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Determining
Functions

Learn to identify whether a relation is a function using tables, graphs, mappings, and equations: the foundation of algebraic thinking.

Pre-Algebra

What Is a Relation?

A relation is any set of ordered pairs \((x, y)\). It connects inputs (the \(x\)-values) to outputs (the \(y\)-values). Every relation has a domain (set of all inputs) and a range (set of all outputs).

Domain (inputs) 1 2 3 Range (outputs) 5 8

What Is a Function?

A function is a special kind of relation where every input has exactly one output. Think of it like a vending machine: you press one button, and you get exactly one item. If pressing the same button gave you different items each time, it would not be a function.

✓ Function
\(\{(1,4),\;(2,7),\;(3,4)\}\)
Each \(x\) appears once.
✕ Not a Function
\(\{(1,4),\;(1,7),\;(3,4)\}\)
\(x = 1\) maps to two outputs.

Why Do Functions Matter?

Functions are the backbone of mathematics and science. They let us predict outputs from inputs: enter a temperature in Celsius and get exactly one Fahrenheit value, enter a time and get exactly one position.

Predictability
One input always gives the same output: reliable results.
Modeling
Distance, cost, temperature: real-world patterns follow function rules.
Foundation
Every equation in algebra, calculus, and beyond builds on the function concept.

The Definition of a Function

✓ Function 1 2 3 a b c ✕ Not a Function 1 2 3 a b c

The Vertical Line Test

On a graph, draw (or imagine) vertical lines sweeping left to right. If any vertical line crosses the graph at more than one point, the relation is not a function.

Passes: Function 1 point Fails: Not a Function 2 points

Relation vs. Function

Relation Function
Definition Any set of \((x,y)\) pairs Each \(x\) maps to exactly one \(y\)
Repeated \(x\)-values Allowed with different \(y\) Not allowed with different \(y\)
Repeated \(y\)-values Allowed Allowed
Vertical Line Test May fail Always passes
Example \(\{(2,3),(2,5)\}\) \(\{(2,3),(4,5)\}\)

Test from a Table

1
Look at the input column (\(x\)-values).
2
Check for repeats. Does any \(x\)-value appear more than once?
3
If a repeated \(x\) has different \(y\)-values: not a function.
4
If every \(x\) is unique, or repeated \(x\)'s share the same \(y\): it is a function.

Test from a Mapping

In a mapping diagram, inputs are on the left and outputs on the right. Arrows show pairings. Count how many arrows leave each input.

✓ Function
Every input has exactly one arrow going out: one output per input.
✕ Not a Function
Any input with two or more arrows going to different outputs breaks the rule.

Test from a Graph

1
Imagine a vertical line sweeping from left to right across the graph.
2
At every position, count how many times the line intersects the graph.
3
If it ever hits more than once: not a function.
4
If every vertical line crosses at most one point: it is a function.

Test from an Equation

For equations, ask: "If I plug in one value for \(x\), do I get exactly one value for \(y\)?" If solving for \(y\) gives two answers (like a \(\pm\)), it is typically not a function.

✓ Function
\(y = 2x + 3\)
Each \(x\) gives one \(y\).
✓ Function
\(y = x^2\)
Each \(x\) gives one \(y\).
✕ Not a Function
\(x^2 + y^2 = 25\)
Solving: \(y = \pm\sqrt{25 - x^2}\).

Example 1: Table

\(x\) -2 0 3 5 3
\(y\) 4 1 7 2 9

Example 2: Graph

Is the graph of \(y = x^2\) a function?

x y -2 -1 1 2 1 2 3 4 1 point

Example 3: Mapping

Input Output 2 5 8 3 7

Common Mistakes

✕ Confusing \(x\) and \(y\)
Checking for repeated outputs instead of repeated inputs. Two \(y\)-values being the same is perfectly fine.
✕ Horizontal Line Test
Using a horizontal line instead of a vertical one. The horizontal line test checks something different (one-to-one).
✕ Ignoring Duplicates
Overlooking a repeated \(x\)-value in a long table or set of ordered pairs. Always scan systematically.

Summary: Four Ways to Test

Representation How to Test Watch For
Table Check if any \(x\) appears twice with different \(y\) Repeated \(x\)-values
Mapping Count arrows leaving each input More than 1 arrow from any input
Graph Vertical Line Test Line hits graph at 2 or more points
Equation Solve for \(y\): get one answer? \(\pm\) in the solution

Question 1 of 5

Which set of ordered pairs represents a function?

Question 2 of 5

A relation contains the pairs \(\{(2,6),\;(3,9),\;(4,6),\;(5,12)\}\). Is it a function?

Question 3 of 5

Which test determines whether a graph represents a function?

Question 4 of 5

In a mapping diagram, an input has two arrows pointing to two different outputs. What can you conclude?

Question 5 of 5

Which equation does not represent \(y\) as a function of \(x\)?