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)
(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

Explanation:

Step1: Recall the definition of an increasing function

A function \(y = f(x)\) is increasing on an interval if for any two points \(x_1\) and \(x_2\) in the interval with \(x_1Looking at the graph, we check the slope of the function. A positive - slope indicates an increasing function. But in the given graph, the slope is negative (for the non - constant part) or zero (for the constant part).

Step2: Recall the definition of a decreasing function

A function \(y = f(x)\) is decreasing on an interval if for any two points \(x_1\) and \(x_2\) in the interval with \(x_1 < x_2\), we have \(f(x_1)>f(x_2)\).
For the non - constant part of the graph (from \(x=-2\) to \(x = 3\)), if we take \(x_1=-2\), \(y_1 = 7\) and \(x_2=3\), \(y_2 = 3\). Since as \(x\) increases from \(-2\) to \(3\), \(y\) decreases. The interval \((-2,3)\) is where the function is decreasing.

Step3: Recall the definition of a constant function

A function \(y = f(x)\) is constant on an interval if for any two points \(x_1\) and \(x_2\) in the interval, \(f(x_1)=f(x_2)\).
For the part of the graph from \(x = 3\) to the right (as \(x\) increases beyond \(3\), \(y\) remains \(3\)), the interval \((3,\infty)\) is where the function is constant.

Answer:

a. B. There is no interval on which the function is increasing
b. The function is decreasing on the interval \((-2,3)\)
c. The function is constant on the interval \((3,\infty)\)