Cashier's Earnings Function A Mathematical Exploration

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In the realm of mathematics, understanding functions is crucial for modeling real-world scenarios. This article delves into a practical example, examining how to represent a cashier's earnings using a mathematical function. We will dissect the problem, explore the given options, and arrive at the correct solution, providing a comprehensive understanding of the underlying concepts.

Problem Statement: Defining the Cashier's Earning Function

The core question we aim to address is: A cashier earns $7 an hour. If x represents the number of hours worked, which function accurately represents the cashier's earnings? The options presented are:

  • A. y = 7x
  • B. y = x^7
  • C. y = 7/x
  • D. y = 7

To solve this, we need to identify the relationship between the number of hours worked (x) and the total earnings (y). This relationship will be expressed as a function.

Dissecting the Problem: Hours Worked and Total Earnings

To accurately represent the cashier's earnings, we must understand the direct relationship between the hourly wage and the number of hours worked. The cashier earns a fixed amount, $7, for each hour of work. This implies a multiplicative relationship: the more hours worked, the higher the earnings, and vice versa. Therefore, the total earnings (y) should increase proportionally with the number of hours worked (x).

Let's consider some examples to illustrate this:

  • If the cashier works 1 hour (x = 1), the earnings would be $7 * 1 = $7.
  • If the cashier works 2 hours (x = 2), the earnings would be $7 * 2 = $14.
  • If the cashier works 4 hours (x = 4), the earnings would be $7 * 4 = $28.

These examples clearly show that the total earnings are obtained by multiplying the hourly wage ($7) by the number of hours worked (x). This leads us to the correct functional representation.

Evaluating the Options: Identifying the Correct Function

Now, let's analyze the given options to determine which one accurately represents the relationship we've established:

  • A. y = 7x: This function states that the total earnings (y) are equal to 7 times the number of hours worked (x). This aligns perfectly with our understanding of the problem. For instance, if x = 3 hours, then y = 7 * 3 = $21, which is the correct earning.
  • B. y = x^7*: This function represents y as x raised to the power of 7. This implies an exponential relationship, which is not relevant in this scenario. The earnings do not increase exponentially with hours worked. If x = 2, then y = 2^7 = $128, which is incorrect for this problem.
  • C. y = 7/x: This function represents y as 7 divided by x. This implies an inverse relationship, meaning the earnings would decrease as the number of hours worked increases, which contradicts the problem statement. If x = 2, then y = 7 / 2 = $3.50, which is an incorrect earning.
  • D. y = 7: This function states that the total earnings (y) are always $7, regardless of the number of hours worked (x). This is a constant function and doesn't accurately model the cashier's earnings, which should vary with the hours worked.

Based on our analysis, only option A, y = 7x, accurately represents the cashier's earnings as a function of the number of hours worked.

The Correct Function: y = 7x

Therefore, the correct function representing the cashier's earnings is y = 7x. This linear function demonstrates that the total earnings (y) are directly proportional to the number of hours worked (x), with the constant of proportionality being the hourly wage ($7).

The Significance of Linear Functions

This problem highlights the practical application of linear functions in everyday scenarios. A linear function, in general, can be represented as y = mx + b, where:

  • y is the dependent variable (in this case, total earnings).
  • x is the independent variable (in this case, hours worked).
  • m is the slope (the rate of change, which is the hourly wage $7).
  • b is the y-intercept (the value of y when x is 0, which is $0 in this case).

In our specific problem, y = 7x is a special case of a linear function where the y-intercept b is 0. This indicates that if the cashier works 0 hours, the earnings will be $0, which is logical.

Real-World Applications of the Earnings Function

Understanding the cashier's earnings function provides valuable insights into financial planning and budgeting. The function can be used to:

  • Calculate potential earnings for a given number of hours.
  • Determine the number of hours needed to work to reach a specific earning goal.
  • Compare earnings across different work schedules.

For instance, if the cashier wants to earn $210 in a week, we can use the function to determine the required hours:

$210 = 7x x = $210 / $7 x = 30 hours

This demonstrates the practical utility of the earnings function in real-life financial scenarios. The use of mathematical functions to model real-world relationships like earnings provides a powerful tool for analysis and decision-making.

Visualizing the Function: The Earnings Graph

To further enhance our understanding, we can visualize the function y = 7x on a graph. The graph will be a straight line passing through the origin (0,0) with a slope of 7. The slope represents the rate at which the earnings increase with each additional hour worked. A steeper slope would indicate a higher hourly wage, while a flatter slope would indicate a lower hourly wage. Understanding the concept of slope is crucial in interpreting linear functions and their graphical representations.

The x-axis of the graph represents the number of hours worked, and the y-axis represents the total earnings. Each point on the line corresponds to a specific combination of hours worked and earnings. For example, the point (3, 21) represents that working 3 hours results in earnings of $21. This visual representation makes it easy to quickly estimate earnings for any given number of hours and vice versa.

Potential Extensions: Expanding the Problem

To further explore this concept, we can introduce additional complexities to the problem. For example:

  1. Overtime Pay: What if the cashier earns 1.5 times the regular hourly wage for hours worked beyond 40 hours per week? How would this be incorporated into the function?
  2. Taxes and Deductions: How would the function change if we considered taxes and other deductions from the cashier's earnings?
  3. Multiple Jobs: What if the cashier has multiple jobs with different hourly wages? How would we model the total earnings from all jobs?

These extensions can be used as exercises to deepen understanding of functions and their applications in real-world financial scenarios. They demonstrate that mathematical models can be adapted and refined to reflect increasing complexities in real-life situations.

Conclusion: The Power of Functional Representation

In conclusion, we have successfully identified the function y = 7x as the accurate representation of a cashier's earnings based on an hourly wage of $7. This problem underscores the power of mathematical functions in modeling real-world relationships and making predictions. By understanding the relationship between variables and expressing them mathematically, we gain valuable insights into various phenomena and can make informed decisions. The process of breaking down the problem, analyzing the options, and verifying the solution is a critical skill in both mathematics and real-life problem-solving. The ability to translate real-world scenarios into mathematical models is an essential skill in various fields, including finance, economics, and engineering.