identify the interval(s) on which the function is increasing. \noa. (-∞,0)\nob. (0,3)\noc. (-∞,-1)\nod. (-1,0)

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_1<x_2), we have (f(x_1)<f(x_2)). Geometrically, the graph of an increasing function rises from left to right.

Step2: Analyze each option

• Option A: For the interval ((-\infty,0)), if we take (x_1=- 2) and (x_2 = -1) ((x_1<x_2)), from the graph, the function is constant (not increasing) for (x\in(-\infty,0)) (before (x = 0) the function has a constant - value part).
• Option B: For the interval ((0,3)), if we take (x_1=a) and (x_2=b) such that (0 < a < b<3). Looking at the graph, as (x) increases from (0) to (3), the (y) - value of the function increases. For example, when (x = 0), (y=0) and when (x = 2), (y = 4).
• Option C: For the interval ((-\infty,-1)), the function is constant (not increasing).
• Option D: For the interval ((-1,0)), the function is constant (not increasing).

Answer:

B. ((0,3))