Vertical Line Test And Graphs Of Functions

The Vertical Line Test And Graphs Of Functions Graph A Quadratic Equa

In this exercise, you are asked to understand the concept of functions through real-world examples and to evaluate a specific function for a given input. The focus is on grasping how functions operate, how to identify them visually in graphs, and to practice evaluating a function with a concrete input. You are to select or be provided with an example of a function, then compute the output for a specified input value. Additionally, there is an emphasis on understanding the role of the vertical line test in determining whether a graph represents a function.

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Functions are fundamental in mathematics and their application extends into real-world scenarios where input-output relationships are of interest. These relationships can be easily understood through graphical representations and algebraic evaluations. One of the key characteristics of a function is that it assigns exactly one output to each input. The vertical line test serves as a visual tool to verify whether a graph depicts a function; if any vertical line intersects the graph more than once, the graph does not represent a function.

For example, a temperature conversion formula from Fahrenheit to Celsius demonstrates a functional relationship. The formula is C = (5/9) (F - 32). When the Fahrenheit temperature is 77°F, calculating the Celsius equivalent involves substituting this value into the formula: C = (5/9) (77 - 32). Simplifying, C = (5/9) * 45, which results in C = 25 degrees Celsius. This calculation exemplifies evaluating a function at a specific input.

Understanding and identifying functions extends beyond formulas to various real-world relationships. For instance, arm length as a function of height is logical as taller individuals tend to have longer arms. The circumference of a circle as a function of diameter follows the formula C = π * d, directly relating the diameter to the circumference. The height of a tree as a function of age reflects growth over time; generally, as the age increases, the height increases, up to a point. Similarly, the duration a person's shadow lasts on the ground depends on their height and the sun's position, demonstrating a functional relationship involving multiple variables.

Other examples include weekly salary as a function of hourly wage and hours worked, which combines these variables into a single output. Compound interest depends on initial investment, interest rate, and time, illustrating a more complex multivariable function. The law of supply and demand also exhibits inverse relationships—typically, as the price of a product increases, demand decreases—though this is more of a qualitative relationship often modeled with functions suitable for economic analysis.

To demonstrate your understanding, choose a specific function, either from the examples provided or one of your own, and evaluate it at a particular input value. For example, using the function for the circumference of a circle, with a diameter of 10 units, the circumference would be C = π 10, which is approximately 31.42 units. If you choose the salary function, with an hourly wage of $15 and 40 hours worked, the weekly salary would be 15 40 = $600. These exercises reinforce the concept of functions as rules that assign a unique output to each input.

It is essential to recognize how graphs of functions can be visually checked using the vertical line test. When analyzing a graph, if any vertical line intersects the graph at more than one point, the graph does not represent a function, because a single input would correspond to multiple outputs—a violation of the definition of a function.

In conclusion, functions are a core concept that can be explored through formulas, graphs, and real-world relationships. Evaluating functions at specific inputs helps build operational understanding, while the vertical line test provides a quick visual check for the functional nature of graphs. Mastery of these concepts is essential for analyzing mathematical models in various disciplines, including science, economics, and engineering.

References

• Anton, H., Bivens, R., & Davis, S. (2016). Calculus: Early Transcendentals (11th ed.). John Wiley & Sons.
• Larson, R., & Edwards, B. H. (2017). Calculus (11th ed.). Cengage Learning.
• Ott, T. (2013). Basic College Mathematics (11th ed.). Cengage Learning.
• Ross, K. A., & Wilson, F. (2020). Mathematical Methods for Economics. Springer.
• Swokowski, E. W., & Cole, J. A. (2016). Algebra and Trigonometry with Analytic Geometry. Cengage Learning.
• Thompson, P. (2018). Precalculus: Mathematics for Calculus (6th ed.). Pearson.
• Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning.
• Lay, D. C. (2016). Linear Algebra and its Applications (5th ed.). Pearson.
• Brigham, E. O. (2000). The Fundamentals of Financial Management. Cengage Learning.
• Morris, M. (2019). Introduction to Mathematical Economics. Routledge.