Determining Functions — I can determine whether a relation...

SECTION 1

Where Did Functions Come From?

Have you ever used a vending machine? You press a button, and you get one specific snack. You wouldn't want a machine that gives you a random item each time you press the same button! That idea — one input always gives one output — is the heart of a function. Mathematicians spent hundreds of years figuring out how to describe this idea.

Long ago, people noticed patterns between numbers. For example, the area of a square depends on the length of its side. As math grew more complex, thinkers needed a way to talk about these relationships clearly. Here are some key moments in the story of functions.

~1670

Leibniz Coins the Word

German mathematician Gottfried Leibniz first used the word "function" to describe a quantity that changes depending on a curve.

1748

Euler's Notation

Leonhard Euler introduced the familiar f(x) notation we still use today. This made it easy to write "f of x" as a shorthand.

1837

Dirichlet's Definition

Peter Dirichlet gave the modern definition: a function pairs each input with exactly one output. This is the rule we use in class today.

Today

Functions Everywhere

Functions power everything from video-game physics to weather forecasts. Learning to spot them is a skill you'll use throughout math and science.

So the big question is: given a set of pairs, a table, a graph, a mapping diagram, or an equation, how can you tell if the relationship is a function? That's exactly what you'll learn in this lesson.

SECTION 2

Core Ideas: What Makes a Function?

Before we test whether something is a function, we need a few building blocks. A relation is just a set of ordered pairs like (x, y). Think of it as a collection of connections between inputs and outputs. A function is a special kind of relation with one extra rule.

Input (Domain)

The input is the value you put in. The set of all inputs is called the domain. In the pair (3, 7), the input is 3.

Output (Range)

The output is the value you get out. The set of all outputs is called the range. In (3, 7), the output is 7.

The Function Rule

A relation is a function if every input has exactly one output. One input going to two different outputs breaks the rule.

Multiple Inputs, Same Output = OK

Two different inputs CAN share the same output. For example, (2, 5) and (4, 5) is fine. The rule only cares that each input has one output.

✦ KEY TAKEAWAY

Think of a function like a toaster. You put in one slice of bread (input), and you always get back one piece of toast (output). If the same slice somehow gave you toast AND a bagel at the same time, your toaster would be broken — and your relation would not be a function.

SECTION 3

Seeing Functions: Mapping Diagrams

One of the clearest ways to see if a relation is a function is a mapping diagram. In a mapping diagram, you draw two ovals. The left oval holds the inputs (domain). The right oval holds the outputs (range). Then you draw arrows from each input to its output.

If any input has two or more arrows leaving it, the relation is NOT a function. Check out the diagram below.

On the left, every input has exactly one arrow leaving it — this is a function. On the right, input 4 has two arrows, so it is not a function.

Notice that on the left side, inputs 1 and 2 both point to 5. That's totally fine! Different inputs can share an output. The rule only says each input must point to exactly one output.

SECTION 4

Testing Functions with Tables and Equations

Testing a Table

A table lists input values (usually x) and output values (usually y). To check if it's a function, look at the x-column. If any x-value appears more than once with different y-values, it is NOT a function.

Quick table tests

x	y	Function?
1, 2, 3, 4	5, 6, 7, 8	✅ Yes — no x repeats
1, 2, 2, 4	5, 6, 7, 8	❌ No — x = 2 gives 6 AND 7
1, 2, 3, 4	5, 5, 5, 5	✅ Yes — same output is OK

Testing an Equation

An equation like y = 2x + 1 is a function because for any x you plug in, you get exactly one answer for y. But an equation like x² + y² = 25 is NOT a function because one x-value can give two y-values (one positive, one negative).

FUNCTION EXAMPLE

y = 2x + 1

For every x, there is exactly one y. This is a function.

NON-FUNCTION EXAMPLE

x² + y² = 25

When x = 3, y could be 4 or −4. Two outputs for one input means not a function.

💡 Quick Tip

If you can solve the equation for y and get only one expression (no ± sign), it's usually a function. If you see a ± or a square root of both sides, check carefully!

SECTION 5

The Vertical Line Test on Graphs

When you have a graph, there's a super-handy trick called the Vertical Line Test. Imagine sliding a vertical line (like a ruler standing upright) across the graph from left to right. If the line ever touches the graph in more than one point at the same time, then the graph is NOT a function.

Why does this work? A vertical line represents one x-value. If it hits the graph twice, that x-value has two y-values — which breaks the function rule.

The dashed yellow vertical line shows the test. On the left (a curve), it hits only one point — function! On the right (an ellipse), it hits two points — not a function!

Straight lines (except vertical lines) always pass the test. They are functions.
Parabolas opening up or down pass the test. They are functions.
Circles and ellipses fail the test. They are NOT functions.
Vertical lines fail the test — they have infinite outputs for one input. NOT a function.

SECTION 6

Worked Example: Is It a Function?

Let's walk through a complete example. We're given this table and asked: Is the relation a function?

x	y
2	5
4	9
6	13
4	11
8	17

Is This Table a Function?

Step 1 — List the inputsThe x-values (inputs) are: 2, 4, 6, 4, 8.

Step 2 — Look for repeated inputsScan the list. The value x = 4 appears twice.

Step 3 — Check the outputs for the repeated inputWhen x = 4, y = 9 the first time and y = 11 the second time. That's two different outputs for the same input.

Step 4 — Apply the function ruleBecause the input 4 produces two different outputs (9 and 11), the relation breaks the function rule.

This relation is NOT a function.

🤔 What If x = 4 Gave the Same y Both Times?

If both rows said (4, 9), the relation would still be a function. Repeating the same pair is fine — it's only a problem when the same input goes to different outputs.

SECTION 7

Comparing the Four Methods

You now know four ways to check if a relation is a function. Each method has its own strengths. The table below compares them.

Method	What You Look For	Best Used When…
Table	Same x-value with different y-values	You have a list of ordered pairs or data
Graph	Vertical line hits the curve more than once	You can see or draw the graph
Mapping	One input with two or more arrows	You have a small set of pairs and want a clear picture
Equation	Solving for y gives more than one expression (like ±)	You're working with a formula

✦ KEY TAKEAWAY

All four methods are checking the same rule: each input must have exactly one output. Think of these methods like four different detectives investigating the same crime. They each use different clues, but they all reach the same verdict!

SECTION 8

From Here to Function Notation and Beyond

Once you can identify functions, the next step is using function notation. Instead of writing y = 2x + 1, you'll write f(x) = 2x + 1. This tells you that the function is named f and the input is x.

What You Know Now	What Comes Next
Identifying whether a relation is a function	Using f(x) notation to name and evaluate functions
Reading ordered pairs from a table	Finding domain and range from equations
The vertical line test on simple graphs	Graphing linear and nonlinear functions
Checking one equation at a time	Comparing two functions and finding where they intersect

Every skill you build on top of functions — graphing, solving systems, even calculus someday — starts with the simple question you learned today: does every input have exactly one output?

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

In your own words, what is the one rule that makes a relation a function?

PROBLEM 2 — BASIC CALCULATION

Is the following set of ordered pairs a function? {(1, 3), (2, 6), (3, 9), (4, 12)}

PROBLEM 3 — INTERMEDIATE

A table shows: x = {3, 5, 7, 3, 9} and y = {10, 14, 18, 10, 22}. Is this relation a function? Explain carefully.

PROBLEM 4 — APPLIED

A teacher records students and their lunch choices: {(Ana, pizza), (Ben, salad), (Ana, tacos), (Cara, pizza)}. Does this relation represent a function from student to lunch choice? Why or why not?

PROBLEM 5 — CRITICAL THINKING

The equation y² = x describes a relation. Pick a positive value for x, find all possible y-values, and use your result to decide whether the equation represents a function. Explain your reasoning using what you know about functions.

SUMMARY

Lesson Summary

A function is a special relation where every input has exactly one output. You can test this using four methods. In a table, check that no x-value appears with different y-values. In a mapping diagram, make sure no input has more than one arrow leaving it. On a graph, use the vertical line test — if a vertical line ever hits the graph in more than one spot, it's not a function.

For an equation, solve for y. If you get only one expression, it's a function. If you get a ± or two possible values, it's not. Remember: different inputs CAN share the same output. The rule is all about each input pointing to exactly one output. Master this idea, and you're ready for function notation and beyond!