Determining Functions, Domain, And Range For Relation {(1, 7), (5, 8), (2, 6), (3, 4)}

July 8, 2025 by Scholario Team 87 views

Determine if the Relation Defines a Function, and State the Domain and Range

Understanding Relations and Functions

In mathematics, a relation is a set of ordered pairs. An ordered pair consists of two elements, typically written as (x, y), where x represents the input and y represents the output. Relations can be represented in various ways, including sets of ordered pairs, graphs, mappings, and equations. A function is a special type of relation that adheres to a specific rule: for every input (x-value), there is exactly one output (y-value). This is often referred to as the vertical line test when examining a graph – if any vertical line intersects the graph more than once, the relation is not a function.

To determine whether a relation defines a function, we need to examine the ordered pairs and check if any input value (x-value) is associated with more than one output value (y-value). If each input has a unique output, the relation is a function. If an input has multiple outputs, then the relation is not a function. The domain of a relation or function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). Identifying the domain and range is crucial for understanding the scope and behavior of the relation or function. In the context of ordered pairs, the domain is simply the set of all first elements in the pairs, and the range is the set of all second elements. These sets can be expressed using set notation, which involves listing the elements within curly braces.

For example, consider the relation {(1, 2), (2, 4), (3, 6), (1, 8)}. In this relation, the input value 1 is associated with two different output values, 2 and 8. This violates the condition for a function, so this relation is not a function. On the other hand, if we have the relation {(1, 2), (2, 4), (3, 6), (4, 8)}, each input has a unique output, so this relation is a function. Understanding these basic concepts is essential for the analysis and manipulation of mathematical relations and functions. The domain and range provide a framework for understanding the input and output values, while the function definition ensures that each input maps to a single, predictable output. This property is fundamental for many mathematical operations and applications, particularly in calculus and analysis, where the behavior of functions is closely examined. Relations, while broader in scope, form the foundation upon which functions are defined, making the distinction between the two crucial for mathematical understanding.

Analyzing the Given Relation: {(1, 7), (5, 8), (2, 6), (3, 4)}

In the given relation, {(1, 7), (5, 8), (2, 6), (3, 4)}, we have a set of ordered pairs. To determine if this relation defines a function, we must carefully examine the input values (x-values) to ensure that each one corresponds to a unique output value (y-value). This is a fundamental principle in mathematics, where functions play a crucial role in modeling and understanding various phenomena. The concept of a function is rooted in the idea of a predictable relationship between inputs and outputs, where each input leads to a specific, well-defined output.

Looking at the ordered pairs, we observe the following:

The input 1 is associated with the output 7.
The input 5 is associated with the output 8.
The input 2 is associated with the output 6.
The input 3 is associated with the output 4.

Each input value (1, 5, 2, and 3) appears only once in the relation. This means that no input value is associated with more than one output value. Therefore, the given relation satisfies the condition for being a function. The absence of any repeated input values with different outputs is the key indicator that this relation adheres to the strict definition of a function. This ensures that for each input, there is a single, predictable output, which is essential for many mathematical operations and applications. For example, in calculus, the concept of a derivative relies heavily on the uniqueness of the function's output for each input, as it measures the instantaneous rate of change at a specific point. Similarly, in computer science, functions are used to define algorithms, where a deterministic relationship between inputs and outputs is crucial for the algorithm's correctness and efficiency.

Understanding this distinction between relations and functions is not just a theoretical exercise; it has practical implications in various fields. In data analysis, for instance, determining whether a relationship between variables is functional can help in building accurate predictive models. In engineering, functions are used to model systems and predict their behavior, ensuring that designs are reliable and consistent. Therefore, the ability to identify functions and understand their properties is a fundamental skill in mathematics and its applications.

Determining the Domain and Range

After establishing that the relation {(1, 7), (5, 8), (2, 6), (3, 4)} defines a function, the next step is to determine its domain and range. The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). Identifying the domain and range provides a comprehensive understanding of the function's scope and limitations. In the context of ordered pairs, the domain and range can be easily determined by collecting the first and second elements of each pair, respectively.

To find the domain, we look at the first element of each ordered pair:

(1, 7) – The first element is 1.
(5, 8) – The first element is 5.
(2, 6) – The first element is 2.
(3, 4) – The first element is 3.

Therefore, the domain of the function is the set of these input values: {1, 5, 2, 3}. It's customary to write sets in ascending order, so we can also express the domain as {1, 2, 3, 5}. The domain represents the set of all possible inputs that the function can accept. Understanding the domain is crucial because it defines the limits within which the function is valid. For example, if a function represents a physical quantity, such as the height of an object over time, the domain would represent the time interval during which the object's height is being considered. Similarly, in computer science, the domain of a function can represent the valid input values for a program, ensuring that the program operates correctly.

To find the range, we look at the second element of each ordered pair:

(1, 7) – The second element is 7.
(5, 8) – The second element is 8.
(2, 6) – The second element is 6.
(3, 4) – The second element is 4.

Therefore, the range of the function is the set of these output values: {7, 8, 6, 4}. Again, writing the set in ascending order, we express the range as {4, 6, 7, 8}. The range represents the set of all possible outputs that the function can produce. It provides insight into the function's behavior and the possible values it can take. In many applications, knowing the range of a function is as important as knowing its domain. For instance, in optimization problems, the range helps determine the possible values of the objective function, allowing for the identification of optimal solutions. In statistics, the range of a data set can provide information about its variability and spread.

In summary, for the relation {(1, 7), (5, 8), (2, 6), (3, 4)}, we have determined that it is a function, and we have identified its domain as {1, 2, 3, 5} and its range as {4, 6, 7, 8}.

Conclusion

In conclusion, by carefully examining the ordered pairs in the relation {(1, 7), (5, 8), (2, 6), (3, 4)}, we have successfully determined that it defines a function. This determination was based on the fundamental principle that for a relation to be a function, each input value must correspond to a unique output value. The absence of any repeated input values with different outputs confirmed that this relation adheres to the strict definition of a function. This process highlights the importance of understanding the core concepts of relations and functions in mathematics, as well as the ability to apply these concepts to specific examples.

Furthermore, we have also identified the domain and range of this function. The domain, which represents the set of all possible input values, was found to be {1, 2, 3, 5}. The range, which represents the set of all possible output values, was found to be {4, 6, 7, 8}. Determining the domain and range provides a comprehensive understanding of the function's scope and behavior, allowing us to analyze its limitations and potential applications. This skill is crucial in various mathematical contexts, as well as in real-world scenarios where functions are used to model and predict phenomena.

The ability to determine whether a relation is a function and to identify its domain and range is a foundational skill in mathematics. It is essential for further studies in calculus, analysis, and other advanced mathematical topics. Moreover, it has practical applications in various fields, including computer science, engineering, and data analysis, where functions are used to model systems, solve problems, and make predictions. By mastering these concepts, one can develop a deeper understanding of mathematical relationships and their significance in the world around us. The process of analyzing relations and functions also reinforces logical reasoning and problem-solving skills, which are valuable assets in any field of study or profession.