Which of These Relations Are Functions? Select All That Apply — A Complete Guide
If you have ever stared at a multiple-choice question that asks "which of these relations are functions? Select all that apply" and felt unsure where to start, you are not alone. This is one of the most common question types in algebra, and mastering it only requires one clear idea: a function is a special kind of relation where every input has exactly one output. In this guide, you will learn the definition, the fastest checking methods, worked examples, and the common traps that make students choose the wrong options.
Understanding Relations First
Before identifying functions, you need to understand what a relation is. A relation is simply any set of ordered pairs — a connection between an input value (usually x) and an output value (usually y).
For example:
- {(1, 2), (3, 4), (5, 6)}
- {(−2, 0), (−2, 5), (4, 7)}
- {(apple, red), (banana, yellow)}
All of these are relations. But not all of them are functions.
The Definition of a Function
A function is a relation in which each input (x-value) is paired with exactly one output (y-value). In other words:
- No x-value may repeat with a different y-value.
- y-values can repeat — that is perfectly fine.
- Every input must have an output; an input with no output breaks the rule too.
Think of it like a vending machine: you press button A1 once, and only one specific snack comes out. If pressing A1 sometimes gave you chips and sometimes gave you juice, the machine would not be a function.
Method 1: Checking Sets of Ordered Pairs
Basically the most common format for "select all that apply" questions.
Example: Which of these relations are functions? Select all that apply.
A. {(1, 3), (2, 5), (3, 7)} B. {(2, 4), (2, 6), (5, 8)} C. {(−1, 0), (0, 1), (1, 0)} D. {(4, 2), (5, 2), (6, 2)}
Step-by-step check:
- List all the x-values.
- Look for repeats.
- If an x-value repeats, check whether it maps to the same y every time.
- A: x-values are 1, 2, 3 — no repeats. ✅ Function
- B: x = 2 appears twice, mapping to 4 and 6. ❌ Not a function
- C: x-values are −1, 0, 1 — no repeats (the repeated 0 is a y-value, which is allowed). ✅ Function
- D: x-values are 4, 5, 6 — no repeats. ✅ Function
Correct answers: A, C, and D.
Notice how option C tricks many students. The repeated number is an output, not an input, so it does not violate the rule.
Method 2: Using Mapping Diagrams
A mapping diagram shows arrows from inputs to outputs The details matter here..
- If every input has exactly one arrow leaving it, the relation is a function.
- If one input splits into two arrows, it is not a function.
For example:
- 1 → 5, 2 → 6, 3 → 7 ✅ Function
- 1 → 5, 1 → 9, 2 → 6 ❌ Not a function (input 1 has two outputs)
Method 3: Using Tables of Values
Tables work the same way as ordered pairs. Scan the input column for duplicates Not complicated — just consistent..
| x | y |
|---|---|
| 0 | 1 |
| 1 | 3 |
| 2 | 5 |
✅ Function — each input appears once.
| x | y |
|---|---|
| 0 | 1 |
| 0 | 4 |
| 2 | 5 |
❌ Not a function — input 0 has two different outputs.
Method 4: The Vertical Line Test (Graphs)
When the relation is shown as a graph, use the vertical line test:
- Imagine drawing vertical lines across the entire graph.
- If any vertical line touches the graph at more than one point, the relation is not a function.
- If every vertical line touches the graph at most once, it is a function.
Examples:
- A straight line such as y = 2x + 1 → ✅ function
- A parabola opening upward, y = x² → ✅ function
- A circle, x² + y² = 9 → ❌ not a function (a vertical line through the center hits two points)
- A sideways parabola, x = y² → ❌ not a function
Method 5: Checking Equations
For equations, solve for y and ask: does each x produce only one y?
- y = 3x − 2 → ✅ function (one output per input)
- y = x² + 1 → ✅ function (squaring still gives one result)
- y² = x → ❌ not a function (x = 4 gives y = 2 and y = −2)
- x² + y² = 25 → ❌ not a function
A helpful shortcut: if y is squared (or appears inside an even root with both signs), the relation usually fails the function test That's the part that actually makes a difference..
A Full "Select All That Apply" Practice Problem
Question: Which of the following relations are functions? Select all that apply Worth keeping that in mind..
- {(3, 1), (4, 1), (5, 1)}
- {(7, 2), (7, 3), (8,
