Let's analyze the problem and the data provided step-by-step.
1. Understanding the Given Data:
- We have a table showing the profits of two businesses over various years.
- [tex]\( x \)[/tex] represents the year.
- [tex]\( f(x) \)[/tex] represents the profits of the first business.
- [tex]\( g(x) \)[/tex] represents the profits of the second business.
The table provided is:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
1995 & \$ 69,682.50 & \$ 72,429.27 \\
\hline
2000 & \$ 78,943.50 & \$ 79,967.77 \\
\hline
2005 & \$ 88,204.50 & \$ 88,290.88 \\
\hline
2006 & \$ 90,056.70 & \$ 90,056.70 \\
\hline
2007 & \$ 91,908.90 & \$ 91,857.83 \\
\hline
2010 & \$ 97,465.50 & \$ 97,480.27 \\
\hline
\end{tabular}
\][/tex]
2. Calculate the Total Increase in Profits:
- For [tex]\( f(x) \)[/tex]:
- Initial profit in 1995: \[tex]$69,682.50
- Final profit in 2010: \$[/tex]97,465.50
- Increase: [tex]\( \$97,465.50 - \$69,682.50 = \$27,783.00 \)[/tex]
- For [tex]\( g(x) \)[/tex]:
- Initial profit in 1995: \[tex]$72,429.27
- Final profit in 2010: \$[/tex]97,480.27
- Increase: [tex]\( \$97,480.27 - \$72,429.27 = \$25,051.00 \)[/tex]
3. Comparison and Justifications:
- Increase in Profits:
- [tex]\( f(x) \)[/tex] increased more than [tex]\( g(x) \)[/tex] over the given period (\[tex]$27,783.00 > \$[/tex]25,051.00).
- Function Characteristics:
- The problem statement notes that [tex]\( f(x) \)[/tex] is exponential.
- In general, exponential functions can grow more quickly than linear functions over long periods, especially as the value of [tex]\( x \)[/tex] increases.
- Conclusion:
- Although both functions show growth in profits, [tex]\( f(x) \)[/tex] (exponential) had a higher total increase in profits (\[tex]$27,783.00) compared to \( g(x) \) (\$[/tex]25,051.00).
Thus the results from our calculations show:
- [tex]\( f(x) \)[/tex] is exponential. Despite typically growing more slowly initially, [tex]\( f(x) \)[/tex] ended up increasing more overall in the given time frame (1995-2010) compared to the linear growth of [tex]\( g(x) \)[/tex]. This showcases the nature of exponential growth where the increments become significantly larger over time.
The answers fit perfectly with our initial expectations, confirming the behavior of the two different profit functions.
