Problem 26 - Algebra 2 and Trigonometry

Question

Let \(c(x)\) represent the cost of an item, \(x,\) plus sales tax and \(d(x)\) represent the cost of an item less a discount of \(\$ 10 .\) a. Write \(c(x)\) using an 8\(\%\) sales tax. b. Write \(\mathrm{d}(x)\) c. Find \(c \circ \mathrm{d}(x)\) and \(\mathrm{d} \circ \mathrm{c}(x) .\) Does \(\mathrm{c} \circ \mathrm{d}(x)=\mathrm{d} \circ \mathrm{c}(x) ?\) If not, explain the difference between the two functions. When does it makes sense to use each function? d. Which function can be used to find the amount that must be paid for an item with a \(\$ 10\) discount and 8\(\%\) tax?

Step-by-Step Solution

Answer

Use \(c \circ d(x) = 1.08x - 10.8\) to find the final price after a \$10 discount and 8% tax.

Step 1: Express Function with Sales Tax

The sales tax is 8\(\%\), which means if the original cost of the item is \(x\), the sales tax is \(0.08 \times x\). So, the cost after adding sales tax is \(x + 0.08x = 1.08x\). Thus, we write \(c(x) = 1.08x\).

Step 2: Express Function with Discount

The original cost of the item \(x\) gets a discount of \$10. Therefore, the function considering the discount is given by \(d(x) = x - 10\).

Step 3: Compute Composition \(c \circ d(x)\)

To find \(c \circ d(x)\), substitute \(d(x)\) into \(c(x)\):\[c(d(x)) = c(x - 10) = 1.08(x - 10) = 1.08x - 10.8\].

Step 4: Compute Composition \(d \circ c(x)\)

To find \(d \circ c(x)\), substitute \(c(x)\) into \(d(x)\):\[d(c(x)) = d(1.08x) = 1.08x - 10\].

Step 5: Compare Compositions \(c \circ d(x)\) and \(d \circ c(x)\)

Compare the function compositions:\(c \circ d(x) = 1.08x - 10.8\) and \(d \circ c(x) = 1.08x - 10\). These are not equal, because the order of applying the sales tax and discount affects the final amount.

Step 6: Determine Appropriate Function for Discounted, Taxed Amount

When you want to apply a discount first and then add tax, use \(c \circ d(x)\). This makes sense as the tax is calculated on the already reduced price. Hence, \(c(d(x))\) represents the actual amount paid after the \$10 discount has been applied, and an 8% tax is added.

Key Concepts

Sales tax calculation

Sales tax is an additional cost added to the price of goods and services, usually expressed as a percentage of the original price. The sales tax amount is calculated by multiplying the item's cost by the sales tax rate. For example, an 8% sales tax on an item costing \(x\) dollars can be calculated as \(0.08 \times x\). This means that the amount added for tax is 8% of the price of the item. In algebra, this is represented in function notation as \(c(x)\), where \(c(x) = x + 0.08x\). This can be simplified to \(c(x) = 1.08x\), indicating the total cost of the item after the sales tax has been applied. Understanding how sales tax calculation works is crucial when managing finances, as it helps in determining the final amount spent on purchases. When planning a budget or recording expenses, always remember to account for the sales tax to avoid surprises.

Algebraic functions

Algebraic functions are mathematical expressions that relate one quantity to another, often involving operations like addition, subtraction, multiplication, and division. In the context of sales tax and discounts, algebraic functions are used to model the relationships and transformations between different costs.In this exercise, two primary algebraic functions are expressed:

• Function \(c(x)\): Represents the total cost of an item after applying an 8% sales tax, given by \(c(x) = 1.08x\).
• Function \(d(x)\): Represents the cost of an item after deducting a fixed discount of $10, shown as \(d(x) = x - 10\).

These functions allow us to perform further operations like function composition, which combines these basic functions to show how items are affected by both discount and tax. Understanding algebraic functions is key to solving problems that involve multiple steps and conditions. By recognizing and applying these functions, you can work through more complex scenarios with ease.

Discount and price reduction

Applying discounts is a common pricing strategy to reduce the cost of items, often encouraging buyers to make a purchase. When a discount is applied, a specific amount is subtracted from the original price, reducing the price paid by the customer. In mathematics, a discount function is typically represented by subtracting a fixed amount from the variable cost.With the discount function \(d(x) = x - 10\), a flat \$10 discount is applied to any item costing \(x\) dollars. It's important to note that discounts can be calculated either before or after adding sales tax, leading to different final costs.If we first subtract the discount and then apply the sales tax (using **\(c(d(x))\)**), you calculate the tax on the reduced price, totaling to \(1.08(x - 10) = 1.08x - 10.8\). Conversely, if you apply the tax before the discount using **\(d(c(x))\)**, you apply the tax on the original price, resulting in \(1.08x - 10\). Understanding these differences helps in making informed financial decisions and assessing the real price paid for an item after discounts and taxes.