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].
