Select All Relations Which Are Not Functions

Navigating the world of relations and functions in mathematics can sometimes feel like traversing a complex maze. While functions are often the stars of the show, understanding relations that aren't functions is equally important. These non-functional relations reveal a broader landscape of mathematical connections and help solidify the defining characteristics of functions themselves.

Understanding Relations and Functions: The Basics

Before we dive into identifying relations that aren't functions, let's briefly revisit the definitions of relations and functions.

• Relation: A relation is simply a set of ordered pairs. These ordered pairs can represent connections between any two sets of objects or values. The first element in each pair is called the domain, and the second element is called the range.

• Function: A function is a special type of relation. It's a relation where each element in the domain maps to exactly one element in the range. Think of it as a well-behaved relation with a clear and unambiguous output for every input.

The key difference lies in this uniqueness requirement. A function can only have one output for each input. A relation, on the other hand, can have multiple outputs for a single input. This distinction is crucial for identifying relations that aren't functions.

Why Study Non-Functional Relations?

You might be wondering why we should bother studying relations that don't qualify as functions. There are several compelling reasons:

• Completeness: Understanding non-functional relations provides a complete picture of mathematical relationships. It shows us the full spectrum of possibilities beyond the strict definition of a function.
• Real-World Modeling: Many real-world relationships are not functions. For instance, a person can have multiple hobbies, or a product can have several customer reviews. These scenarios are naturally modeled by relations that are not functions.
• Foundation for Advanced Concepts: Non-functional relations form the basis for more advanced mathematical concepts like inverse relations, multi-valued functions (in complex analysis), and set-valued mappings.
• Sharpening Understanding of Functions: By contrasting relations with functions, we develop a deeper appreciation for the specific properties that make a function a function. We understand why the uniqueness requirement is so important.

Identifying Relations That Are Not Functions

Now, let's get to the heart of the matter: how do we identify relations that are not functions? Here are several methods and examples to guide you:

1. The Vertical Line Test (for Graphical Representations)

The vertical line test is a simple yet powerful visual tool for determining whether a graph represents a function.

• The Rule: If any vertical line drawn on the graph intersects the graph at more than one point, then the relation represented by the graph is not a function.

• Why it Works: A vertical line represents a specific x-value (the input). If the vertical line intersects the graph at multiple points, it means that the same x-value is associated with multiple y-values (outputs), violating the uniqueness requirement of a function.

Examples:

• Circle: A circle is a classic example of a relation that is not a function. A vertical line drawn through the center of the circle will intersect it at two points, demonstrating that a single x-value corresponds to two different y-values.

• Parabola Opening Sideways: A parabola that opens to the left or right (instead of up or down) also fails the vertical line test.

• Squiggly Line with Loops: Any graph with a loop or a section that curves back on itself is likely to fail the vertical line test.

Conversely:

• Straight Line (non-vertical): A straight line that is not vertical will always pass the vertical line test. Each x-value has a unique y-value.

• Parabola Opening Upwards or Downwards: A standard parabola that opens upwards or downwards also passes the vertical line test.

2. Examining Sets of Ordered Pairs

When a relation is presented as a set of ordered pairs, you can identify if it's not a function by checking for repeated x-values with different y-values.

• The Rule: If you find two or more ordered pairs with the same first element (x-value) but different second elements (y-values), then the relation is not a function.

• Why it Works: This directly violates the definition of a function, which requires each input to have only one output.

Examples:

• {(1, 2), (2, 4), (3, 6), (1, 5)}: This relation is not a function because the x-value 1 is paired with both 2 and 5.

• {(a, b), (c, d), (e, f), (a, g)}: This relation is not a function because the x-value a is paired with both b and g.

Conversely:

• {(1, 2), (2, 3), (3, 4), (4, 5)}: This relation is a function because each x-value is unique.

• {(a, x), (b, x), (c, x)}: This relation is a function. It's perfectly acceptable for multiple x-values to map to the same y-value; the only restriction is that each x-value must map to only one y-value.

3. Using Mapping Diagrams

A mapping diagram visually represents a relation by drawing arrows from elements in the domain to elements in the range. This representation makes it easy to see whether a relation is a function.

• The Rule: If any element in the domain has more than one arrow originating from it, then the relation is not a function.

• Why it Works: Each arrow represents a mapping from an input to an output. If an input has multiple arrows, it means it's mapping to multiple outputs, violating the uniqueness requirement.

Examples:

Imagine a mapping diagram with the domain {1, 2, 3} and the range {a, b, c}.

• If 1 has arrows pointing to both a and b, the relation is not a function.

• If 2 has arrows pointing to c and 3 has an arrow pointing to c, the relation is a function. (Multiple inputs can map to the same output).

4. Analyzing Equations

Sometimes, a relation is defined by an equation. In these cases, you need to analyze the equation to determine if it represents a function.

• The Rule: If, for a given x-value, the equation produces more than one possible y-value, then the relation is not a function.

• Why it Works: This aligns with the fundamental definition of a function. If a single input can lead to multiple outputs, it's not a function.

Examples:

• x = y^2: This equation defines a relation that is not a function. For example, if x = 4, then y could be either 2 or -2.

• x^2 + y^2 = 9: This equation represents a circle, which we already know is not a function. For any x-value between -3 and 3, there are two corresponding y-values.

• y = ±√x: This equation explicitly shows that for each positive x-value, there are two y-values (one positive and one negative). Therefore, it's not a function.

Conversely:

• y = 3x + 2: This equation represents a straight line and is a function. For each x-value, there is only one corresponding y-value.

• y = x^3: This equation is a function. Each x-value has a unique cube.

5. Considering Real-World Scenarios

Many real-world relationships can be modeled as relations. Thinking about these scenarios can help you understand the difference between functional and non-functional relationships.

Examples:

• Students and Their Courses: Consider a relation between students and the courses they are taking. A student can take multiple courses. Therefore, this relation is not a function (from students to courses). However, if we consider the relation from courses to students, it's definitely not a function, as a course has multiple students.

• Employees and Their Salaries: Consider a relation between employees and their salaries in a company. Assuming each employee has only one salary, this relation is a function (from employees to salaries).

• Addresses and Residents: A relation between street addresses and residents is not a function, as multiple people can live at the same address.

• States and Their Capitals: A relation between US states and their capitals is a function. Each state has only one capital city.

Common Mistakes to Avoid

When identifying relations that are not functions, be mindful of these common mistakes:

• Assuming All Equations Are Functions: Just because a relationship is expressed as an equation doesn't automatically make it a function. Always analyze the equation to see if it produces multiple y-values for a single x-value.

• Confusing the Roles of x and y: The vertical line test applies specifically when x is the independent variable (input) and y is the dependent variable (output). If you switch the axes, the test will give you incorrect results.

• Overlooking Implicit Definitions: Sometimes, a relation is defined implicitly (e.g., x^2 + y^2 = 9). You need to manipulate the equation to understand the relationship between x and y fully.

• Ignoring the Domain: The domain of a relation can affect whether it's a function. For example, if we restrict the domain of x = y^2 to only non-negative values of y, it could become a function (but it's generally understood to include both positive and negative roots).

• Thinking "One-to-Many" Automatically Means Not a Function: It's true that if one input maps to many outputs, it's not a function. But remember, many inputs mapping to the same output is perfectly acceptable for a function.

Examples and Practice Problems

Let's solidify your understanding with some examples and practice problems.

Example 1:

Is the relation defined by the set of ordered pairs {(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)} a function?

Solution:

Yes, this is a function. Each x-value is unique. The y-values are repeated, but that doesn't violate the definition of a function.

Example 2:

Is the relation defined by the equation y^2 = x + 4 a function?

Solution:

No, this is not a function. We can rewrite the equation as y = ±√(x + 4). For any x > -4, there will be two possible values of y (one positive and one negative). For instance, if x = 0, then y = ±2.

Example 3:

Consider the relation between cars and their colors. Is this relation a function (from cars to colors)?

Solution:

Yes, assuming each car has only one primary color (we're not considering multi-colored cars here). Each car maps to only one color.

Practice Problems:

1. Which of the following sets of ordered pairs represent functions?

   • {(1, 2), (2, 3), (3, 4), (4, 5)}
   • {(1, 2), (2, 2), (3, 2), (4, 2)}
   • {(1, 2), (1, 3), (2, 4), (3, 5)}
   • {(a, b), (b, c), (c, a)}
   • {(a, b), (a, c), (a, d)}

2. Which of the following equations represent relations that are not functions?

   • y = 2x - 5
   • x^2 + y = 4
   • x = y^3
   • y = |x|
   • x^2 + y^2 = 16
   • x = 5

3. Sketch a graph of a relation that is not a function. Explain why it's not a function using the vertical line test.

4. Describe a real-world scenario that can be modeled as a relation that is not a function.

The Importance of Precise Definitions

The distinction between relations and functions highlights the importance of precise definitions in mathematics. Functions, with their strict uniqueness requirement, are essential building blocks for calculus, analysis, and many other areas of mathematics and science. They allow us to make predictions and build models with confidence. Relations, on the other hand, offer a more flexible way to represent relationships where ambiguity or multiplicity is inherent.

Conclusion

Identifying relations that are not functions is a fundamental skill in mathematics. By understanding the definition of a function and applying techniques like the vertical line test, examining ordered pairs, and analyzing equations, you can confidently distinguish between functional and non-functional relationships. This knowledge not only deepens your understanding of functions but also broadens your perspective on the diverse ways mathematical connections can be expressed. Remember that while functions are essential, relations offer a richer and more complete view of the mathematical landscape.