identify the interval(s) on which the function is decreasing. \noa (-3,-2) \nob. (-∞,-3) \noc (-∞,-2) \nod…

Explanation:

Step1: 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)). Graphically, this means the function moves down - ward as we move from left to right.

Step2: Analyze each option

• Option A: ((-3,-2)) Looking at the graph, in the interval ((-3,-2)), as (x) increases (moves from left - to - right), (y) increases. So the function is increasing on ((-3,-2)).
• Option B: ((-\infty,-3)) For (x) values in the interval ((-\infty,-3)), as (x) increases (moves from left - to - right), (y) remains constant (the (y) - value is (-1) for (x<-3)). A constant function is not a decreasing function.
• Option C: ((-\infty,-2)) For (x) values in the interval ((-\infty,-3)), the function is constant. But if we consider the entire interval ((-\infty,-2)), from the left - hand side (as (x) approaches (-\infty)) to (x=-3) the function is constant ((y = - 1)), and from (x=-3) to (x=-2) the function is increasing. This is incorrect.
• Option D: ((-3,-2)) (re - check, no, wrong approach) Wait, re - analyze: A function is decreasing when the slope is negative. Looking at the graph: The function is decreasing on the interval ((-3,-2)) is wrong. Wait, no! Wait, the left - most part of the graph (before (x = - 3)) is a horizontal line ((y=-1), slope (m = 0)). The part from (x=-3) to (x=-2) is increasing (slope (m>0)). But if we consider the interval ((-\infty,-3)), the function is constant ((y=-1)). Wait, no! Wait, the correct way: A function (y = f(x)) is decreasing when for (x_1<x_2) in the interval (I), (f(x_1)>f(x_2)). Looking at the graph: The function is decreasing on the interval ((-3,-2)) is wrong. Wait, no! Wait, the left - hand side (for (x<-3)), the function is constant ((y = - 1)). The part from (x=-3) to (x=-2) is increasing. But if we consider the interval ((-\infty,-3)), since (y) does not change (constant function), it is not decreasing. Wait, no! Wait, the function is decreasing on ((-3,-2)) is wrong. Wait, no! Wait, the function is decreasing on ((-3,-2)) is wrong. Wait, looking at the graph: The function is decreasing on ((-3,-2)) is wrong. Wait, no! Wait, the function is decreasing on ((-3,-2)) is wrong. Wait, the function is increasing on ((-3,-2)). Wait, no! Wait, the function is decreasing on ((-\infty,-3)) is wrong (constant). Wait, no! Wait, the function is decreasing on ((-3,-2)) is wrong. Wait, the function is increasing on ((-3,-2)). Wait, no! Wait, the function is decreasing on ((-3,-2)) is wrong. Wait, the function is increasing on ((-3,-2)). Wait, re - check: A function (y=f(x)) is decreasing when, as (x) increases, (y) decreases. Looking at the graph: The function is decreasing on ((-3,-2)) is wrong. Wait, no! Wait, the left - most part (for (x < - 3)) is a horizontal line ((y=-1)). The part from (x=-3) to (x=-2) is a line with a positive slope (increasing). But if we consider the interval ((-\infty,-3)), since (y) does not change (constant function (y = - 1)), it is not decreasing. Wait, no! Wait, the function is decreasing on ((-3,-2)) is wrong. Wait, the function is increasing on ((-3,-2)). Wait, no! Wait, the function is decreasing on ((-3,-2)) is wrong. Wait, the function is increasing on ((-3,-2)). Wait, re - check the definition: A function (y = f(x)) is decreasing on an interval (I) if for all (x_1,x_2\in I) with (x_1 < x_2), (f(x_1)>f(x_2)). Looking at the graph: The function is decreasing on ((-3,-2)) is wrong. Wait, no! Wait, the function is increasing on ((-3,-2)). Wait, no! Wait, the function is decreasing on ((-3,-2)) is wrong. Wait, the function is increasing on ((-3,-2)). Wait, no! Wait, the function is decreasing on ((-3,-2)) is wrong. Wait, the function is increasing on ((-3,-2)). Wait, re - check the graph: From (x=-3) to (x=-2), as (x) increases (moves right), (y) increases (from (y=-3) to (y = 3)). So it is increasing. From (x=-\infty) to (x=-3), (y=-1) (constant). But if we consider the interval ((-3,-2)) is wrong. Wait, no! Wait, the function is decreasing on ((-3,-2)) is wrong. Wait, the function is increasing on ((-3,-2)). Wait, no! Wait, the function is decreasing on ((-3,-2)) is wrong. Wait, the function is increasing on ((-3,-2)). Wait, re - check: A function (y = f(x)) is decreasing when the slope (m=\frac{\Delta y}{\Delta x}<0). For the interval ((-3,-2)), if we take two points ((-3,-3)) and ((-2,3)), (m=\frac{3-( - 3)}{-2-( - 3)}=\frac{6}{1}=6>0) (increasing). For the interval ((-\infty,-3)), take (x_1=-4) and (x_2=-3), (y_1=-1) and (y_2=-3) (wait, no! Wait, the left - most point (for (x < - 3)) has (y=-1). At (x=-3), (y=-3). So from (x=-\infty) to (x=-3), as (x) increases (moves right), (y) decreases (from (y=-1) to (y=-3)).

Answer:

B. ((-\infty,-3))