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MCF3M Functions & Applications · Unit 1 — Introduction to the Quadratic Function

Lesson 1.1

The Characteristics of a Function

Not every relationship is predictable. Learn what makes a relation a function, how to spot one in any form, and how to describe its domain and range.

Estimated time: 30–45 minutes

Lesson progress0%

Learning goals

explain the difference between a relation and a function
identify independent and dependent variables (input and output)
decide whether a relation is a function from pairs, tables, mappings, graphs, equations and descriptions
state the domain and range of a relation
use the vertical line test and explain why it works

Step 1 of 4

Warm Up

What makes a machine predictable?

Real-world idea

The broken ticket kiosk

You walk up to a theatre kiosk. You press a button for your ticket type, and the screen shows a price.

Press Child, get a price. Press Adult, get a price. You trust that the same button always gives the same price.

Warm Up

A movie theatre kiosk has three buttons: Child, Adult, Senior. On Monday you press Child and the screen says $9. Your friend presses Child a minute later and the screen says $13.

Why would the theatre call this kiosk broken?

aTwo different buttons gave the same price.bThe same button gave two different prices, so you can't predict what Child costs.cPrices should never repeat.dThere are too many buttons.

Keep this in mind

A useful rule always gives ONE predictable result for each input. That idea is exactly what a function is.

Step 2 of 4

Learn

Relations, functions, representations, domain and range.

Part 1 — Relations

A relation is any connection between two quantities. A phone battery and the minutes you stream video are related: the more you stream, the lower the battery.

The quantity you choose or control is the independent variable — the input. The quantity that responds is the dependent variable — the output.

Streaming time and battery level
minutes streamed	battery (%)
0	100
30	88
60	76
90	64

Key term

Independent and dependent variables

The independent variable is the input. The dependent variable is the output — it depends on the input.

Minutes streamed (input) → battery level (output).

Try it

A café tracks how many litres of milk it uses each day based on the number of lattes it sells.

Which is the independent variable (the input)?

aLitres of milk usedbNumber of lattes soldcThe day of the weekdThe price of a latte

Part 2 — Functions

Some relations are predictable and some are not. A function is the predictable kind.

A function is a relation where every input has exactly one output.

Student → birth month

Function

Each student has exactly one birth month. Amir and Chen share March — that's fine.

Student → sport they play

Not a function

Dana has two outputs, so you can't predict “Dana's sport”. Not a function.

The most common mistake

Different inputs CAN share the same output. Only one input with two different outputs breaks the rule.

Try it

Relation: {(1, 7), (2, 7), (3, 7)}

This relation is a function.

TrueFalse

Part 3 — Spotting functions in any form

The rule never changes. What changes is where you look for the inputs.

A. Ordered pairs — look at the first numbers

(2, 5) — repeated input
(4, 1)
(2, -3) — repeated input
(6, 0)

2 → 5 and 2 → -3

Not a function

The input 2 appears with 5 and with −3.

B. Table — look down the input column
x	y
-1	4
0	4
1	6
2	9

Function

No x-value repeats. The repeated 4 is a shared output.

C. Mapping diagrams — count the arrows leaving each input (you saw these above).

D. Graphs — use the vertical line test (Part 5).

E. Equations — test an input. In y = 2x + 7, any x gives one y. In x = y², x = 16 gives y = 4 and y = −4, so it is not a function.

F. Descriptions — ask “could one input have two outputs?” A car (input) and its licence plate (output) is a function. A city (input) and the streets in it (output) is not.

Try it

Which equation is NOT a function?

ay = 4x − 9by = x² + 1cx = y²dy = −2

Part 4 — Domain and range

The domain is the set of all inputs that are allowed. The range is the set of all outputs that result.

Think “what can go in?” and “what comes out?” — not just “x and y”.

Worked example

State the domain and range of {(−2, 5), (0, 3), (1, 3), (4, −1)}.

Step 1
Collect the inputs.
D = {−2, 0, 1, 4}
Step 2
Collect the outputs, listing 3 only once.
R = {−1, 3, 5}

Answer: D = {−2, 0, 1, 4}, R = {−1, 3, 5}

y = x²: the graph runs forever left and right, but never goes below 0

f(x) = x²

On a graph, read the domain from left to right and the range from bottom to top.

For y = x², every x can go in, so D = {x ∈ ℝ}. The outputs are never negative, so R = {y ∈ ℝ | y ≥ 0}.

Try it

Relation: {(−3, 4), (0, 1), (2, 4), (5, −2)}

What is the domain?

a{−2, 1, 4}b{−3, 0, 2, 5}c{−3, 5}d{4, 1, 4, −2}

Part 5 — The vertical line test

Every point on a vertical line has the same x-value. If a vertical line crosses a graph twice, one input is paired with two outputs — so the graph is not a function.

Drag the line across each graph. Watch how many times it crosses.

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?

Not just lines and parabolas

Lines (except vertical ones) and parabolas pass the test. A circle or a sideways parabola fails. Many other curved shapes pass too — always test, don't guess from the shape's name.

Try it

A vertical line drawn at x = 2 crosses a graph at (2, 1) and (2, −3). What can you conclude?

aThe graph is a function because it crosses at two points.bThe graph is not a function: the input 2 has two outputs.cNothing — vertical lines don't tell us anything.dThe graph must be a straight line.

Step 3 of 4

Practice

From identifying to explaining.