Describe A Real-Life Situation That Can Be Modeled By A Func
Describe A Real Life Situation That Can Be Modeled By Afunctiona Rela
Describe a real-life situation that can be modeled by a function (a relation in which every input value corresponds to only one output value). The situation must be different from the examples provided in the reading and in the homework. Identify the input value and the output value of the function, and provide justification that your relation does, in fact, represent a function. Create a function in the form y = f(x) that models the relation described. You may choose variables in place of y or x that better represent the relation, as seen in the provided example. Identify the domain and range of your function, using proper mathematical notation. Cite any sources you may have used in formulating your response.
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A practical example of a situation that can be modeled by a function is the relationship between hours worked and earnings in a part-time job. In this scenario, the input variable is the number of hours an individual works, and the output variable is the total earnings received, which depend directly on the pay rate. This relation is suitable for modeling with a function because each number of hours worked corresponds to exactly one amount of earnings, assuming a fixed hourly wage and no additional bonuses or deductions.
The input value in this case is the number of hours worked, denoted as h, which can be any non-negative real number (h ∈ [0, ∞)). The output value is the total earnings, denoted as E, which can be expressed as a function of h. Assuming an hourly wage of $15 per hour, the relationship can be represented as E = 15h. This equation satisfies the criteria of a function since each input h gives exactly one output E.
The justification that this relation is a function hinges on the fact that each value of h corresponds to one and only one value of E. For instance, if someone works 5 hours, their earnings would be E = 15 × 5 = $75. No matter how many hours are worked, the total earnings are uniquely determined by the linear equation. There are no inputs h that produce multiple outputs E, which confirms the relation's validity as a function.
The domain of the function represents all possible input values, that is, the hours worked. Since no one can work negative hours, the domain is h ∈ [0, ∞). The range of the function accounts for all possible earnings, which are proportional to hours worked, starting at $0 (when no hours are worked) and increasing without bound as hours increase. Mathematically, the range is E ∈ [0, ∞), corresponding to the domain through the function E = 15h.
This model accurately captures the linear relationship between hours worked and earnings, a common scenario in many employment contexts. It is straightforward, easy to analyze, and applicable in budgeting, payroll processing, and financial planning. Another reason this relation demonstrates a function is because it adheres to the definition that for every input (hours worked), there is only one output (earnings).
In conclusion, modeling the relationship between hours worked and earnings as a function allows for clear analysis and predictions. It exemplifies the core concept of a function in mathematics—each input yields exactly one output, reinforcing the importance of this structure in real-world applications. Similar models can be extended to other contexts, such as calculating fuel consumption based on distance traveled, further illustrating the versatility of functions in modeling everyday situations.
References
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6. U.S. Census Bureau. (2020). Employment and Wages. https://www.census.gov
7. Bureau of Labor Statistics. (2022). Hourly wages and employment data. https://bls.gov
8. Khan Academy. (2023). Functions and their graphs. https://khanacademy.org/math/algebra
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10. National Geographic Society. (2020). Mathematical modeling in real life. https://natgeo.com