Question

the table above shows selected values for the continuous fur
for which interval of x is the average rate of change of f(x) ec
-4 ≤ x ≤ 1
1 ≤ x ≤ 3
3 ≤ x ≤ 5
-4 ≤ x ≤ 5

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is given by \(\frac{f(b)-f(a)}{b - a}\).

Step2: Calculate the average rate of change for \(-4\leq x\leq1\)

Here \(a=-4\), \(b = 1\), \(f(a)=7\), \(f(b)=2\).

$$ \frac{f(1)-f(-4)}{1-(-4)}=\frac{2 - 7}{1 + 4}=\frac{-5}{5}=-1 $$

Step3: Calculate the average rate of change for \(1\leq x\leq3\)

Here \(a = 1\), \(b=3\), \(f(a)=2\), \(f(b)=8\).

$$ \frac{f(3)-f(1)}{3 - 1}=\frac{8-2}{2}=\frac{6}{2}=3 $$

Step4: Calculate the average rate of change for \(3\leq x\leq5\)

Here \(a = 3\), \(b = 5\), \(f(a)=8\), \(f(b)=10\).

$$ \frac{f(5)-f(3)}{5-3}=\frac{10 - 8}{2}=\frac{2}{2}=1 $$

Step5: Calculate the average rate of change for \(-4\leq x\leq5\)

Here \(a=-4\), \(b = 5\), \(f(a)=7\), \(f(b)=10\).

$$ \frac{f(5)-f(-4)}{5-(-4)}=\frac{10 - 7}{5+4}=\frac{3}{9}=\frac{1}{3} $$

Answer:

B. \(1\leq x\leq3\)