Explain What A Function Is In Your Own Words (2 Pts)

Explain what a function is in your own words 2 pts

Explain what a function is in your own words. (2 pts)

A function is a relationship between a set of inputs and a set of outputs where each input is related to exactly one output. In other words, for every element in the domain, there is a unique corresponding element in the range. Functions are like machines that take an input, process it, and produce a single output.

Paper For Above instruction

A function is a fundamental concept in mathematics that describes a specific type of relationship between two sets of elements. It can be understood as a rule or a machine that assigns each input from the domain to exactly one output in the range. For example, imagine a vending machine that dispenses a specific snack when a certain button is pressed. Here, pressing the button is the input, and the snack received is the output. There is a clear one-to-one relationship—each button (input) corresponds to a particular snack (output). In mathematical terms, this relationship must satisfy the condition that no input corresponds to more than one output. This concept underpins much of algebra, calculus, and higher-level mathematics because it provides a structured way to analyze how different quantities relate to each other.

Mapping example

An example of a mapping is the function that assigns each person’s name to their age. For instance, John → 25, Alice → 30, Bob → 22. In this mapping, each name (input) corresponds to a single age (output). It demonstrates a mapping from the set of names (domain) to the set of ages (range).

Algebraic example

An algebraic example of a function is y = 3x + 4. Here, for each value of x, you calculate y using this formula. For instance, if x = 2, then y = 3(2) + 4 = 10. This formula ensures that every x value produces exactly one y value, satisfying the definition of a function.

Domain and range for both examples

For the mapping example (people’s names to ages), the domain is the set of all people’s names, and the range is the set of their ages, such as {25, 30, 22}. The domain could be limited to specific individuals, and the range could be limited to certain age groups.

For the algebraic example y = 3x + 4, if x can be any real number, then the domain is all real numbers (ℝ). The range, depending on the domain, is also all real numbers (ℝ) because the linear function y = 3x + 4 covers all real y-values as x varies over ℝ.

“All Relations are Functions”. Explain whether this statement is true or false. (4 pts)

The statement "All relations are functions" is false. A relation between two sets can associate each element in the domain with one or more elements in the range, or even with none. Unlike functions, which require that each input has exactly one output, relations can be more general, including many-to-many or one-to-none connections. For example, the relation "has friends" could associate a person with multiple friends (one-to-many), which is not a function. Therefore, not all relations are functions, but all functions are relations because they are specific types of relations with additional restrictions.

Paper For Above instruction

Relations and functions are fundamental concepts in set theory and mathematics. A relation between two sets is simply a collection of ordered pairs, where each pair consists of an element from the first set (domain) and an element from the second set (range). Relations can include any kind of pairing, such as one-to-one, one-to-many, or many-to-many, without restrictions. For example, the relation "is friends with" might associate one person with multiple others, which does not qualify as a function because a function requires that each input corresponds to exactly one output.

On the other hand, a function is a special type of relation that enforces a rule: each input must have exactly one output. This means that for every element in the domain, there is a single, well-defined element in the range. If a relation pairs an element with multiple outputs, it is not a function. Consider the relation "is a sibling of"—it could relate a person to multiple siblings, which means this relation is not a function. Hence, the statement "All relations are functions" is false, because the set of relations includes many that do not meet the criteria for functions.

In summary, while all functions are relations, not all relations qualify as functions. Relations are broader and include any pairing between sets, whereas functions impose a strict one-to-one (or one-to-many) rule. This distinction is essential for understanding mathematical structures and their properties.

“All Functions are Relations”. Explain whether this statement is true or false. (4 pts)

The statement "All functions are relations" is true. By definition, a relation between two sets is any set of ordered pairs where the first element belongs to the first set, and the second element belongs to the second set. Since a function is a specific type of relation where each element in the domain is related to exactly one element in the range, every function is inherently a relation. Functions obey the broader concept of relations but with additional rules that restrict how elements are paired.

Paper For Above instruction

In the context of set theory and mathematical definitions, a relation represents a collection of ordered pairs, which associate elements from one set with elements from another set. All functions are relations because they can be viewed as particular relations that follow stricter rules. Specifically, a function must assign exactly one output to each input, which means every function is a set of ordered pairs with this property. The key difference is that relations can be arbitrary pairings without constraints, whereas functions require the uniqueness of outputs per input. Therefore, the statement "All functions are relations" is accurate, as functions are special cases within the larger set of all possible relations. This understanding underscores the hierarchical relationship where the set of all functions is a subset of the set of all relations.

In conclusion, recognizing that functions are relations with specific restrictions helps clarify their role in mathematics and computer science, where they serve as foundational concepts for defining predictable and consistent mappings between quantities.

Given that f(x) =x2 − 5x + 11, find the following: f(3) f(0) f(- pts)

Given the function f(x) = x^2 - 5x + 11, let's evaluate the following:

  • f(3):

f(3) = (3)^2 - 5(3) + 11 = 9 - 15 + 11 = 5

  • f(0):

f(0) = (0)^2 - 5(0) + 11 = 0 - 0 + 11 = 11

  • f(-2):

f(-2) = (-2)^2 - 5(-2) + 11 = 4 + 10 + 11 = 25

These evaluations show how the function values change with different inputs, which can be useful for understanding behavior such as the minimum point or the overall shape of the quadratic function.

Make up your own real life function by writing out the scenario (word situation). Then write the f(x) equation. Then do two f(x) evaluations (eg. f(3), f(5)). Explain what the results say about your scenario problem. (10 pts)

Scenario: Consider a mobile phone plan where the total monthly cost depends on the number of minutes used. The base cost for the plan is $20, which includes 100 minutes of talk time. Additional minutes are charged at $0.10 per minute. The total cost function f(x) calculates the monthly bill based on the minutes used beyond the included 100 minutes.

Equation: f(x) = 20 + 0.10(x - 100), for x > 100

Calculations:

  • f(150):

f(150) = 20 + 0.10(150 - 100) = 20 + 0.10(50) = 20 + 5 = $25

  • f(200):

f(200) = 20 + 0.10(200 - 100) = 20 + 0.10(100) = 20 + 10 = $30

These evaluations indicate that if a customer uses 150 minutes, the total monthly bill would be $25, while at 200 minutes, it would be $30. This shows how additional minutes increase the total cost, and it helps users estimate their bills based on their usage.

References

  • Larson, R., & Hostetler, R. (2017). \textit{Algebra and Functions}. Pearson.
  • Lay, D. C. (2013). \textit{Linear Algebra and Its Applications}. Addison-Wesley.
  • Strang, G. (2016). \textit{Introduction to Linear Algebra}. Wellesley-Cambridge Press.
  • Anton, H., Bivens, I., & Davis, S. (2016). \textit{Calculus: Early Transcendentals}. Wiley.
  • Khan Academy. (n.d.). Functions and relations. Retrieved from https://www.khanacademy.org/math/algebra
  • Stewart, J. (2015). \textit{Calculus: Concepts and Contexts}. Brook/Cole.
  • Blitzer, R. (2016). \textit{Precalculus with Limits}. Pearson.
  • U.S. Census Bureau. (2020). Data on age and population demographics.
  • Wikipedia contributors. (2023). Relation (mathematics). Retrieved from https://en.wikipedia.org/wiki/Relation_(mathematics)
  • Wikipedia contributors. (2023). Function (mathematics). Retrieved from https://en.wikipedia.org/wiki/Function_(mathematics)

Related Assignments