Question

find the average rate of change from -1 < x < 1

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a=-1\) and \(b = 1\).

Step2: Find \(f(-1)\) and \(f(1)\) from the graph

From the graph, when \(x=-1\), \(y=-3\) (so \(f(-1)=-3\)), and when \(x = 1\), \(y=3\) (so \(f(1)=3\)).

Step3: Substitute into the formula

Substitute \(a=-1\), \(b = 1\), \(f(a)=-3\), and \(f(b)=3\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{3-(-3)}{1-(-1)}=\frac{3 + 3}{1+1}\).

Step4: Simplify the expression

\(\frac{6}{2}=3\).

Answer:

\(3\)