Problem 97 - College Algebra with Modeling and Visualization

Question

Sale Price \(\mathbf{A}\) store is discounting all regularly priced merchandise by \(25 \% .\) Find a function \(f\) that computes the sale price of an item having a regular price of \(x .\) If an item normally costs \(\$ 56.24\), what is its sale price?

Step-by-Step Solution

Answer

The sale price of the item is $42.18.

Step 1: Understanding the Problem

We need to find a function that computes the sale price of an item when there is a 25% discount on the regular price. Additionally, we will calculate the sale price for an item that costs $56.24.

Step 2: Define the Function

To find the sale price, we need to apply a 25% discount to the regular price, which means the item is sold at 75% of its regular price. Therefore, the function can be defined as: \[ f(x) = 0.75x \] where \( x \) is the regular price of the item.

Step 3: Apply the Function to a Specific Case

Now, we apply our function to find the sale price of an item that normally costs $56.24. By substituting \( x = 56.24 \) into the function:\[ f(56.24) = 0.75 \times 56.24 \]

Step 4: Calculate the Sale Price

Perform the multiplication to find the sale price:\[ f(56.24) = 42.18 \] Thus, the sale price of the item is $42.18.

Key Concepts

Function Notation

In algebra, function notation is a way to express relationships between two variables in a clear and concise form. It involves defining a function, which acts like a machine taking an "input" (usually represented by \(x\)) and delivering an "output" (often denoted as \(f(x)\)). For example, the function we defined to compute the sale price of an item with a regular price \(x\) was \( f(x) = 0.75x \). This notation helps us see directly how the sale price depends on the regular price.

• "\( f \)" identifies the function.
• "\( x \)" is our input variable, representing the regular price.
• "0.75x" shows the operation performed on \(x\) (in this case, multiplying by 0.75 to account for the 25% discount).

Using function notation makes it easy to plug in different values for \(x\) and immediately see the outcome, which in the case of sales, is the discounted price. This method is not only helpful for this problem but is also a foundational concept that spans across many areas of mathematics.

Percentages

Understanding percentages is crucial, especially when dealing with sales and discounts. A percentage is essentially a way of expressing a number as a fraction of 100. For example, a 25% discount means that 25 out of every 100 parts of the original price are removed.
To work with percentages in calculations, we convert them into decimal form. This involves dividing the percentage by 100. So, 25% becomes 0.25 when you divide 25 by 100. This decimal form can then be used to perform various arithmetic operations.

• To find out what remains after removing the percentage, subtract the decimal from 1. In our example, 1 - 0.25 = 0.75.
• Multiply this result by the regular price to find the sale price.

In this exercise, understanding how to convert percentages and apply them as a multiplier provided us with the function \( f(x) = 0.75x \), reflecting a 25% discount.

Discount Calculations

Calculating discounts is a common real-world application of percentages and function notation. The core idea is to reduce the regular price of an item by a certain percentage, known as the discount rate. In this example, we dealt with a discount of 25%, meaning every item was sold at 75% of its original price.
To calculate the discount:

• First, convert the discount percentage to its decimal form, which is 0.25 in this case.
• Then subtract this from 1 to find out what percentage of the price remains (0.75).
• Finally, multiply the regular price by this value. For example, for an item costing $56.24, the calculation was \(f(56.24) = 0.75 \times 56.24 = 42.18\).

By understanding these steps, one can efficiently compute any discounts, ensuring proper financial transactions. Discount calculations are not only an essential part of shopping but a vital skill in budgeting and planning.