Question

Tobias is shopping at a local sports store that is having a \(25 \%\) -off sale on apparel. Where he lives, the federal tax adds \(5 \%\) to the selling price. a) Write the function, \(s(p),\) that relates the regular price, \(p,\) to the sale price, \(s,\) both in dollars. b) Write the function, \(t(s),\) that relates the sale price, \(s,\) to the total cost including taxes, \(t,\) both in dollars. c) Write a composite function that expresses the total cost in terms of the regular price. How much did Tobias pay for a jacket with a regular price of \$89.99?

Short answer

Tobias paid approximately \$70.86 for the jacket.

Step-by-step solution

Define the Sale Price Function

First, determine the function that calculates the sale price based on the regular price, considering a 25% discount. The regular price is denoted by \( p \), and the sale price after a 25% discount is given by: \[ s(p) = p - 0.25p \]Simplify this to: \[ s(p) = 0.75p \]

Define the Tax-Included Price Function

Next, determine the function that calculates the total price including the 5% tax based on the sale price. The sale price is denoted by \( s \), and after adding a 5% tax, the total price is given by: \[ t(s) = s + 0.05s \]Simplify this to: \[ t(s) = 1.05s \]

Write the Composite Function

To express the total cost in terms of the regular price, \( p \), combine the functions from Step 1 and Step 2. Substitute the function \( s(p) \) into \( t(s) \): \[ t(s(p)) = t(0.75p) \]Using the tax-included function: \[ t(0.75p) = 1.05(0.75p) \]Simplify this to: \[ t(s(p)) = 0.7875p \]

Calculate the Total Cost for the Jacket

Finally, use the composite function to calculate the total cost for a jacket with a regular price of \$89.99: \[ t(89.99) = 0.7875 \times 89.99 \]Perform the multiplication: \[ t(89.99) \approx 70.86 \]Therefore, Tobias paid approximately \$70.86 for the jacket.

Key concepts

sale price function

When a store offers a discount, the sale price function helps us determine the new price after the discount is applied. In this exercise, Tobias's store provides a 25% discount on apparel. This discount means we reduce the original price by 25%. This can be expressed using the function:

• Let the regular price be denoted by \( p \).
• The sale price function, \( s(p) \), is then \( p - 0.25p \).
• We simplify it to \( s(p) = 0.75p \).

This means that, for any given regular price, multiplying it by 0.75 gives the sale price after applying the 25% discount. The sale price function is crucial when we need an adjusted price before tax calculation.

tax-included price function

After determining the sale price, we need to include any applicable taxes to find the total cost. Tobias's region adds a 5% tax to the sale price. Let's express this:

• Let the sale price be denoted by \( s \).
• The tax-included price function, \( t(s) \), is \( s + 0.05s \).
• Simplifying, we get \( t(s) = 1.05s \).

This indicates that to include the 5% tax, you multiply the sale price by 1.05. This function is essential for understanding the final price you pay after taxes are added.

discount calculation

Calculating discounts is a common scenario in shopping, and it involves basic percentage operations. In our case, Tobias enjoys a significant 25% discount on his purchase. Let's break down how this calculation works:

• Identify the original price, \( p \).
• Find the discount amount using: \( 0.25 \times p \).
• Subtract the discount from the original price: \( p - 0.25p \).
• Alternatively, as shown earlier, this simplifies directly to \( 0.75p \).

Understanding how discounts are calculated can help shoppers determine the reduced price quickly and ensure they are getting the expected deal.

percentages in functions

Percentages play a vital role in various mathematical functions, especially in scenarios involving discounts and taxes. In Tobias's case:

• The 25% discount represents a reduction, calculated as a fraction of the original price (\( 0.25 \times p \)).
• The 5% tax represents an increase, calculated as a fraction of the sale price (\( 0.05 \times s \)).

To work with percentages in functions, you need to convert the percentage to a decimal by dividing by 100. For instance, 25% becomes 0.25, and 5% becomes 0.05. This allows you to perform multiplication directly. Combining these steps in composite functions, like finding the total cost after both discount and tax, involves applying each percentage operation sequentially. By understanding percentages in functions, you can handle real-world financial calculations accurately.