Which Relation Does Not Represent A Function

Which Relation Does Not Represent a Function? Understanding Functions and Relations in Mathematics

Understanding functions and relations is fundamental to grasping many mathematical concepts. While closely related, they possess key differences. This article delves into the specifics of what defines a function, explores various types of relations, and clearly illustrates which relations fail to meet the criteria of a function. We'll use numerous examples and visual aids to ensure a comprehensive understanding.

What is a Relation?

A relation is simply a set of ordered pairs. These ordered pairs show a connection or correspondence between elements of two sets, often denoted as set A and set B. Each ordered pair (x, y) indicates that element x from set A is related to element y from set B in some way. The relation doesn't necessarily imply any specific type of connection; it merely states that a link exists.

Example:

Let's consider the relation "is taller than" between two people. If we have three people: Alice (A), Bob (B), and Charlie (C), and their heights are: A = 5'4", B = 6'0", and C = 5'8", then the relation could be represented by the set of ordered pairs: {(A, B), (A, C), (C, B)}. This shows that Alice is taller than Bob, Alice is taller than Charlie, and Charlie is taller than Bob.

What is a Function?

A function, a special type of relation, is a set of ordered pairs where each element from the first set (the domain) is associated with exactly one element from the second set (the range or codomain). This is often referred to as the vertical line test. If you can draw a vertical line anywhere on the graph of a relation and it intersects the graph more than once, the relation is not a function.

The crucial difference between a function and a general relation lies in this uniqueness. In a function, every input (x-value) maps to only one output (y-value). If a single input has multiple outputs, it's not a function.

Example:

The relation {(1, 2), (2, 4), (3, 6)} is a function because each input (1, 2, and 3) has only one corresponding output (2, 4, and 6, respectively). However, the relation {(1, 2), (1, 3), (2, 4)} is not a function because the input 1 has two outputs, 2 and 3.

Identifying Relations That Are Not Functions

Several ways exist to determine if a given relation is not a function:

1. The Vertical Line Test (Graphical Method)

This is the most intuitive method. If you graph the relation on a Cartesian coordinate system and a vertical line intersects the graph at more than one point, the relation is not a function. This signifies that at least one x-value is paired with multiple y-values.

Example: Consider the equation x² + y² = 4 (a circle). If you draw vertical lines through the circle, many of them will intersect the circle at two points. Therefore, the equation x² + y² = 4 does not represent a function.

2. Examining Ordered Pairs (Analytical Method)

Carefully examine the ordered pairs that comprise the relation. If any x-value appears multiple times with different y-values, the relation fails to be a function.

Example: The set {(1, 2), (3, 4), (1, 5)} is not a function because the x-value 1 is associated with both 2 and 5.

3. Using Set Notation

If the relation is defined using set notation, check if any element in the domain (the set of x-values) is mapped to more than one element in the range (the set of y-values).

Example: The relation A = {(a, 1), (b, 2), (a, 3)} is not a function because the element 'a' in the domain is mapped to both 1 and 3 in the range.

Types of Relations that are NOT Functions

Several common types of relations often fail the function test:

1. Circles and Ellipses

Equations representing circles and ellipses generally do not represent functions. A vertical line drawn through most points will intersect the curve at two points.

Example: x² + y² = 25 (a circle centered at the origin with a radius of 5) is not a function.

2. Parabolas Opening Horizontally

Parabolas that open horizontally (e.g., x = y²) fail the vertical line test because a vertical line will intersect the parabola at two points for most x-values. Note that parabolas opening vertically (e.g., y = x²) are functions.

3. Relations Defined by Equations with Multiple Solutions for y

Any equation where solving for 'y' produces multiple solutions indicates a relation that is not a function. For example, equations involving even powers of 'y' often fall into this category.

Example: y² = x. Solving for 'y', we get y = ±√x. For any positive x-value, we get two distinct y-values, making this relation not a function.

Real-World Examples of Relations that are NOT Functions

Many real-world scenarios illustrate relations that aren't functions.

• People and their phone numbers: A single person might have multiple phone numbers (home, work, mobile). This is a relation, but not a function because one person maps to multiple phone numbers.
• Students and their grades in a class: This represents a relation where a student is linked to their grade. If a student has multiple grades on different assignments, this is not a function (one student, multiple grades).
• Cities and their zip codes: While many cities have a single zip code, some might have multiple (spanning different areas of a large city). Therefore, this isn't a strict function.

Advanced Concepts: One-to-One and Many-to-One Functions

While this article focuses on relations that are not functions, it's beneficial to briefly touch upon function classifications:

• Many-to-one function: A function where multiple x-values map to the same y-value (e.g., y = x²).
• One-to-one function (or injective function): A function where each x-value maps to a unique y-value, and vice-versa. This is a stricter condition than a general function. For example, y = x is a one-to-one function.

Understanding these distinctions is critical for more advanced mathematical studies involving inverse functions and transformations.

Conclusion: Mastering the Function vs. Relation Distinction

The difference between a function and a relation is subtle yet critical. A function is a specific type of relation with the defining characteristic that each element in the domain maps to only one element in the range. Using the vertical line test, examining ordered pairs, and analyzing the equation defining a relation are effective methods to determine whether a given relation represents a function. Mastering this distinction is essential for progressing through various mathematical disciplines and applying these concepts to practical real-world situations. By understanding which relations do not represent functions, one can gain a deeper and more nuanced understanding of functional relationships within mathematics and beyond. Remember, practice is key; work through numerous examples and test your understanding to solidify your comprehension.