Which Of The Relations Are Functions

Which of the Relations Are Functions? A Clear, Step‑by‑Step Guide

Introduction

When you first encounter the idea of a “relation” in mathematics, it can feel a little abstract. A relation is simply any set of ordered pairs that connects elements from one set (the domain) to another set (the codomain). A function, on the other hand, is a special kind of relation that follows a very specific rule: every input must be paired with exactly one output.

Understanding the difference between a general relation and a function is crucial because functions are the building blocks of calculus, algebra, and many applied fields. If you can tell whether a given relation is a function, you’ll be able to read graphs, interpret tables, and work with formulas more confidently.

In this guide we’ll walk through the definition of a relation, unpack the precise definition of a function, and then look at a variety of examples—tables, graphs, mapping diagrams, and equations—to see which relations qualify as functions and which do not. By the end, you’ll have a clear, practical checklist you can apply to any relation you encounter.

What Exactly Is a Relation?

Definition in Plain Language

A relation is just a collection of ordered pairs. So each pair consists of an input (often called the x‑value or the element from the domain) and an output (the y‑value or the element from the codomain). There’s no restriction on how many times an input can appear; it can be paired with zero, one, or many outputs.

Formal Notation

Mathematically, if we have two sets (A) (the domain) and (B) (the codomain), a relation (R) from (A) to (B) is a subset of the Cartesian product (A \times B). In symbols:

[ R \subseteq { (a,b) \mid a \in A,\ b \in B } ]

Simple Examples

• Set of points on a circle: ({(x,y) \mid x^2 + y^2 = 1}) – every (x) (except (-1) and (1)) appears with two different (y) values.
• List of student IDs and their favorite colors: ({(101,\text{blue}), (102,\text{green}), (103,\text{blue}), (104,\text{red})}). Here each ID appears once, but the same color can appear multiple times.
• Parent‑child pairs: ({(\text{Alice},\text{Bob}), (\text{Alice},\text{Carol}), (\text{David},\text{Eve})}). The parent “Alice” appears twice because she has two children.

Notice that none of these examples automatically guarantee that each input has a single output. That extra condition is what turns a relation into a function.

What Makes a Relation a Function?

The Core Rule

A relation (F) from a set (A) to a set (B) is a function if and only if every element of (A) appears exactly once as the first component of an ordered pair in (F). In other words:

• For each input (x) there is one and only one output (y).

If an input appears with two different outputs, the relation fails the function test.

The Vertical Line Test (Graphical View)

When a relation is displayed as a graph in the coordinate plane, the vertical line test gives a quick visual answer:

• Imagine drawing a vertical line (a line of constant (x)) anywhere across the graph.
• If that line ever intersects the graph in more than one point, the relation is not a function.
• If every vertical line touches the graph at most once, the relation is a function.

This test works because a vertical line corresponds to fixing an input (x) and checking how many outputs (y) appear.

Mapping Diagrams

A mapping diagram draws two columns—one for inputs, one for outputs—and draws arrows from each input to its output(s). Practically speaking, a relation is a function exactly when each input has exactly one arrow leaving it. If you see an input with two or more arrows, or with none at all, the relation fails the function test.

Tabular Form

When a relation is presented as a table with two columns (input, output), scan the input column. If any input value repeats with different output values, the relation is not a function. Repeats with the same output are fine; they still satisfy the “exactly one output” rule because the output is identical.

Examples: Which Relations Are Functions?

Let’s walk through several common representations and apply the rules above.

Example 1: A Set of Ordered Pairs

[ R_1 = { (1,2), (2,4), (3,6), (4,8) } ]

Check:* Each input (1,2,3,4) appears once, and each has a single output.
Verdict: Function (specifically, (f(x)=2x)).

Example 2: Repeated Input with Different Outputs

[ R_2 = { (1,2), (1,5), (2,3), (3,7) } ]

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Check:* The input 1 appears twice, paired with 2 and 5.
Verdict: Not a function (violates the “exactly one output” rule).

Example 3: Repeated Input with Same Output

[ R_3 = { (1,4), (2,4), (3,4), (4,4) } ]

Check:* Each input appears once; the output happens to be the same for all, but that’s allowed.
Verdict: Function (a constant function (f(x)=4)).

Example 4: A Circle (Graphical Form)

Equation: (x^2 + y^2 = 25).

Graph:* A circle centered at the origin with radius 5.
Vertical line test:* A vertical line at (x=0) hits the circle at ((0,5)) and ((0,-5)) – two points.
Verdict: Not a function because some (x) values give

Example 4 (continued): The Circle Revisited

The equation

[ x^{2}+y^{2}=25 ]

produces a circle of radius 5 centered at the origin. Even so, because a single (x)-value can correspond to two distinct (y)-values (the upper and lower halves of the circle), the vertical line test fails. In functional language, the relation defined by the circle is not a function; it is a multivalued* relation.

If we restrict the circle to its upper semicircle, however, the situation changes:

[ y=\sqrt{25-x^{2}},\qquad -5\le x\le 5 . ]

Now every vertical line intersects the graph at exactly one point, so the restricted relation is a function. Its domain is ([-5,5]) and its range is ([0,5]).

Example 5: A Piecewise‑Defined Relation

Consider the set

[ R_5=\bigl{(x,y)\mid y=\begin{cases} x+1 &\text{if }x<0,\[4pt] 2x &\text{if }0\le x\le 3,\[4pt] 5 &\text{if }x>3 \end{cases} \bigr}. ]

Each input (x) falls into exactly one of the three cases, and the corresponding formula yields a single output (y). Therefore (R_5) satisfies the “exactly one output per input’’ condition and is a function. Its graph consists of three line segments that meet at (x=0) and (x=3); the vertical line test is passed because no vertical line crosses more than one segment at a single (x)-coordinate.

Example 6: A Relation with Gaps in the Domain

Let

[ R_6={(1,10),;(2,20),;(4,40),;(5,50)}. ]

Here the input values are not consecutive; the number 3 is missing. What matters is that each listed input has a unique output. On top of that, this omission does not disqualify the relation from being a function. Since every listed (x) appears only once, (R_6) is a perfectly valid function, albeit one whose domain is ({1,2,4,5}).

Example 7: A Relation That Fails the Horizontal Test (But Not the Function Test)

[ R_7={(a,1),;(b,1),;(c,2),;(d,2)}. ]

Every input (a,b,c,d) maps to exactly one output, so (R_7) is a function. The fact that two different inputs share the same output is irrelevant to the definition of a function; that property belongs to the concept of injectivity* (one‑to‑one), not to the basic function test.

Summary of the Decision Process

1. Identify the set of inputs.
2. Check whether each input appears exactly once.
   • If an input appears more than once with different outputs → not a function.
   • If an input appears more than once with the same* output → still a function (the output is unique).
3. Apply the vertical line test (graphical view).
   • A vertical line intersecting the graph more than once signals a failure of the function condition.
4. Examine tables or mapping diagrams.
   • Scan the input column; duplicate inputs with differing outputs break the rule.

When any of these checks is satisfied, the relation qualifies as a function; otherwise it does not.

Conclusion

A relation is a function precisely when each permissible input is paired with one and only one output. This requirement can be verified through several equivalent lenses: a set of ordered pairs, a mapping diagram, a tabular list, or a graph in the plane. The vertical line test provides a quick visual cue for graphs, while algebraic or tabular scrutiny confirms the same principle in discrete settings.

Understanding this definition enables students to classify a wide variety of relations—linear equations, circles, piecewise formulas, and even sparse data sets—by systematically applying the “one input, one output’’ rule. Mastery of this concept forms the foundation for deeper topics such as injectivity, surjectivity, inverses, and functional composition, all of which build upon the simple yet powerful idea that a function assigns a unique output to each input.