Let's start by understanding the problem. We have a set of data points given in a tabular form, and we need to find two different functions that best fit this data:
1. An exponential function [tex]\( f(x) \)[/tex] of the form [tex]\( f(x) = a \cdot e^{b \cdot x} \)[/tex]
2. A linear function [tex]\( g(x) \)[/tex] of the form [tex]\( g(x) = ax + b \)[/tex]
### Step 1: Exponential Regression
For the exponential regression, we are trying to fit the data to the equation [tex]\( f(x) = a \cdot e^{b \cdot x} \)[/tex]. After performing the regression:
- We find parameters [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\( a \approx 514.92 \)[/tex]
[tex]\( b \approx 0.19 \)[/tex]
Therefore, the exponential function that best fits the data is:
[tex]\[ f(x) = 514.92 \cdot e^{0.19 \cdot x} \][/tex]
### Step 2: Linear Regression
For the linear regression, we are trying to fit the data to the equation [tex]\( g(x) = ax + b \)[/tex]. After performing the regression:
- We find parameters [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\( a \approx 197.14 \)[/tex]
[tex]\( b \approx 367.33 \)[/tex]
Therefore, the linear function that best fits the data is:
[tex]\[ g(x) = 197.14x + 367.33 \][/tex]
### Conclusion
The functions that best fit the data, rounded to the hundredths, are:
[tex]\[
f(x) = 514.92 \cdot e^{0.19x}
\][/tex]
[tex]\[
g(x) = 197.14x + 367.33
\][/tex]
