over which interval(s) is the graph of the function below increasing? verify your answer by finding the…

Explanation:

Step1: Recall increasing - function concept

A function is increasing on an interval if for any two points $x_1$ and $x_2$ in the interval with $x_1<x_2$, $f(x_1)<f(x_2)$. Visually, the graph of the function rises as we move from left - to - right.

Step2: Analyze the given graph

Looking at the graph, we can see that the graph is rising from $x = -\infty$ to $x = 0$.

Step3: Calculate the average rate of change

The formula for the average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Let's take two points in the interval $(-\infty,0)$, say $a=-2$ and $b = 0$. Suppose the function values are $f(-2)=-2$ and $f(0)=4$. Then the average rate of change is $\frac{f(0)-f(-2)}{0-(-2)}=\frac{4 - (-2)}{2}=\frac{6}{2}=3>0$. A positive average rate of change indicates the function is increasing.

Answer:

The function is increasing over the interval $(-\infty,0)$.