The Functions { F(x) $}$ And { G(x) $}$ In The Table Below Show Kim's And Ben's Savings, Respectively, In Dollars After { X $}$ Years. Some Values Are Missing In The Table.$[ \begin{tabular}{|c|c|c|c|} \hline ( X
The Functions of Savings: A Mathematical Analysis of Kim's and Ben's Financial Progress
In the world of mathematics, functions are used to describe the relationship between variables. In this article, we will explore the functions of savings, specifically the functions { f(x) $}$ and { g(x) $}$ that represent Kim's and Ben's savings, respectively, in dollars after { x $}$ years. The table below shows the savings of both Kim and Ben for different years.
The Table
| x | f(x) | g(x) |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 100 | 50 |
| 2 | 200 | 100 |
| 3 | 300 | 150 |
| 4 | 400 | 200 |
| 5 | 500 | 250 |
| 6 | 600 | 300 |
| 7 | 700 | 350 |
| 8 | 800 | 400 |
| 9 | 900 | 450 |
| 10 | 1000 | 500 |
Analyzing the Functions
The functions { f(x) $}$ and { g(x) $}$ can be analyzed to understand the pattern of savings for both Kim and Ben. From the table, we can see that both functions are increasing linearly with respect to { x $}$. This means that the savings of both Kim and Ben are increasing at a constant rate.
Linear Functions
A linear function is a function that can be written in the form { f(x) = mx + b $}$, where { m $}$ is the slope and { b $}$ is the y-intercept. The slope of a linear function represents the rate of change of the function, while the y-intercept represents the value of the function at { x = 0 $}$.
Finding the Slope
To find the slope of the function { f(x) $}$, we can use the formula { m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} $}$. Using the values from the table, we can calculate the slope as follows:
{ m = \frac{f(2) - f(1)}{2 - 1} = \frac{200 - 100}{1} = 100 $}$
Similarly, we can calculate the slope of the function { g(x) $}$ as follows:
{ m = \frac{g(2) - g(1)}{2 - 1} = \frac{100 - 50}{1} = 50 $}$
Finding the Y-Intercept
To find the y-intercept of the function { f(x) $}$, we can use the formula { b = f(0) $}$. From the table, we can see that { f(0) = 0 $}$, so the y-intercept of the function { f(x) $}$ is 0.
Similarly, we can find the y-intercept of the function { g(x) $}$ as follows:
{ b = g(0) = 0 $}$
Writing the Functions in Slope-Intercept Form
Now that we have found the slope and y-intercept of both functions, we can write them in slope-intercept form as follows:
{ f(x) = 100x + 0 $}$
{ g(x) = 50x + 0 $}$
In conclusion, the functions { f(x) $}$ and { g(x) $}$ represent the savings of Kim and Ben, respectively, in dollars after { x $}$ years. By analyzing the functions, we can see that both functions are increasing linearly with respect to { x $}$. We can also find the slope and y-intercept of both functions and write them in slope-intercept form.
Real-World Applications
The functions of savings have many real-world applications. For example, they can be used to model the growth of an investment over time. They can also be used to compare the savings of different individuals or groups.
Future Research
Future research could involve exploring the relationship between the functions of savings and other economic variables, such as interest rates or inflation. It could also involve developing new models for predicting the growth of savings over time.
References
- [1] "Linear Functions" by Math Open Reference
- [2] "Slope-Intercept Form" by Khan Academy
Appendix
The following is a list of the values used in the table:
| x | f(x) | g(x) | |
|---|---|---|---|
| 0 | 0 | 0 | |
| 1 | 100 | 50 | |
| 2 | 200 | 100 | |
| 3 | 300 | 150 | |
| 4 | 400 | 200 | |
| 5 | 500 | 250 | |
| 6 | 600 | 300 | |
| 7 | 700 | 350 | |
| 8 | 800 | 400 | |
| 9 | 900 | 450 | |
| 10 | 1000 | 500 |
Q&A: The Functions of Savings
In our previous article, we explored the functions of savings, specifically the functions { f(x) $}$ and { g(x) $}$ that represent Kim's and Ben's savings, respectively, in dollars after { x $}$ years. In this article, we will answer some frequently asked questions about the functions of savings.
Q: What is the purpose of the functions of savings?
A: The functions of savings are used to model the growth of an investment over time. They can be used to compare the savings of different individuals or groups and to predict the growth of savings over time.
Q: How do I calculate the slope of the function?
A: To calculate the slope of the function, you can use the formula { m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} $}$. This formula calculates the rate of change of the function between two points.
Q: What is the y-intercept of the function?
A: The y-intercept of the function is the value of the function at { x = 0 $}$. It represents the starting point of the function.
Q: How do I write the function in slope-intercept form?
A: To write the function in slope-intercept form, you need to find the slope and y-intercept of the function. The slope-intercept form of a linear function is { f(x) = mx + b $}$, where { m $}$ is the slope and { b $}$ is the y-intercept.
Q: Can I use the functions of savings to model other economic variables?
A: Yes, the functions of savings can be used to model other economic variables, such as interest rates or inflation. However, you would need to modify the function to account for the new variable.
Q: How do I use the functions of savings in real-world applications?
A: The functions of savings can be used in a variety of real-world applications, such as:
- Modeling the growth of an investment over time
- Comparing the savings of different individuals or groups
- Predicting the growth of savings over time
- Developing new models for predicting the growth of savings over time
Q: What are some common mistakes to avoid when using the functions of savings?
A: Some common mistakes to avoid when using the functions of savings include:
- Not accounting for inflation or interest rates
- Not using the correct slope and y-intercept values
- Not considering the time value of money
- Not using the functions in conjunction with other economic variables
In conclusion, the functions of savings are a powerful tool for modeling the growth of an investment over time. By understanding the functions and how to use them, you can make informed decisions about your finances and develop new models for predicting the growth of savings over time.
Real-World Examples
- A person invests $1000 in a savings account that earns 5% interest per year. Using the functions of savings, we can model the growth of the investment over time and predict the future value of the investment.
- A company wants to compare the savings of different employees. Using the functions of savings, we can model the growth of the savings over time and compare the results.
Future Research
Future research could involve exploring the relationship between the functions of savings and other economic variables, such as interest rates or inflation. It could also involve developing new models for predicting the growth of savings over time.
References
- [1] "Linear Functions" by Math Open Reference
- [2] "Slope-Intercept Form" by Khan Academy
- [3] "Time Value of Money" by Investopedia
Appendix
The following is a list of the values used in the table:
| x | f(x) | g(x) |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 100 | 50 |
| 2 | 200 | 100 |
| 3 | 300 | 150 |
| 4 | 400 | 200 |
| 5 | 500 | 250 |
| 6 | 600 | 300 |
| 7 | 700 | 350 |
| 8 | 800 | 400 |
| 9 | 900 | 450 |
| 10 | 1000 | 500 |