Which of these relations is a function is a question that appears frequently in algebra classrooms, standardized tests, and everyday problem‑solving scenarios. Understanding the difference between a general relation and a function is essential because functions describe predictable, rule‑based relationships that we can rely on for calculations, modeling, and further mathematical analysis. In this article we will explore what makes a relation qualify as a function, examine practical methods for testing this property, work through illustrative examples, and provide practice problems that reinforce the concept. By the end, you’ll be able to look at any set of ordered pairs, graph, or rule and confidently answer “which of these relations is a function?
Understanding Relations and Functions
What is a Relation?
In mathematics, a relation is simply a set of ordered pairs ((x, y)) that connects elements from one set (the domain) to elements of another set (the codomain). There is no restriction on how many times an (x)-value may appear; a single input can be paired with multiple outputs. Take this case: the set
[ {(1,2), (1,3), (2,4), (3,5)} ]
is a relation because it groups numbers together, but the input (1) is associated with both (2) and (3).
What Makes a Relation a Function?
A function is a special type of relation with an additional rule: each input value from the domain must correspond to exactly one output value. Put another way, no (x)-coordinate may be repeated with different (y)-coordinates. If we think of a function as a machine, you feed it an input, and it always returns the same output—no surprises.
Formally, a relation (R) is a function iff
[ \forall x \in \text{Domain}(R),; \exists! y \text{ such that } (x,y) \in R. ]
The symbol (\exists!) reads “there exists exactly one.”
How to Determine If a Relation Is a Function
There are several practical techniques to test the “one output per input” condition, depending on how the relation is presented.
Using Ordered Pairs
When the relation is given as a list of pairs, simply scan the first components (the (x)-values). If any (x)-value appears more than once with different second components, the relation fails the function test Simple as that..
Example:
[ {(0,1), (2,3), (2,5), (4,0)} ]
Here the input (2) maps to both (3) and (5); therefore this set is not a function.
Using Mapping Diagrams
A mapping diagram draws two columns—one for inputs, one for outputs—and connects each input to its output(s) with arrows. If any input has more than one arrow leaving it, the relation is not a function. This visual method is especially helpful for learners who benefit from seeing the “many‑to‑one” versus “one‑to‑many” distinction.
Vertical Line Test (Graphs)
When a relation is displayed as a graph in the coordinate plane, the vertical line test provides a quick graphical check. So imagine drawing vertical lines (lines of constant (x)) across the graph. If any vertical line intersects the graph at more than one point, the relation assigns multiple (y)-values to a single (x)-value and thus is not a function. Conversely, if every vertical line touches the graph at most once, the relation is a function Easy to understand, harder to ignore..
Common Examples and Non‑Examples
Example 1: Set of Ordered Pairs
Consider the relation
[ R_1 = {(-2,4), (-1,1), (0,0), (1,1), (2,4)}. ]
Each (x)-value (-2, -1, 0, 1, 2) appears exactly once, so (R_1) is a function. In fact, it represents the rule (y = x^2) restricted to those five inputs And it works..
Example 2: Equation (y = x^2)
The equation (y = x^2) defines a relation consisting of all points ((x, x^2)). Applying the vertical line test to the parabola confirms that any vertical line cuts the curve at most once. No matter which real number you choose for (x), squaring it yields a single result. Hence, (y = x^2) is a function.
Real talk — this step gets skipped all the time.
Example 3: Circle (x^2 + y^2 = 1)
The set of points satisfying (x^2 + y^2 = 1) forms a circle centered at the origin with radius 1. The same input (x = 0) yields two different outputs, violating the function condition. Even so, pick (x = 0); the equation gives (y^2 = 1), so (y = 1) or (y = -1). A vertical line through (x = 0) intersects the circle at two points, confirming that this relation is not a function.
Practice Problems: Which of These Relations Is a Function?
Below are five relations. Determine whether each one is a function, and justify your answer using the appropriate test.
Problem Set
- (A = {(3,7), (3,9), (5,2), (8,0)})
- (B = {(t, t^3) \mid t \in \mathbb{R}})
- (C = {(x, y) \mid y = \sqrt{x}})
- (D = {(p, q) \mid p^2 + q^2 = 9})
- (E = {(m, n) \mid n = 2m + 1})
Solutions and Explanations
- Relation A
