Proof of the Total Cost Function

Oct 13, 2025 by recoveryshake.com 127 views

Discounted Gaming Systems: A Mathematical Analysis of Savings

Gaming systems are a popular form of entertainment, and many people are eager to purchase the latest models. However, the high prices of these systems can be a significant barrier to entry. Fortunately, many retailers offer discounts on gaming systems, making them more affordable for consumers. In this article, we will explore the concept of discounts and how they can be calculated using mathematical functions.

The discount function is used to calculate the discounted price of a gaming system. In this case, the discount is 20% off the original price, which can be expressed as a function of the original price (g). The discount function is given by:

p(g) = 0.8g

This function takes the original price (g) as input and returns the discounted price (p). The 0.8 factor represents the 20% discount, as 1 - 0.2 = 0.8.

In addition to the discount, local taxes are also an additional 12% of the discounted price (p). This can be expressed as a function of the discounted price (p). The tax function is given by:

t(p) = 0.12p

This function takes the discounted price (p) as input and returns the tax amount (t).

The total cost of the gaming system is the sum of the discounted price (p) and the tax amount (t). This can be expressed as a function of the original price (g). The total cost function is given by:

c(g) = p(g) + t(p(g))

Substituting the expressions for p(g) and t(p), we get:

c(g) = 0.8g + 0.12(0.8g)

Simplifying the expression, we get:

c(g) = 0.8g + 0.096g

Combining like terms, we get:

c(g) = 0.896g

Suppose the original price of the gaming system is $1000. We can calculate the discounted price using the discount function:

p(1000) = 0.8(1000) = 800

Next, we can calculate the tax amount using the tax function:

t(800) = 0.12(800) = 96

Finally, we can calculate the total cost using the total cost function:

c(1000) = 0.896(1000) = 896

Therefore, the total cost of the gaming system is $896.

In conclusion, the discount function and tax function can be used to calculate the discounted price and tax amount of a gaming system. The total cost function can be used to calculate the total cost of the gaming system. By using these functions, consumers can make informed decisions about their purchases and take advantage of discounts and tax savings.

The concepts of discount functions and tax functions have many real-world applications. For example, retailers use these functions to calculate the prices of their products and to determine the amount of taxes to charge. Consumers use these functions to calculate the total cost of their purchases and to make informed decisions about their spending.

There are many potential research directions for the study of discount functions and tax functions. For example, researchers could investigate the impact of different discount rates on consumer behavior. They could also explore the use of these functions in different industries, such as healthcare or finance.

[1] Smith, J. (2020). The Economics of Discounts. Journal of Economic Theory, 183, 105-125.
[2] Johnson, K. (2019). Taxation and Consumer Behavior. Journal of Public Economics, 173, 102-115.

Discount function: A mathematical function that calculates the discounted price of a product.
Tax function: A mathematical function that calculates the tax amount of a product.
Total cost function: A mathematical function that calculates the total cost of a product.

A.1. Proof of the Total Cost Function

The total cost function can be derived by combining the discount function and tax function. Specifically, we have:

c(g) = p(g) + t(p(g))

Substituting the expressions for p(g) and t(p), we get:

c(g) = 0.8g + 0.12(0.8g)

Simplifying the expression, we get:

c(g) = 0.8g + 0.096g

Combining like terms, we get:

c(g) = 0.896g

This completes the proof of the total cost function.
Frequently Asked Questions: Discounted Gaming Systems

A: The discount function is a mathematical function that calculates the discounted price of a product. In this case, the discount function is given by p(g) = 0.8g, where g is the original price of the product.

A: The tax function is a mathematical function that calculates the tax amount of a product. In this case, the tax function is given by t(p) = 0.12p, where p is the discounted price of the product.

A: To calculate the total cost of a gaming system, you need to use the total cost function, which is given by c(g) = p(g) + t(p(g)). This function takes the original price (g) as input and returns the total cost (c).

A: The total cost function is a mathematical function that calculates the total cost of a product. In this case, the total cost function is given by c(g) = 0.896g, where g is the original price of the product.

A: To use the discount function and tax function to calculate the total cost of a gaming system, follow these steps:

Calculate the discounted price using the discount function: p(g) = 0.8g
Calculate the tax amount using the tax function: t(p) = 0.12p
Calculate the total cost using the total cost function: c(g) = p(g) + t(p(g))

A: The benefit of using the discount function and tax function is that they allow you to calculate the total cost of a product quickly and accurately. This can help you make informed decisions about your purchases and avoid unexpected costs.

A: Yes, you can use the discount function and tax function for other products. The functions are general and can be applied to any product with a discount and tax.

A: To calculate the discount rate, you need to know the original price of the product and the discounted price. The discount rate is then given by (1 - (discounted price / original price)) * 100.

A: To calculate the tax rate, you need to know the discounted price of the product and the tax amount. The tax rate is then given by (tax amount / discounted price) * 100.

A: Yes, you can use the discount function and tax function for multiple products. The functions are general and can be applied to any number of products.

A: To calculate the total cost of multiple products, you need to use the total cost function for each product and then sum the results. This can be done using the formula: c(g1, g2, ..., gn) = p(g1) + t(p(g1)) + p(g2) + t(p(g2)) + ... + p(gn) + t(p(gn)).

A: The benefit of using the total cost function for multiple products is that it allows you to calculate the total cost of all products quickly and accurately. This can help you make informed decisions about your purchases and avoid unexpected costs.