Training Relations and Functions Relations, Functions, and the Vertical Line Test
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Relations, Functions, and the Vertical Line Test

22 min Relations and Functions

Relations and Functions

A relation is any set of ordered pairs — a rule that connects inputs to outputs. A function is a special kind of relation where every input produces exactly one output. This distinction is foundational to all higher mathematics.

The vertical line test is a quick visual check: if every vertical line crosses a graph at most once, the graph represents a function. If any vertical line crosses twice, the relation fails to be a function because one input maps to two different outputs.

This lesson introduces the formal definitions, gives you practice classifying relations as functions or non-functions, and begins building the vocabulary — domain, range, input, output — that you will use throughout algebra and calculus.

Definition — Relation

A relation is any set of ordered pairs $(x, y)$. The set of all $x$-values is the domain; the set of all $y$-values is the range.

Definition — Function

A function is a relation where each input ($x$) maps to exactly one output ($y$). Think: "one $x$, one $y$."

Vertical Line Test

If any vertical line crosses a graph more than once, the graph is not a function.

  • Straight line (non-vertical) → function ✓
  • Parabola (opening up or down) → function ✓
  • Circle → not a function ✗
  • Sideways parabola → not a function ✗
Example 1

Is $\{(1,3),(2,5),(3,7),(1,9)\}$ a function?

No. The input $1$ maps to two outputs ($3$ and $9$).

Example 2

Is $\{(1,4),(2,4),(3,5)\}$ a function?

Yes. Each $x$ has exactly one $y$. Repeated $y$-values are fine.

Example 3

Is $y = x^2 + 1$ a function?

Yes. For every $x$ there is exactly one $y$. Passes VLT.

Example 4

Is $x^2 + y^2 = 25$ a function?

No. This is a circle of radius 5. For $x = 0$: $y = \pm 5$ (two outputs). Fails VLT.

Example 5

Determine domain and range from the set $\{(-2,1),(0,3),(4,7),(5,3)\}$.

Domain $= \{-2, 0, 4, 5\}$. Range $= \{1, 3, 7\}$.

Practice Problems

1. Is $\{(3,1),(4,2),(5,3)\}$ a function?
2. Is $\{(1,2),(1,3),(2,4)\}$ a function?
3. Does $y = 3x - 7$ pass the VLT?
4. Does $x = y^2$ represent a function of $x$?
5. Domain and range of $\{(0,0),(1,1),(2,4),(3,9)\}$.
6. Is a horizontal line a function?
7. Is a vertical line a function?
8. $\{(-1,5),(0,5),(1,5)\}$ — function?
9. Does the equation $y = |x|$ define a function?
10. Does $|y| = x$ define $y$ as a function of $x$?
Show Answer Key

1. Yes

2. No (input 1 gives two outputs)

3. Yes — it is a function

4. No — e.g. $x = 4$ gives $y = \pm 2$

5. Domain $= \{0,1,2,3\}$, Range $= \{0,1,4,9\}$

6. Yes (each $x$ gives one $y$)

7. No (one $x$, many $y$'s)

8. Yes (same $y$ is fine)

9. Yes — each $x$ gives one $|x|$

10. No — e.g. $x = 3$ gives $y = 3$ or $y = -3$

Function Notation and Evaluation