Question

given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval 3 ≤ x ≤ 7.
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$$\begin{tabular}{|c|c|} \\hline $x$ & $f(x)$ \\\\ \\hline 1 & 2 \\\\ \\hline 3 & 2 \\\\ \\hline 5 & 10 \\\\ \\hline 7 & 26 \\\\ \\hline 9 & 50 \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) over the interval \( [a, b] \) is given by \( \frac{f(b) - f(a)}{b - a} \). Here, \( a = 3 \) and \( b = 7 \).

Step2: Find \( f(3) \) and \( f(7) \) from the table

From the table, when \( x = 3 \), \( f(3) = 2 \); when \( x = 7 \), \( f(7) = 26 \).

Step3: Substitute into the formula

Substitute \( a = 3 \), \( b = 7 \), \( f(3) = 2 \), and \( f(7) = 26 \) into the formula:

$$ \frac{f(7) - f(3)}{7 - 3} = \frac{26 - 2}{7 - 3} $$

Step4: Simplify the expression

First, calculate the numerator and the denominator:

• Numerator: \( 26 - 2 = 24 \)
• Denominator: \( 7 - 3 = 4 \)

Then, divide the numerator by the denominator: \( \frac{24}{4} = 6 \)

Answer:

6