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Given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval [tex]4 \leq x \leq 8[/tex].

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline
3 & 3 \\
\hline
4 & 7 \\
\hline
5 & 13 \\
\hline
6 & 21 \\
\hline
7 & 31 \\
\hline
8 & 43 \\
\hline
\end{tabular}


Sagot :

To find the average rate of change of the function [tex]\( f(x) \)[/tex] over the interval [tex]\( 4 \leq x \leq 8 \)[/tex], we can use the formula for the average rate of change:

[tex]\[ \text{Average rate of change} = \frac{f(x_{\text{end}}) - f(x_{\text{start}})}{x_{\text{end}} - x_{\text{start}}} \][/tex]

In this problem, the interval given is from [tex]\( x = 4 \)[/tex] to [tex]\( x = 8 \)[/tex]. So, [tex]\( x_{\text{start}} = 4 \)[/tex] and [tex]\( x_{\text{end}} = 8 \)[/tex].

From the table, we find the corresponding [tex]\( f(x) \)[/tex] values:
- [tex]\( f(4) = 7 \)[/tex]
- [tex]\( f(8) = 43 \)[/tex]

Now, substituting these values into our formula:

[tex]\[ \text{Average rate of change} = \frac{f(8) - f(4)}{8 - 4} \][/tex]

Substitute [tex]\( f(8) = 43 \)[/tex] and [tex]\( f(4) = 7 \)[/tex]:

[tex]\[ \text{Average rate of change} = \frac{43 - 7}{8 - 4} = \frac{36}{4} = 9 \][/tex]

Therefore, the average rate of change of the function [tex]\( f(x) \)[/tex] over the interval [tex]\( 4 \leq x \leq 8 \)[/tex] is [tex]\( 9 \)[/tex].