MCR3U Functions · Unit 1 — Introduction to Functions

Lesson 1.1

Relations & Functions

Press a button, get one item. That simple idea is what separates a relation from a function — and you will spot it in pairs, tables, diagrams and graphs.

Estimated time: 30–45 minutes

Lesson progress0%

Learning goals

identify inputs and outputs
explain what a relation is
explain what makes a relation a function
identify functions from ordered pairs, tables, mapping diagrams, graphs, and equations
use the vertical-line test
explain why a relation is or is not a function

Step 1 of 4

Warm Up

Quick Check — a couple of minutes on what you already know.

You’ve worked with inputs and outputs before. Let’s see what you remember.

Question 1

y = 2x + 3

If the input is x = 4, what is the output?

Your answer

Question 2

(5, 12)

Which number is the input?

a5b12

Putting something in and getting something out is going to be important in this lesson.

Step 2 of 4

Learn

What’s a function? Start with a machine you already know — then see it, work through an example, and try it yourself.

Real-world idea

Think About a Vending Machine

Vending machine

A1Chips
A2Water
A3Granola Bar

Press a button, one item drops.

You press A2. What should come out? Water.

Think of A2 as the INPUT and water as the OUTPUT.

INPUT → OUTPUT

Quick check

Imagine pressing A2 could give you water OR juice. Would A2 have one clear output?

aYesbNo

That idea leads us to two important math terms.

Key term

Relation

A relationship between a set of inputs and outputs.

(x, y)

Relations are often written as ordered pairs. The first value is the input. The second value is the output.

So (2, 7) means 2 → 7.

Quick check

(5, 12)

In this ordered pair, what is the input?

a5b12

Key term

Function

A relation where each input has exactly one output.

ONE INPUT → ONE OUTPUT

This one line decides every question in this lesson.

Different inputs CAN have the same output

Back at the vending machine, imagine the buttons are set up like this.

Vending machine buttons

Function

This is still okay. Each button has only one output. The problem happens when ONE input points to TWO different outputs.

Curriculum connection

These are the words your teacher and the Ontario curriculum use. Learn them here and use them when you explain your thinking.

Curriculum language · A1.1Relation: What this meansA relation is any connection between a set of inputs and a set of outputs.
Curriculum language · A1.1Function: What this meansA function is a relation where each input is connected to exactly one output.

Does every input have exactly one output?

No matter how a relation is shown, this is the only question you ask.

A. Ordered pairs

Read the first number in each pair — that is the input.

Example 1

(1, 4)
(2, 6)
(3, 8)
(4, 10)

Function

Every input has exactly one output.

Example 2

(1, 4)
(2, 6) — repeated input
(2, 9) — repeated input
(3, 11)

2 → 6 and 2 → 9

Not a function

The same input has two different outputs.

Quick check

(1, 5), (2, 8), (3, 8), (4, 10)

Function or not a function?

aFunctionbNot a function

Keep these two straight

Repeated outputs = OK. One input with different outputs = NOT a function.

B. Tables

Same question — read down the x-column first.

Table 1
x	y
1	5
2	7
3	9
4	11

Function

Each x-value appears once, with one y-value.

Table 2
x	y
1	5
2 — repeated input	7
2 — repeated input	10
3	12

Not a function

Input 2 has two different outputs.

Your turn
x	y
-2	4
-1	1
0	0
1	1
2	4

Quick check

x: −2, −1, 0, 1, 2 · y: 4, 1, 0, 1, 4

Is this table a function?

aFunctionbNot a function

Try it

(2, 5), (3, 7), (4, 9), (5, 11)

Function or not a function?

aFunctionbNot a function

C. Mapping diagrams

Now the arrows do the work. Count how many arrows leave each input.

Example 1

Function

Two different inputs can point to the same output.

Example 2

Not a function

One input has two different outputs.

Quick check

A mapping diagram shows: 2 → 5, 4 → 5, 6 → 9.

Function or not a function?

aFunctionbNot a function

Try it

(1, 4), (1, 6), (2, 8), (3, 10)

Function or not a function?

aFunctionbNot a function

D. Graphs and the vertical-line test

Vertical-line test: a graph represents a function if no vertical line crosses the graph more than once.

Here is why that works. A vertical line represents one x-value. If the line touches the graph twice, that one x-value has two y-values. That breaks the function rule: ONE INPUT → ONE OUTPUT.

Drag the dashed line across each graph, or use the slider and arrow keys. Watch the marked points, then decide.

Graph 1 — a straight liney = x + 1

Move the vertical line (x = -1)

The line touches the graph 1 time here.

Is this a function?

Graph 2 — a parabola opening upy = 0.4x² − 3

Move the vertical line (x = -1)

The line touches the graph 1 time here.

Is this a function?

Graph 3 — a circlex² + y² = 16

Move the vertical line (x = -1)

The line touches the graph 2 times here.

Is this a function?

Graph 4 — a sideways parabolax = 0.3y² − 3

Move the vertical line (x = -1)

The line touches the graph 2 times here.

Is this a function?

A Function at Work

Suppose you earn $18 per hour. Your pay follows a rule, and that rule is a function.

Here h is the hours you worked and E(h) is what you earn.

E(h) = 18h

E(5) = 18(5)Work 5 hours — put 5 in for h.
E(5) = 905 hours → $90
E(10) = 18010 hours → $180

For each number of hours worked, this rule gives one amount of pay. So earnings are a function of hours worked.

Quick check

You earn $18 per hour, so E(h) = 18h.

At this same rate, could 5 hours give both $90 and $150?

aYesbNo

Try it — putting it together

One more set before you practise on your own.

Try it

A mapping diagram shows: 1 → 7, 2 → 7, 3 → 7.

Function or not a function?

aFunctionbNot a function

Step 3 of 4

Practice

Mixed forms, more on your own. Hints are here if you want them — tap “Show me a hint”.

Build it

Spot functions in three different forms.

Ordered pairs

(0, 3), (1, 5), (2, 7), (0, 9)

Function or not a function?

aFunctionbNot a function

Table

x: 1, 2, 3, 4 · y: 6, 6, 6, 6

Function or not a function?

aFunctionbNot a function

Mapping diagram

A mapping diagram shows: 5 → 1, 5 → 2, 6 → 3.

Function or not a function?

aFunctionbNot a function

Apply it

Same rule, real situation.

Profile pictures on a streaming service

Apply it

A streaming service lets each account choose one profile picture. Account A → Dog, Account B → Cat, Account C → Dog, Account D → Panda

Does this represent a function from account → selected profile picture?

aYesbNo

Account B is now listed twice

Apply it

Now the records show: Account B → Cat AND Account B → Lion

Is this still a function from account → selected profile picture?

aYesbNo

Challenge

Take your time with this one.

Explore the relation x = y². Fill in some values before you decide.

Some pairs that fit x = y²
x	y
0	0
1	1
1	-1
4 — repeated input	2
4 — repeated input	-2

Challenge

x = y²

Is y a function of x?

aYesbNo

Step 4 of 4

Mastery Check

Five questions, on your own. No hints here — answer them all, then see how you did.

Think About a Vending Machine

Relation

Function

Different inputs CAN have the same output

A. Ordered pairs

B. Tables

C. Mapping diagrams

D. Graphs and the vertical-line test

A Function at Work

Try it — putting it together

Build it

Apply it

Challenge

Lesson 1.1 mastery check

Relations & Functions