Problem 25 - Algebra 2 and Trigonometry

Question

A candy store sells candy by the piece for 10 cents each. The amount that a customer pays for candy, \(y,\) is a function of the number of pieces purchased, \(x .\) a. Describe, in set-builder notation, the cost in cents of each purchase as a function of the number of pieces of candy purchased. b. Yesterday, no customer purchased more than 8 pieces of candy. List the ordered pairs that describe possible purchases yesterday. c. What is the domain for yesterday's purchases? d. What is the range for yesterday's purchases?

Step-by-Step Solution

Answer

a. \( f(x) = \{ y \mid y = 10x, \ x \in \mathbb{N} \} \). b. \( (1,10), (2,20), (3,30), (4,40), (5,50), (6,60), (7,70), (8,80) \). c. \( \{1, 2, 3, 4, 5, 6, 7, 8\} \). d. \( \{10, 20, 30, 40, 50, 60, 70, 80\} \).

Step 1: Define the Function

The cost per piece of candy is 10 cents, so the cost function is defined as: \( y = 10x \). In set-builder notation, this function can be described as \( f(x) = \{ y \mid y = 10x, \ x \in \mathbb{N} \} \), where \( x \) is a natural number.

Step 2: Identify Possible Purchases

Since no customer bought more than 8 pieces, the possible number of pieces \( x \) a customer could purchase is from 1 to 8. This means the combinations of \( x \) and \( y \) are ordered pairs where \( y = 10x \).

Step 3: List Ordered Pairs

For each possible value of \( x \) from 1 to 8, calculate \( y \) using the function:\ 1. \( x = 1, y = 10 \) 2. \( x = 2, y = 20 \) 3. \( x = 3, y = 30 \) 4. \( x = 4, y = 40 \) 5. \( x = 5, y = 50 \) 6. \( x = 6, y = 60 \) 7. \( x = 7, y = 70 \) 8. \( x = 8, y = 80 \) Thus, the ordered pairs are: (1,10), (2,20), (3,30), (4,40), (5,50), (6,60), (7,70), (8,80).

Step 4: Determine the Domain

The domain is the set of all possible \( x \) values. Since customers bought between 1 and 8 pieces, the domain is \( \{ 1, 2, 3, 4, 5, 6, 7, 8 \} \).

Step 5: Determine the Range

The range is the set of corresponding \( y \) values, which are the costs in cents. From the ordered pairs, these values are \( \{ 10, 20, 30, 40, 50, 60, 70, 80 \} \).

Key Concepts

Understanding Set-Builder Notation

Set-builder notation is a concise way to describe a set by specifying a property that its members must satisfy. For example, if we're interested in the cost of candy in terms of how many pieces are bought, set-builder notation helps us define this relationship clearly.
In the candy store example, each piece costs 10 cents. If we let the number of pieces purchased be represented by \( x \), we can express the cost \( y \) with the function \( f(x) = 10x \).
This can be stated in set-builder notation as:

• \( \{ y \mid y = 10x, \ x \in \mathbb{N} \} \)

Here, \( \mid \) means "such that," which indicates a condition on elements in the set. This notation tells us that \( y \) is the cost when \( x \) is any natural number.

Domain and Range Explained

In function terminology, the domain and range are crucial concepts:

• The domain is the set of all possible input values (\( x \)) for the function.
• The range is the set of all possible output values (\( y \)) that result from applying the function to the domain.

For the candy store:

• The domain is from 1 to 8, representing the number of pieces anyone could have bought yesterday, thus: \( \{ 1, 2, 3, 4, 5, 6, 7, 8 \} \).
• The range, based on these inputs, is the set of possible cost values: \( \{ 10, 20, 30, 40, 50, 60, 70, 80 \} \).

The function maps each possible number of pieces to a specific cost.

What Are Ordered Pairs?

Ordered pairs are used to express a relationship between two quantities, typically seen in the format \((x, y)\).
In our candy example, each purchased quantity has a corresponding cost:

• When \( x = 1 \), \( y = 10 \), so the ordered pair is \((1,10)\).
• When \( x = 2 \), \( y = 20 \), yielding \((2,20)\).
• And so forth, up to \((8,80)\).

These ordered pairs illustrate how pairs of numbers relate in a function, providing a clear representation of the connection between different variables.

Understanding Natural Numbers

Natural numbers are the basic counting numbers starting from 1 and moving upwards: \( \{ 1, 2, 3, \, ... \} \).
These numbers are integral for functions, especially when defining inputs like in our candy store scenario.
The importance of natural numbers in this exercise stems from their role as potential counts of candy pieces purchased. Since no fractional or negative pieces could be bought, natural numbers perfectly encapsulate the possible quantities customers purchased.