What Makes Something Not A Function

Let's delve into the core concept of functions in mathematics and computer science. Understanding what defines a function also necessitates knowing what violates its fundamental principles. When a relationship fails to meet the criteria of a function, it's crucial to recognize those deviations to avoid errors in reasoning and application.

What is a Function?

At its heart, a function is a well-defined relationship between two sets. One set is called the domain, and the other is called the codomain. Think of it as a machine: you feed it an input from the domain, and it produces a unique output in the codomain.

For a relationship to be considered a function, it must satisfy two key conditions:

Every element in the domain must be mapped to an element in the codomain. In simpler terms, you can't have any input that doesn't produce an output.
Each element in the domain must be mapped to exactly one element in the codomain. This is the most crucial rule: one input cannot produce multiple, different outputs.

Violations of the Function Definition: What Makes Something NOT a Function

Now, let's explore the scenarios that violate the function definition. These violations fall into two broad categories:

Unmapped Elements in the Domain: When one or more elements in the domain do not have a corresponding element in the codomain.
Multiple Mappings from a Single Element in the Domain: When one or more elements in the domain are associated with more than one element in the codomain.

Let's examine each of these violations with examples.

1. Unmapped Elements in the Domain

This occurs when you have an input value for which the function doesn't provide any output. Consider the following scenarios:

Division by Zero: The function f(x) = 1/x is not defined for x = 0. There is no value that 1/0 can take. Thus, if the domain is all real numbers, this is not a function. However, if the domain excludes 0, then it is a function. This highlights that the definition of the domain is crucial to determining if a relationship is a function.
Square Root of a Negative Number (in Real Numbers): The function f(x) = √x (where √ represents the square root) is not defined for negative values of x if we are working within the realm of real numbers. You cannot take the square root of a negative number and get a real number as a result. Therefore, if the domain is all real numbers, this is not a function. If, however, the domain is defined to be all non-negative real numbers (x ≥ 0), then it is a function.
Logarithm of Zero or a Negative Number: The function f(x) = log(x) is not defined for x ≤ 0. Logarithms are only defined for positive numbers. Therefore, if the domain is all real numbers, this is not a function. If, however, the domain is defined to be all positive real numbers (x > 0), then it is a function.
Partial Functions: In computer science, a partial function is a function that is not defined for all possible input values of its declared type. For example, a function might be defined to take an integer as input but only produce a valid output for positive integers. In such cases, the function is only "partial" because it doesn't cover the entire potential domain. These are generally not considered mathematical functions unless the domain is explicitly restricted to the values for which the function is defined.

Why does this violate the function definition?

The fundamental requirement of a function is that every element in the domain must have a corresponding element in the codomain. When an element in the domain is "unmapped," it means the function doesn't provide an output for that specific input, thus breaking the core definition.

2. Multiple Mappings from a Single Element in the Domain

This is the most common reason why a relationship fails to be a function. It occurs when a single input value produces more than one distinct output value. Consider these scenarios:

The Inverse of a Parabola (Without Restriction): Consider the equation x = y². If we try to express y as a function of x, we get y = ±√x. For every positive value of x, there are two possible values of y: a positive square root and a negative square root. For instance, if x = 4, then y = 2 or y = -2. Since one input (x=4) gives two outputs (y=2 and y=-2), this is not a function. However, if we restrict the range of y to non-negative values (y ≥ 0), then y = √x is a function. Similarly, if we restrict the range of y to non-positive values (y ≤ 0), then y = -√x is a function. The key is the restriction!
A Relation Defined by a Circle: Consider the equation of a circle: x² + y² = r², where r is the radius. Solving for y, we get y = ±√(r² - x²). Again, for most values of x within the range -r ≤ x ≤ r, there are two corresponding values of y. One above the x-axis and one below. Therefore, this is not a function.
Functions Returning Sets: Suppose you have a "function" defined as f(x) = {y | y² = x}. This attempts to define a function where the output is the set of all numbers whose square is x. For example, f(4) = {2, -2}. This is not a function because a function must return a single value, not a set of values.
A Real-World Example: Student to Courses (Without Restriction): Imagine a relationship between students and the courses they are taking. If a student can enroll in multiple courses, then mapping a student to the set of courses they are taking is not a function because one student (input) maps to multiple courses (outputs). However, if we consider the relationship of mapping a student to their student ID number, this is a function, assuming each student has a unique ID.

Why does this violate the function definition?

The core principle of a function is that each input must have exactly one output. When a single input produces multiple outputs, it creates ambiguity and violates the deterministic nature required for a function to be considered well-defined. The "machine" is acting unpredictably.

Visualizing Functions: The Vertical Line Test

A helpful way to visually determine if a graph represents a function is the vertical line test. If you can draw a vertical line that intersects the graph at more than one point, then the graph does not represent a function. This is because the points of intersection represent different y-values (outputs) for the same x-value (input), violating the single-output rule.

Parabola on its Side (x = y²): A parabola opening to the side (horizontally) will fail the vertical line test. Any vertical line to the right of the vertex will intersect the parabola at two points.
Circle (x² + y² = r²): A circle will also fail the vertical line test. Any vertical line within the circle's diameter will intersect the circle at two points.
A Standard Function (y = x²): A standard upward-opening parabola (y = x²) will pass the vertical line test. Any vertical line will intersect the parabola at only one point.

Piecewise Functions

Piecewise functions are defined by different formulas on different intervals of their domain. It's important to ensure that the pieces "meet" correctly at the boundaries of these intervals to maintain the function property.

Discontinuity: If the different pieces of a piecewise function do not meet at a boundary point, resulting in a "jump" or "gap," it can still be a function as long as the function is defined for only one of the pieces at that boundary point. One piece would have a closed circle and the other an open circle at the point, indicating the value is defined for only one of the interval pieces at the point.
Overlapping Definitions: If, at a boundary point, the function is defined by both pieces and the pieces have different values, then it is not a function because the input has multiple outputs. If the two pieces do have the same value at the boundary point, there is no problem.

Functions in Computer Science

The concept of a function is central to programming. In programming, a function (also often called a subroutine, procedure, or method) is a block of code that performs a specific task.

While the mathematical definition of a function emphasizes a deterministic mapping from input to output, the term "function" in programming is sometimes used more loosely.

Functions with Side Effects: In imperative programming languages, functions can have side effects, meaning they can modify the state of the program outside of their return value. For example, a function might update a global variable or write data to a file. While these are often called functions, they deviate from the pure mathematical concept because their behavior isn't solely determined by their input arguments.
Functions that Don't Return a Value: Some programming languages allow functions that don't explicitly return a value (often called "void" functions or procedures). These functions are primarily used for their side effects. While technically they could be considered to return a null or "undefined" value, the focus is on their actions rather than their output.
Functions that Throw Exceptions: A function might encounter an error during its execution and throw an exception. This is a way of signaling that the function was unable to produce a valid output. From a mathematical perspective, throwing an exception could be seen as analogous to the function being undefined for a particular input (similar to division by zero).

In functional programming, there is a strong emphasis on pure functions. A pure function is a function that:

Always returns the same output for the same input.
Has no side effects.

Pure functions align perfectly with the mathematical definition of a function and are easier to reason about and test.

Common Mistakes and Misconceptions

Confusing Relations with Functions: All functions are relations, but not all relations are functions. A relation is simply any set of ordered pairs. A function is a special type of relation that satisfies the single-output rule.
Assuming All Equations Represent Functions: Many equations can be rearranged to express one variable in terms of another. However, as we've seen with the examples of x = y² and x² + y² = r², not all such expressions define y as a function of x (unless restrictions are applied).
Ignoring the Domain: The domain is a crucial part of the function definition. A relationship might not be a function with one domain but could become a function with a more restricted domain.

Examples to Test Your Understanding

Let's test your understanding with a few examples:

f(x) = x³: Is this a function?
The mapping of a country to its capital city: Is this a function?
The mapping of a person to their mother: Is this a function?
f(x) = tan(x): Is this a function if the domain is all real numbers?
A vending machine that sometimes dispenses two items when you press one button: Does this represent a function in the mathematical sense?

Answers:

Yes, f(x) = x³ is a function. For every real number x, there is exactly one value of x³.
Yes, the mapping of a country to its capital city is a function (assuming each country has only one official capital).
Yes, the mapping of a person to their mother is a function (assuming each person has only one biological mother).
No, f(x) = tan(x) is not a function if the domain is all real numbers because tan(x) is undefined at x = π/2 + nπ, where n is any integer (due to division by zero in the definition of tangent).
No, the vending machine does not represent a function because pressing one button (input) sometimes results in multiple items (multiple outputs).

Conclusion

Understanding what makes something not a function is just as important as understanding what is a function. The core principles of a function – that every element in the domain must be mapped and that each element must be mapped to exactly one element in the codomain – are crucial for ensuring mathematical rigor and consistency. By recognizing the violations of these principles, you can avoid errors in reasoning and apply the concept of functions correctly in various fields, from mathematics and computer science to data analysis and engineering. Always remember to consider the domain, the potential for multiple outputs, and the visual aid of the vertical line test to determine whether a relationship truly qualifies as a function.