Question

approximate the intervals where each function is increasing and decreasing. 16) a) increasing: $(-1,0),(1,\infty)$ decreasing: $(-\infty,-1),(0,1)$ b) increasing: $(2,\infty)$ decreasing: $(-\infty,2)$ c) increasing: $(-\infty,2),(2,\infty)$ decreasing: no intervals exist. d) increasing: $(-\infty,0),(2,\infty)$ decreasing: $(0,2)$

Explanation:

Step1: Recall the definition of increasing and decreasing functions

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_1

Step2: Analyze the graph

Looking at the graph, we can see that the function has a local minimum at \(x=- 1\) and \(x = 1\).
For \(x\in(-\infty,-1)\), as \(x\) increases, \(y = f(x)\) decreases.
For \(x\in(-1,0)\), as \(x\) increases, \(y = f(x)\) increases.
For \(x\in(0,1)\), as \(x\) increases, \(y = f(x)\) decreases.
For \(x\in(1,\infty)\), as \(x\) increases, \(y = f(x)\) increases.

Answer:

A. Increasing: \((-1,0),(1,\infty)\) Decreasing: \((-\infty,-1),(0,1)\)