Problem 49 - Algebra 2

Question

For Exercises \(48-50,\) use the following information. Liluye wants to buy a pair of inline skates that are on sale for 30\(\%\) off the original price of \(\$ 149 .\) The sales tax is 5.75\(\% .\) Which composition of functions represents the price of the inline skates, \(p[s(x)]\) or \(s[p(x)]\) ? Explain your reasoning.

Step-by-Step Solution

Answer

The correct composition of functions is \(s[p(x)]\): first apply the discount, then sales tax.

Step 1: Understanding the Problem

To solve the problem, we need to determine the correct sequence of applying the discount and sales tax functions to find the final purchase price. We have two options for the order: first apply the discount and then add sales tax, or vice versa.

Step 2: Define the Functions

Let's define the functions:- Let \( p(x) \) represent the function for applying the discount. Therefore, \( p(x) = x - 0.30x = 0.70x \).- Let \( s(x) \) represent the function for applying the sales tax. Therefore, \( s(x) = x + 0.0575x = 1.0575x \).

Step 3: Determine Composition Order

To get the final price after the discount and the addition of sales tax, you should first apply the discount, then apply the sales tax on the discounted price. This means first use \( p(x) \), then apply \( s(x) \).

Step 4: Formulate the Composition

Start with the original price \( x = 149 \). Compute \( p(x) = 0.70 \times 149 \) to find the discounted price. Next, apply the tax function \( s(p(x)) = 1.0575 \times p(x) \) to find the final price after tax.

Step 5: Evaluate Both Compositions

If you were to mistakenly apply the tax before the discount, \( s[p(x)] = s(0.70 \times 149) = 1.0575 \times (0.70 \times 149) \). Thus, applying \( p[s(x)] \) is incorrect because it would mean applying the sales tax to the original price and then reducing it by the discount, which misrepresents the sale situation.

Key Concepts

Discount Function

A discount function is a way to calculate the reduced price of an item when a discount is applied. It's essential because it tells you how much you will save and helps you plan your purchases better. For example, when an object is sold at a discount, the new price is calculated by reducing the original price by a certain percentage.
To calculate the discounted price for the inline skates, you use the function \( p(x) = x - 0.30x = 0.70x \). Here, \( x \) represents the original price, \($149\), and the number 0.30 signifies the 30% discount. This tells us that the skates cost 70% of the original price. Applying the function first gives us the discounted price.

Sales Tax Function

After calculating the discounted price, a sales tax function is used to determine the tax that must be added to this price. The sales tax is a percentage of the purchase amount that goes to the government in exchange for the right to conduct transactions.
In our example, the sales tax function is defined as \( s(x) = x + 0.0575x = 1.0575x \). This equation means that when you multiply \( x \) by 1.0575, you add 5.75% of \( x \) as sales tax. The new price, therefore, becomes the discounted price plus the extra tax.

Function Composition

Function composition is combining two functions in a specific order to obtain a desired outcome. It is a powerful mathematical tool that lets you simplify processes by nesting functions.
The order you apply the functions matters a lot. For buying the skates with discount and tax, you first find the price after the discount using \( p(x) \), and then you apply the tax with \( s(p(x)) \). In this way, you calculate the final price correctly. If you did it the other way, you would calculate the full price with tax before reducing for the discount, which doesn’t represent the true cost.

Order of Operations

Order of operations describes the sequence in which mathematical operations are carried out, and it’s crucial to solving problems correctly. In this scenario, the order of operations means deciding whether you apply the discount or the tax first, which can completely change the result.
For Liluye’s skates, the discount should first be applied to find the reduced price before the tax is added. This is because the discount is taken from the original price, reducing it, before calculating the tax on the lower amount. Applying the operations in this order ensures that Liluye pays the right amount. Proper order of operations prevents overpaying or misunderstanding sales offers, making sure calculations reflect real-world transactions accurately.