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Which function increases at the fastest rate between [tex]x=0[/tex] and [tex]x=8[/tex]?

Linear Function

[tex]\[
\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{$f(x)=2x+2$} \\
\hline $x$ & $f(x)$ \\
\hline 0 & 2 \\
\hline 2 & 6 \\
\hline 4 & 10 \\
\hline 6 & 14 \\
\hline 8 & 18 \\
\hline
\end{tabular}
\][/tex]

Exponential Function

[tex]\[
\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{$f(x)=2^x+2$} \\
\hline $x$ & $f(x)$ \\
\hline 0 & 3 \\
\hline 2 & 6 \\
\hline
\end{tabular}
\][/tex]



Answer :

GinnyAnswer
To determine which function increases at the fastest rate over the interval from [tex]\(x = 0\)[/tex] to [tex]\(x = 8\)[/tex], we need to analyze both provided functions and their outputs.

Linear Function: [tex]\( f(x) = 2x + 2 \)[/tex]

| [tex]\(x\)[/tex] | [tex]\( f(x) \)[/tex] |
|------|----------|
| 0 | 2 |
| 2 | 6 |
| 4 | 10 |
| 6 | 14 |
| 8 | 18 |

The rate of increase for a linear function is constant and equal to its slope. The slope [tex]\( m \)[/tex] for the function [tex]\( f(x) = 2x + 2 \)[/tex] is [tex]\( 2 \)[/tex]. Therefore, the rate of increase for the linear function is [tex]\( 2 \)[/tex].

Exponential Function: [tex]\( f(x) = 2^x + 2 \)[/tex]

| [tex]\(x\)[/tex] | [tex]\( f(x) \)[/tex] |
|------|----------|
| 0 | 3 |
| 2 | 6 |
| 4 | 18 |
| 6 | 66 |
| 8 | 258 |

To find the rate of increase for the exponential function over the interval [tex]\( x = 0 \)[/tex] to [tex]\( x = 8 \)[/tex], we calculate the average rate of change by comparing the values of the function at the endpoints:

- Initial value at [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = 3 \)[/tex]
- Final value at [tex]\( x = 8 \)[/tex]: [tex]\( f(8) = 258 \)[/tex]

The total change in [tex]\( f(x) \)[/tex] over the interval is:
[tex]\[ 258 - 3 = 255 \][/tex]

The length of the interval is:
[tex]\[ 8 - 0 = 8 \][/tex]

The average rate of change for the exponential function is:
[tex]\[ \frac{255}{8} = 31.875 \][/tex]

Conclusion:

- The linear function [tex]\( f(x) = 2x + 2 \)[/tex] has a constant rate of increase of [tex]\( 2 \)[/tex].
- The exponential function [tex]\( f(x) = 2^x + 2 \)[/tex] has an average rate of increase of [tex]\( 31.875 \)[/tex].

Thus, the exponential function increases at the fastest rate between [tex]\( x = 0 \)[/tex] and [tex]\( x = 8 \)[/tex].

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