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Given the tables for [tex]\(f\)[/tex] and [tex]\(g\)[/tex] below, find the following:

\begin{tabular}{|r|r|r|}
\hline
[tex]$x$[/tex] & [tex]$f(x)$[/tex] & [tex]$g(x)$[/tex] \\
\hline
0 & 2 & 1 \\
\hline
1 & 0 & 7 \\
\hline
2 & 5 & 8 \\
\hline
3 & 9 & 0 \\
\hline
4 & 7 & 3 \\
\hline
5 & 1 & 6 \\
\hline
6 & 3 & 2 \\
\hline
7 & 8 & 5 \\
\hline
8 & 4 & 4 \\
\hline
9 & 6 & 9 \\
\hline
\end{tabular}

The average rate of change of [tex]\(g\)[/tex] from [tex]\(x=0\)[/tex] to [tex]\(x=7\)[/tex] is [tex]\(\square\)[/tex].



Answer :

GinnyAnswer
To find the average rate of change of [tex]\(g(x)\)[/tex] from [tex]\(x=0\)[/tex] to [tex]\(x=7\)[/tex], we need to follow a few steps.

1. Identify the values of [tex]\(g(x)\)[/tex] at [tex]\(x=0\)[/tex] and [tex]\(x=7\)[/tex]:
- From the table, at [tex]\(x=0\)[/tex], [tex]\(g(0) = 1\)[/tex].
- At [tex]\(x=7\)[/tex], [tex]\(g(7) = 5\)[/tex].

2. Calculate the change in [tex]\(g(x)\)[/tex]:
- Change in [tex]\(g(x)\)[/tex] = [tex]\(g(7) - g(0)\)[/tex].
- Substituting the values, Change in [tex]\(g(x)\)[/tex] = [tex]\(5 - 1 = 4\)[/tex].

3. Determine the change in [tex]\(x\)[/tex]:
- The change in [tex]\(x\)[/tex] from [tex]\(x=0\)[/tex] to [tex]\(x=7\)[/tex] is [tex]\(7 - 0 = 7\)[/tex].

4. Calculate the average rate of change:
- The average rate of change of [tex]\(g\)[/tex] is given by the formula: [tex]\(\frac{\text{Change in } g(x)}{\text{Change in } x}\)[/tex].
- Substituting the changes:
[tex]\[ \frac{\text{Change in } g(x)}{\text{Change in } x} = \frac{4}{7} \][/tex]

So, the average rate of change of [tex]\(g(x)\)[/tex] from [tex]\(x=0\)[/tex] to [tex]\(x=7\)[/tex] is approximately [tex]\(0.5714285714285714\)[/tex].

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