the function ( f ) is graphed below. determine the intervals on which ( f ) increasing and…

the function ( f ) is graphed below. determine the intervals on which ( f ) increasing and decreasing.\nanswer attempt 1 out of 2\nthe function is ( ) on the interval ( square<x<square ) (segment 1), ( ) on the interval ( square<x<square ) (segment 2),\nand ( ) on the interval ( square<x<square ) (segment 3).

the function ( f ) is graphed below. determine the intervals on which ( f ) increasing and decreasing.\nanswer attempt 1 out of 2\nthe function is ( ) on the interval ( square<x<square ) (segment 1), ( ) on the interval ( square<x<square ) (segment 2),\nand ( ) on the interval ( square<x<square ) (segment 3).

Answer

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<x_2), we have (f(x_1)<f(x_2)). 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)).

Step2: Analyze the graph

Looking at the graph:

  • For segment 1: As (x) increases (from left - to - right), (y) increases. The left - most (x) value of this segment is (x=- 6) and the right - most (x) value (where the function changes its behavior) is (x=-1).
  • For segment 2: As (x) increases (from left - to - right), (y) decreases. The left - most (x) value of this segment is (x = - 1) and the right - most (x) value (where the function changes its behavior) is (x = 2).
  • For segment 3: As (x) increases (from left - to - right), (y) decreases. The left - most (x) value of this segment is (x=2) and the right - most (x) value is (x = 7).

Answer:

The function is increasing on the interval (-6<x<-1) (segment 1), decreasing on the interval (-1 < x < 2) (segment 2), and decreasing on the interval (2<x<7) (segment 3).