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. the function is never 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. the function is never decreasing.

(c) select the correct choice below and, if necessary, fill in the answer box to complete your choice.
○ a. the function is constant on the interval(s)
(type your answer in interval notation. use a comma to separate answers as needed.)
○ b. the function is never constant.

Explanation:

Step1: Recall the definitions

• A function \(y = f(x)\) is increasing on an open interval \((a,b)\) if for any \(x_1,x_2\in(a,b)\) with \(x_1 < x_2\), \(f(x_1)<f(x_2)\).
• A function \(y = f(x)\) is decreasing on an open interval \((a,b)\) if for any \(x_1,x_2\in(a,b)\) with \(x_1 < x_2\), \(f(x_1)>f(x_2)\).
• A function \(y = f(x)\) is constant on an open interval \((a,b)\) if for any \(x_1,x_2\in(a,b)\) with \(x_1 < x_2\), \(f(x_1)=f(x_2)\).

Step2: Analyze the graph

• Looking at the graph, we note that for \(x\in(3,\infty)\), as \(x\) increases, \(y\) increases.
• For \(x\in(-\infty,3)\), as \(x\) increases, \(y\) does not decrease (since it is not a line with negative slope) and for \(x\in(-\infty,3)\) the function is not constant (because the \(y -\) value is not the same for all \(x\) in a non - singleton open interval).

Answer:

(a) A. The function is increasing on the interval(s) \((3,\infty)\)
(b) B. The function is never decreasing
(c) B. The function is never constant