Limitations of the Other Options

Oct 13, 2025 by recoveryshake.com 119 views

When it comes to saving money, individuals often have a specific goal in mind, whether it's to save for a short-term expense or a long-term investment. In this scenario, we're tasked with determining the function that best describes the amount an individual can put into savings each year. This problem requires a mathematical approach, and we'll explore the given options to find the most suitable function.

Analyzing the Options

Let's examine each option carefully to determine which function best describes the amount an individual can put into savings each year.

Option A: $$h(x)=(40 x+10)(25 x+5)$$

This option represents a quadratic function, which is a polynomial function of degree two. The function is the product of two linear functions, each with a positive coefficient. When we multiply these two functions, we get a quadratic function with a positive leading coefficient. This means that the function will increase as x increases, indicating that the amount saved will increase over time.

However, this function may not accurately represent the amount saved each year, as it does not take into account the individual's income or expenses. A more realistic function would likely involve a linear relationship between the amount saved and the individual's income or expenses.

Option B: $$h(x)=15 x+5$$

This option represents a linear function, which is a polynomial function of degree one. The function has a positive slope, indicating that the amount saved will increase as x increases. This function is more realistic than option A, as it suggests that the amount saved is directly proportional to the individual's income or expenses.

However, this function may not accurately represent the amount saved each year, as it does not take into account any potential deductions or expenses that may affect the individual's savings.

Option C: $$h(x)=-15 x-5$$

This option represents a linear function with a negative slope, indicating that the amount saved will decrease as x increases. This function is less realistic than options A and B, as it suggests that the amount saved is inversely proportional to the individual's income or expenses.

Option D: $$h(x)=65$$

This option represents a constant function, which is a polynomial function of degree zero. The function has a constant value of 65, indicating that the amount saved will remain the same regardless of the individual's income or expenses.

Conclusion

Based on our analysis, option B, $$h(x)=15 x+5$$, is the most suitable function to describe the amount an individual can put into savings each year. This function represents a linear relationship between the amount saved and the individual's income or expenses, making it a more realistic representation of the problem.

Why This Function?

This function is more realistic because it suggests that the amount saved is directly proportional to the individual's income or expenses. This is a common scenario in real-life savings, where individuals aim to save a certain percentage of their income each year.

Limitations of the Other Options

While options A and C may seem plausible at first glance, they have significant limitations. Option A represents a quadratic function, which may not accurately represent the amount saved each year. Option C represents a linear function with a negative slope, which is less realistic than options A and B.

Real-World Applications

This problem has real-world applications in personal finance and budgeting. Individuals can use this function to determine how much they can save each year based on their income or expenses. This can help them create a realistic savings plan and achieve their financial goals.

Conclusion

In our previous article, we explored the function that best describes the amount an individual can put into savings each year. We analyzed the given options and determined that $$h(x)=15 x+5$$ is the most suitable function. In this article, we'll address some frequently asked questions related to this topic.

Q: What is the significance of the coefficient 15 in the function $$h(x)=15 x+5$$?

A: The coefficient 15 in the function $$h(x)=15 x+5$$ represents the rate at which the amount saved increases with respect to the individual's income or expenses. In this case, the amount saved increases by $15 for every $1 increase in income or expenses.

Q: How does the function $$h(x)=15 x+5$$ account for deductions or expenses?

A: The function $$h(x)=15 x+5$$ does not account for deductions or expenses. However, this can be modified to include these factors by subtracting the deductions or expenses from the function. For example, if the individual has a deduction of $10 per year, the modified function would be $$h(x)=15 x-10$$.

Q: Can the function $$h(x)=15 x+5$$ be used for individuals with variable income?

A: The function $$h(x)=15 x+5$$ assumes a linear relationship between the amount saved and the individual's income or expenses. While this may be a reasonable assumption for individuals with stable income, it may not accurately represent individuals with variable income. In such cases, a more complex function may be needed to account for the fluctuations in income.

Q: How can the function $$h(x)=15 x+5$$ be used in real-world applications?

A: The function $$h(x)=15 x+5$$ can be used in various real-world applications, such as:

Creating a savings plan for individuals with stable income
Determining the amount saved for a specific financial goal, such as a down payment on a house
Calculating the impact of deductions or expenses on savings
Developing a budget that takes into account the individual's income and expenses

Q: Can the function $$h(x)=15 x+5$$ be used for individuals with multiple sources of income?

A: The function $$h(x)=15 x+5$$ assumes a single source of income. While it can be modified to account for multiple sources of income, it may become more complex and difficult to interpret. In such cases, a more advanced function or a separate function for each source of income may be needed.

Q: How can the function $$h(x)=15 x+5$$ be used in conjunction with other financial tools?

A: The function $$h(x)=15 x+5$$ can be used in conjunction with other financial tools, such as:

Budgeting software to track income and expenses
Investment calculators to determine the impact of savings on investments
Retirement planning tools to determine the impact of savings on retirement goals

Conclusion

In conclusion, the function $$h(x)=15 x+5$$ is a useful tool for determining the amount an individual can put into savings each year. By understanding the significance of the coefficient 15 and how the function accounts for deductions or expenses, individuals can use this function to create a realistic savings plan and achieve their financial goals.