Which Relation Is Not A Function

Which Relation is Not a Function? A Deep Dive into Functions and Relations

Understanding the difference between a relation and a function is fundamental in mathematics, particularly in algebra and calculus. While all functions are relations, not all relations are functions. This article will delve deep into the concept of functions and relations, exploring various examples and providing a clear understanding of when a relation fails to qualify as a function. We'll also touch upon the vertical line test, a handy graphical method for identifying functions.

Defining Relations and Functions

Before we can discuss which relations are not functions, we need to clearly define both terms.

A relation is simply a set of ordered pairs. These ordered pairs can represent any kind of connection or association between two sets of values. For instance, consider the relation representing the relationship between the number of hours worked and the amount of money earned: {(2, 20), (4, 40), (6, 60)}. Here, each ordered pair (x, y) indicates that 'x' hours of work resulted in 'y' dollars earned.

A function, on the other hand, is a special type of relation where each input (x-value) has only one output (y-value). In simpler terms, for every x, there's only one corresponding y. Using the previous example, the relation representing hours worked and money earned is a function because each number of hours worked corresponds to only one amount earned.

Identifying Relations That Are Not Functions

The key characteristic that distinguishes a function from a non-function relation is the uniqueness of the output for each input. If any input value has more than one associated output value, the relation is not a function.

Let's explore some scenarios where relations fail to be functions:

Scenario 1: Multiple Outputs for a Single Input

Consider the relation: {(1, 2), (2, 4), (3, 6), (1, 5)}. Notice that the input value '1' is associated with two different output values: '2' and '5'. This violates the fundamental rule of functions—one input, one output. Therefore, this relation is not a function.

Scenario 2: Relations Defined by Equations

Not all relations are presented as sets of ordered pairs. Sometimes, they are defined by equations. Consider the equation x² + y² = 25. This equation represents a circle with a radius of 5 centered at the origin. If we try to solve for 'y' in terms of 'x', we get y = ±√(25 - x²). For many x-values (except for x = ±5), we'll get two corresponding y-values. For example, if x = 3, y = ±4. This indicates that the equation x² + y² = 25 does not represent a function.

Scenario 3: Real-World Examples

Let's examine some real-world scenarios to illustrate relations that are not functions.

• The relationship between a person's name and their age: Many people may share the same name, but have different ages. Therefore, this relation is not a function because a single name (input) can be associated with multiple ages (outputs).

• The relationship between a city and its population: Different cities can have the same population. Thus, this is not a function since one population (output) can correspond to several cities (inputs).

• Mapping students to their favorite subjects: Several students may share the same favorite subject. Hence, this relation is not a function because one favorite subject (output) can be associated with multiple students (inputs).

The Vertical Line Test: A Graphical Method

The vertical line test provides a simple visual method to determine whether a graph represents a function.

The Vertical Line Test: If any vertical line intersects the graph at more than one point, the graph does not represent a function.

Why does this work? Because a vertical line at a given x-value represents all possible y-values for that specific x. If the vertical line intersects the graph at more than one point, it means there are multiple y-values for that single x-value, violating the function definition.

Example: Consider the graph of a circle (x² + y² = 25). Draw a vertical line through the circle. You'll find it intersects the circle at two points. Therefore, by the vertical line test, the circle does not represent a function.

In contrast, if you consider the graph of a simple line like y = 2x + 1, any vertical line will intersect the graph at only one point. This confirms that y = 2x + 1 represents a function.

Functions as Mappings

Another way to think about functions is as mappings between sets. Let's say we have two sets, A and B. A function, f, from set A to set B (denoted f: A → B) is a rule that assigns to each element in set A exactly one element in set B. The elements of set A are called the domain, and the elements of set B that are assigned by the function are called the range.

If our mapping assigns more than one element in B to a single element in A, it's not a function. This reiterates the core principle: one input, one output.

Advanced Concepts and Extensions

The concept of functions extends far beyond the basic examples. We encounter various types of functions in higher mathematics:

• One-to-one functions (Injective): Each element in the range is mapped to by only one element in the domain.

• Onto functions (Surjective): Each element in the range is mapped to by at least one element in the domain.

• Bijective functions: Functions that are both one-to-one and onto. These functions have inverses.

Understanding these different types of functions is crucial for more advanced mathematical concepts.

Conclusion: Differentiating Functions from Relations

The difference between a relation and a function hinges on the uniqueness of the output for each input. If a relation assigns multiple outputs to a single input, it is not a function. The vertical line test provides a convenient visual method for determining whether a graph represents a function. Mastering the distinction between relations and functions is essential for success in further mathematical studies. By understanding the fundamental principles and applying the vertical line test, you can confidently identify relations that do not meet the criteria for being a function. Remember to always check for multiple outputs for a single input to identify those relations that fail the function test. This thorough understanding will serve as a strong foundation for more complex mathematical concepts.