interval on which the function is constant