What Is Not A Function On A Graph

Graphs are powerful tools for visualizing relationships between variables, but not every graph represents a function. Understanding the criteria that define a function is crucial for interpreting data accurately and making informed decisions. Let's delve into the intricacies of what constitutes a function on a graph, exploring the key concepts, common scenarios, and methods for determining whether a given graph qualifies as a function.

What is a Function?

At its core, a function is a relationship between a set of inputs and a set of permissible outputs with the attribute that each input is related to exactly one output. The inputs are known as the domain, and the set of possible outputs are called the range. Functions are used extensively in mathematics, science, and engineering to model real-world phenomena.

In simpler terms, think of a function as a machine. You put something in (the input), and the machine gives you something back (the output). The critical part is that for the same input, you always get the same output. If you put the same input in and sometimes get one output and sometimes get another, it's not a function.

Representing Functions

Functions can be represented in various ways:

• Equations: For example, y = x + 2 is a function where for every x value (input), there is a unique y value (output).
• Tables: A table can show the correspondence between inputs and outputs. Each input value should have only one output value associated with it.
• Graphs: A graph visually represents the relationship between inputs and outputs, with the input typically plotted on the x-axis and the output on the y-axis.
• Mappings: Illustrating the function as arrows from a set of inputs to a set of outputs.

The Vertical Line Test

The vertical line test is the primary visual method for determining whether a graph represents a function. If any vertical line drawn through the graph intersects the graph at more than one point, then the graph is not a function. This is because a vertical line represents a single x-value (input), and if it intersects the graph at multiple points, it means that this single x-value is associated with multiple y-values (outputs), violating the definition of a function.

What is Not a Function on a Graph: Common Scenarios

Several types of graphs commonly fail the vertical line test and therefore are not considered functions. Here are a few notable examples:

1. Circles

The equation of a circle centered at the origin is x² + y² = r², where r is the radius. If you graph this equation, you'll see a circle. A circle is not a function because for most x-values within the circle's diameter, there are two corresponding y-values. For example, if r = 5 and x = 3, then y² = 25 - 9 = 16, so y = ±4. This means that the x-value of 3 is associated with both y = 4 and y = -4, failing the function test.

2. Ellipses

Similar to circles, ellipses generally do not represent functions. An ellipse's equation is (x²/a²) + (y²/b²) = 1, where a and b are the semi-major and semi-minor axes, respectively. Unless the ellipse is degenerate (a line segment), a vertical line will typically intersect the ellipse at two points for many x-values, thus not representing a function.

3. Parabolas Opening Sideways

A parabola that opens upwards or downwards (with the equation y = ax² + bx + c) is a function. However, a parabola that opens sideways (with the equation x = ay² + by + c) is not a function. In this case, for a given x-value, there are generally two y-values that satisfy the equation, causing it to fail the vertical line test.

4. Relations with Multiple Y-Values for a Single X-Value

Any graph that explicitly shows a single x-value mapping to multiple y-values is not a function. This could occur in various ways, such as a graph with a vertical line segment, a squiggly line that doubles back on itself vertically, or a scatter plot that does not show a functional relationship.

5. Piecewise "Functions" with Overlapping Domains

A piecewise function can be a function if its pieces are carefully defined so that for any x-value in the overall domain, there is only one corresponding y-value. However, if the domains of the pieces overlap and produce different y-values for the same x-value, then the overall relation is not a function.

For example, consider a piecewise relationship:

• y = x for x ≤ 2
• y = x + 1 for x ≥ 2

At x = 2, the first piece gives y = 2, and the second piece gives y = 3. Since there are two different y-values for the same x-value, this is not a function.

Why Does This Matter? Implications of Non-Functions

Understanding the distinction between functions and non-functions is crucial for several reasons:

• Mathematical Integrity: Functions are fundamental to mathematical analysis. Many theorems and techniques rely on the unique mapping property of functions. Applying these techniques to non-functions can lead to incorrect or nonsensical results.
• Modeling Reality: When using mathematical models to represent real-world phenomena, it's essential to ensure that the model accurately reflects the underlying relationships. If a situation inherently has a one-to-many relationship (one input can produce multiple outputs), then modeling it as a function will be inappropriate.
• Data Analysis: In data analysis, identifying whether a relationship is functional or not affects how you interpret and use the data. Confusing a non-functional relationship with a functional one can lead to incorrect predictions and flawed decision-making.
• Computer Science: In programming, functions are the building blocks of code. Ensuring that your functions behave predictably and consistently (i.e., always return the same output for the same input) is critical for creating reliable software.

More In-Depth Examples of Graphs That Are Not Functions

Let's dive deeper into some examples to illustrate the concepts discussed:

Example 1: The Absolute Value Inverse

Consider the graph represented by the equation x = |y|. This is not a function.

To see why, consider x = 4. This implies |y| = 4, which means y = 4 or y = -4. Therefore, the single x-value of 4 corresponds to two y-values, +4 and -4, violating the definition of a function. Graphically, this results in a sideways "V" shape, which clearly fails the vertical line test.

Example 2: A Spiral

A spiral, such as an Archimedean spiral, often does not represent a function if plotted with standard Cartesian coordinates. While its equation can be complex, the key issue is that a vertical line will intersect the spiral multiple times as it coils around the origin, indicating multiple y-values for a single x-value.

Example 3: A "Wobbly" Line

Imagine drawing a line that oscillates back and forth horizontally as it moves upward. If the oscillations are frequent enough, a vertical line will intersect the "wobbly" line in multiple places, meaning the graph is not a function.

Example 4: Disconnected Points or Clusters

A scatter plot of disconnected points or clusters may show a relationship between x and y, but it usually doesn't represent a function unless each x-value has only one corresponding y-value. If there are multiple points with the same x-value but different y-values, the vertical line test fails.

Transforming Non-Functions into Functions (Sometimes)

Sometimes, it's possible to restrict the domain of a non-function to create a new relation that is a function. This is often done with circles and ellipses.

Restricting the Domain of a Circle

The equation x² + y² = r² is not a function. However, if we solve for y, we get y = ±√(r² - x²). We can create two separate functions:

• y = √(r² - x²), which represents the upper half of the circle.
• y = -√(r² - x²), which represents the lower half of the circle.

Both of these are functions, as each x-value (within the interval [-r, r]) now has only one corresponding y-value. The domain of both functions is restricted to [-r, r] to avoid taking the square root of a negative number.

Restricting the Domain of a Sideways Parabola

The equation x = ay² + by + c is not a function. However, by restricting the range (y-values), we can create two separate functions. We can complete the square to rewrite the equation as x = a(y - k)² + h. Solving for y gives:

• y = k + √((x - h)/a)
• y = k - √((x - h)/a)

If a > 0, these are functions for x ≥ h. The first equation gives the upper half of the parabola, and the second equation gives the lower half.

Common Misconceptions

Several common misconceptions can lead to confusion about what constitutes a function on a graph:

• Thinking every graph is a function: It's easy to assume that any line or curve on a graph represents a function, but this is incorrect. The vertical line test is the definitive way to check.
• Confusing relations with functions: A relation is a general association between two sets of information. A function is a specific type of relation that adheres to specific rules. Not every relationship is a function.
• Assuming a graph with a "formula" is automatically a function: Just because you can write an equation for a graph doesn't automatically make it a function. The equation must satisfy the condition of unique output for each input. For example, x = y² has a formula but is not a function.
• Believing that a function must be continuous: A function doesn't have to be continuous (unbroken) to be a function. It can have breaks or jumps and still be a function as long as it passes the vertical line test.

Advanced Considerations

Beyond the basics, more nuanced scenarios can arise:

Parametric Equations

Parametric equations define both x and y in terms of a third variable, often denoted as t (time). While a parametric equation as a whole defines a curve, that curve may or may not represent y as a function of x. To determine if it does, you would need to analyze the relationship between x(t) and y(t) to see if, for each x, there is only one corresponding y.

Implicit Functions

Implicit functions are defined implicitly by an equation, where y is not explicitly isolated as a function of x. For instance, x³ + y³ + 6xy = 0 is the folium of Descartes, an implicit curve. In some regions, this curve does define y as a function of x, but in other regions, it does not. The vertical line test would need to be applied carefully to different portions of the graph.

Multivalued Functions

In some advanced mathematical contexts, the concept of a multivalued function is used. This is technically not a function in the standard sense, as it allows a single input to have multiple outputs. Examples include the complex logarithm and the inverse trigonometric functions (when not restricted to principal values). These are often treated as collections of single-valued functions or as functions mapping to sets of values.

Conclusion

Distinguishing between graphs that represent functions and those that do not is crucial for accurate mathematical analysis, modeling, and data interpretation. The vertical line test provides a simple yet powerful visual method for making this determination. By understanding the definition of a function, recognizing common non-functional graphs, and avoiding common misconceptions, you can confidently navigate the world of graphs and functions. Remember that a function requires a unique output for each input; anything less falls outside the realm of what mathematicians consider a function.