Problem 94 - College Algebra with Modeling and Visualization

Question

Write a symbolic representation (formula) for a function \(g\) that calculates the given quantify. Then evaluate \(g(10)\) and interpret the result. The number of quarters in \(x\) dollars

Step-by-Step Solution

Answer

The function is \(g(x) = 4x\); for 10 dollars, there are 40 quarters.

Step 1: Understand the Problem

To find the number of quarters in a given amount of dollars, we must consider that each dollar is equivalent to 4 quarters. Therefore, the function should express the total number of quarters for any given amount of dollars, denoted by \(x\).

Step 2: Define the Function

To formulate the function, note that for every 1 dollar, there are 4 quarters. Thus, the number of quarters in \(x\) dollars is given by the formula \(g(x) = 4x\). This formula calculates the total number of quarters in \(x\) dollars.

Step 3: Evaluate the Function at x=10

Substitute \(x = 10\) into the function \(g(x)\). \[ g(10) = 4 imes 10 = 40 \]This calculation shows that there are 40 quarters in 10 dollars.

Step 4: Interpret the Result

The result of \(g(10) = 40\) indicates that when you have 10 dollars, you will have a total of 40 quarters. This confirms the relationship between dollars and quarters (4 quarters per dollar).

Key Concepts

Evaluating Functions

When you're asked to evaluate a function, you're essentially tasked with finding out what the function equals when you plug a specific number into it.
This concept might seem tricky at first, but think of it like following a simple recipe, where your ingredients are numbers and the function is your set of instructions.
To evaluate the function provided in our example, which is denoted as \( g(x) = 4x \), you simply substitute \(x\) with any given number—in this case, 10.
Substituting, we perform the calculation:

• Insert 10 in place of \(x\) giving us \( g(10) = 4 \times 10 \).
• Calculate the product: \( 40 \).

This means when \(x\) is 10, \(g(10)\) gives you 40. This is the value of the function at \(x = 10\) expressing how many quarters fit into 10 dollars.

Symbolic Representation

Symbolic representation in mathematics allows you to express real-world problems using symbols, like letters or numbers.
This lets you convey complex ideas in a simple, easy-to-understand format. For example, the symbolic representation for calculating the number of quarters in \(x\) dollars is written as \( g(x) = 4x \).
The letter \(g\) denotes the function name, helping to identify the operation you're describing. The \(x\) symbolizes the variable or the number of dollars.
And the "equals \(4x\)" is your rule or formula that shows how to calculate it:

• "4" tells you for each dollar, there are 4 quarters.
• "x" allows you to replace it with any dollar amount.

This concise expression simplifies your task, showing quickly and efficiently how the quarters relate to dollars.

Quarters Calculation

Understanding quarters calculation requires recognizing the relationship between dollars and quarters. Each dollar is equivalent to 4 quarters because a quarter is worth \(0.25 and four times \)0.25 equals one dollar.
This knowledge is essential for forming and understanding the function \( g(x) = 4x \).
Here, you use this information to calculate how many quarters make up a certain number of dollars:

• Start with the total dollars as your input.
• Apply the rule from the function, multiplying by 4.
• Get the number of quarters as output.

So, when you have 10 dollars and plug it into the function as \( g(10) = 40 \), it tells you that those 10 dollars consist of 40 quarters, showing how money gets broken into smaller units.