Which Of The Following Relations Can Be A Function?

In mathematics, the concept of a function is fundamental. A function establishes a unique relationship between two sets, where each element from the first set (the domain) is associated with exactly one element in the second set (the codomain). This uniqueness is the hallmark of a function. To determine whether a relation qualifies as a function, we meticulously examine how elements are paired. In this article, we will explore the question of which relation defines a function among the provided options. This entails understanding the underlying principles of functions and their application in various scenarios. We will delve into the specifics of each relation, analyzing whether they adhere to the functional requirement of unique mapping. Our examination will provide a comprehensive understanding of why some relations can be classified as functions while others cannot, reinforcing the foundational principles of functions in mathematics.

Understanding Functions

Before diving into the specifics of the given relations, it's crucial to solidify our understanding of what a function truly is. A function, in mathematical terms, is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Imagine a function as a machine: you put something in (the input), and the machine gives you something specific out (the output). The crucial part is that for every input, you always get the same, single output. The domain of a function is the set of all possible inputs, and the codomain is the set of all possible outputs. The range, on the other hand, is the set of actual outputs the function produces. To determine if a relation is a function, we often use the vertical line test in a graphical representation. If any vertical line intersects the graph more than once, the relation is not a function because it means one input has multiple outputs. In essence, the concept of a function is about predictability and uniqueness. For every input, there is one, and only one, output. This principle underpins many mathematical and real-world applications, from simple equations to complex algorithms. Without the certainty that each input yields a unique output, the reliability and predictability of mathematical models would be severely compromised. Thus, understanding and identifying functions are pivotal skills in mathematics and its applications.

Analyzing the Given Relations

Now, let's dissect each relation provided in the question to determine whether it qualifies as a function. We'll apply our understanding of functions to each case, focusing on whether the uniqueness criterion is met. This involves careful consideration of the relationships described and their potential implications for multiple outputs from a single input.

A. R = (x,y) y is a sister of x

This relation describes a scenario where y is a sister of x. Think about this in terms of individuals. Can a person (x) have more than one sister (y)? Absolutely. In many families, it's common for someone to have multiple sisters. Therefore, for a single input x, there can be multiple outputs y. This violates the fundamental definition of a function, which requires each input to have only one output. Imagine a family with one brother and two sisters. If the brother is x, then both sisters would be y values related to that x. This clearly demonstrates that this relation cannot be a function. The possibility of multiple outputs for a single input makes this relation unsuitable for functional representation. The non-uniqueness in the sisterhood relationship breaks the strict mapping rule that functions adhere to.

B. R = (x, y) x is a father of y

This relation states that x is a father of y. Again, we consider if a single x can be related to multiple y values. Can a father (x) have more than one child (y)? Yes, a father can have multiple children. Therefore, for one father (x), there can be several children (y). This directly contradicts the requirement of a function, where each input must correspond to exactly one output. Consider a father with three children. The input x (the father) would be associated with three different y values (his children). This multi-output scenario disqualifies this relation from being a function. The potential for multiple offspring from a single father makes this relation non-functional, highlighting the importance of the one-to-one mapping in function definitions.

C. R = (x, y) y is father of x

Here, the relation states that y is the father of x. Can a person (x) have more than one father (y)? Biologically, no. A person has only one biological father. Therefore, for each x (a person), there is only one y (their father). This relation adheres to the fundamental property of a function: each input (x) maps to exactly one output (y). There is no possibility of multiple fathers for a single person, ensuring a unique mapping. This satisfies the condition for a functional relationship. The singularity of fatherhood makes this relation a valid function, demonstrating the core principle of unique output for each input.

D. R = (x, y) y is grandfather of x

In this relation, y is the grandfather of x. Can a person (x) have more than one grandfather (y)? Yes, a person has two grandfathers – their paternal grandfather and their maternal grandfather. Thus, for a single x, there are two possible y values. This violates the principle that a function must have a unique output for each input. Consider a person and their two grandfathers. The single input x (the person) would map to two different y values (their grandfathers). This dual-mapping scenario means this relation cannot be a function. The existence of two grandfathers for each person prevents this relation from being a function, highlighting the crucial role of unique mapping in functional relationships.

Conclusion

After analyzing each relation, it's clear that only one adheres to the definition of a function. The relation C. R = (x, y) y is father of x is the only one where each input (x) has exactly one output (y). A person has only one biological father, ensuring a unique mapping. The other relations (sisterhood, fatherhood, and grandfatherhood) allow for multiple outputs from a single input, thus disqualifying them as functions. Understanding the nuances of function definitions is crucial in mathematics. This analysis underscores the importance of unique mapping in defining a function and provides a clear understanding of how real-world relationships can be assessed in mathematical terms. The exercise of evaluating these relations strengthens our grasp of functional principles and their applicability in various contexts.

• A function requires a unique output for each input.
• Relations like