Describe domain and range and apply contextual restrictions

Ontario Grade 11 Mathematics

Characteristics of Functions

MCR3U · Functions · Expectation A1.3

Every function you meet in this course takes input values and produces output values. Two simple questions help you understand any function: What inputs are allowed? What outputs can actually come out? The answers to those two questions are the domain and the range. Once you can read domain and range from an equation or a graph, you will also learn how to tighten them when the function models something real — because not every mathematically valid input makes sense in a real-world story. This lesson builds those skills step by step.

What you will learn

Explain what domain and range mean for a function in plain language and with correct notation.
Identify the domain and range of a function from its equation, a table of values, or a graph.
Recognize why real-world situations force restrictions on domain and range that the bare equation would not impose.
Write domain and range using set notation and number-line inequalities.

Prerequisite Bridge: Functions and Function Notation

Before Grade 11, you learned that a function is a special relationship in which each input produces exactly one output. The notation

f(x)

is shorthand for 'the output of function

f

when the input is

x

.' You do not need anything beyond this idea to start today's lesson.

Think of a function as a vending machine. You press a button (the input), and one specific item drops out (the output). The set of all buttons that actually work is like the domain. The set of all items the machine can deliver is like the range.

A function assigns exactly one output to each input.
$f(x)$ names the output when the input is $x$ .
Domain = all valid inputs; Range = all possible outputs.

What Domain and Range Mean

The domain of a function is the complete set of input values (

x

-values) for which the function is defined — meaning it actually produces a real-number output. The range is the complete set of output values (

y

-values, or

f(x)

-values) that the function can produce when you use inputs from the domain.

When you look at a graph, the domain is everything the curve covers left to right along the horizontal axis, and the range is everything it covers up and down along the vertical axis. When you have an equation, you look for any

x

-value that would cause a problem — for example, creating a zero in a denominator or a negative number under a square root — and you exclude those values from the domain.

We write domain and range using inequality notation or set notation. For example, 'all real numbers greater than or equal to zero' can be written as

x \geq 0

or as \{x ∈

\mathbb{R}

\mid x ≥ 0\}. Both forms are acceptable in this course. The symbol

\mathbb{R}

stands for the set of all real numbers.

\{x ∈ \mathbb{R} \mid a ≤ x ≤ b\}

Domain: the set of all $x$ -values for which $f(x)$ is defined.
Range: the set of all $f(x)$ -values the function actually produces.
On a graph, domain reads left–right and range reads up–down.
Exclude any $x$ -value that makes the expression undefined (e.g., division by zero).
Use inequality or set notation to record your answer.

Finding Domain and Range from Equations and Graphs

For a linear function such as

f(x) = 3x - 5

, every real number can be substituted for

x

and produces a real output. So the domain is all real numbers, written x ∈

\mathbb{R}

, and the range is also all real numbers, f(x) ∈

\mathbb{R}

.

For a quadratic function such as

g(x) = x^2 - 4

, every real number is still a valid input, so the domain remains x ∈

\mathbb{R}

. The output, however, is always

x^2

minus

4

. Because

x^2 \geq 0

for all real

x

, the smallest possible output is

0 - 4 = -4

, achieved when

x = 0

. The parabola opens upward, so the range is

g(x) \geq -4

.

For a square-root function such as

h(x) = \sqrt{x + 3}

, the expression under the root must be non-negative for

h(x)

to be a real number. Setting

x + 3 \geq 0

gives

x \geq -3

. So the domain is \{x ∈

\mathbb{R}

\mid x ≥ -3\}. Since a square root always produces a non-negative result, the range is

h(x) \geq 0

.

Reading a graph: find the leftmost and rightmost

x

-values the curve reaches (or extends to) — that span is the domain. Find the lowest and highest

y

-values — that span is the range. An open circle at an endpoint means that point is excluded; a closed circle means it is included; an arrow means the graph continues without bound.

x + 3 ≥ 0 \implies x ≥ -3

Linear functions: domain and range are both all real numbers.
Quadratic (upward parabola): domain is all reals; range has a minimum value.
Square-root function: set the radicand $\geq 0$ to find the domain restriction.
Open circle = endpoint excluded; closed circle = endpoint included; arrow = unbounded.

Contextual Restrictions

An equation taken on its own lives in a purely mathematical world where any valid input is fair game. But when a function models a real situation, the story adds extra rules that the algebra alone cannot see. These are called contextual restrictions.

Consider a function that models the height, in metres, of a ball thrown upward. The mathematical formula might allow negative time values, but physically this is impossible. Time cannot be negative, and height below the ground is usually impossible once the ball has landed. You restrict the domain to

t \geq 0

and to the interval before the ball hits the ground.

Consider a business model where

C(n) = 50n + 200

gives the cost, in CAD, of producing

n

custom T-shirts. Mathematically,

n

could be any real number. In context, you cannot produce a fraction of a shirt or a negative number of shirts, so the domain is limited to whole numbers: n ∈ \{0, 1, 2, 3, \dots\}.

Always ask three questions when applying a contextual restriction: (1) Can the input be negative? (2) Can the input be a non-integer (decimal)? (3) Is there a physical or logical maximum on the input?

Contextual restrictions come from the real-world meaning of the variables.
Time, length, and count are common sources of restrictions.
Narrowing the domain always narrows or keeps the range the same.
State restrictions explicitly; do not assume the reader knows the context.

Function Types Quick Reference

Type	Domain	Range
Linear	x in R	y in R
Quadratic (Up)	x in R	y >= min
Quadratic (Down)	x in R	y <= max
Square Root	x >= boundary	y >= 0

Worked example

Example 1 — Quadratic Function with a Contextual Restriction

A soccer ball is kicked from ground level. Its height in metres after

t

seconds is

H(t) = -5t^2 + 20t

. State the mathematical domain and range, then apply contextual restrictions.

Determine mathematical domain
Quadratic functions are defined for all real numbers.
$t ∈ \mathbb{R}$
Find vertex
The vertex of a quadratic occurs at the time calculated by the formula $-b/(2a)$ .
$t = -20 / (2 × -5) = 2$
Determine mathematical range
The maximum height is the function value at the vertex, $H(2) = -5(4) + 20(2) = 20$ .
$H(t) ≤ 20$
Apply contextual domain
Time starts at $0$ . The ball hits the ground when $H(t) = 0$ , which occurs at $t = 4$ .
$0 ≤ t ≤ 4$
Determine contextual range
Since the ball rises to $20$ m and falls back to $0$ m, the height is restricted.
$0 ≤ H(t) ≤ 20$

Answer: Math: t ∈

\mathbb{R}

,

H(t) \leq 20

. Contextual:

0 \leq t \leq 4

,

0 \leq H(t) \leq 20

.

Check: At

t=0, H=0

; at

t=2, H=20

; at

t=4, H=0

.

Worked example

Example 2 — Square-Root Function

A biologist models leaf width

W(A) = \sqrt{2A - 6}

where

A

is area. Sample areas are between

3

and

23

. Find mathematical and contextual domains.

Find mathematical domain
The radicand must be non-negative.
$2A - 6 ≥ 0$
Solve domain
Solving the inequality shows the smallest possible area.
$A ≥ 3$
Define contextual domain
The sample constraints add an upper bound to the mathematical domain.
$3 ≤ A ≤ 23$
Determine contextual range
Evaluate the function at both domain endpoints.
$W(3)=0, W(23)=\sqrt{40}$

Answer: Math:

A \geq 3

,

W \geq 0

. Contextual:

3 \leq A \leq 23

,

0 \leq W \leq \sqrt{40}

.

Check: For

A=3

,

W=0

. For

A=23

,

W=\sqrt{40} \approx 6.32

.

Common mistakes and how to avoid them

Using range for domain.

Correction: Always remember domain is horizontal (

x

-axis) and range is vertical (

y

-axis).

Ignoring context.

Correction: Always apply real-world constraints like non-negative time or quantity.

Lesson summary

Domain is the set of all valid input values.
Range is the set of all resulting output values.
Mathematical restrictions involve denominators and roots.
Contextual restrictions include physical limits like time and count.
Always state the variable when writing notation.

Check your understanding

Question 1

What is the domain of f(x) = sqrt(x - 5)?

x > 5
x >= 5
x <= 5
x in R

Show answer and explanation

x >= 5

The expression inside the root must be >= 0.

Question 2

Which set notation is correct for x >= 0?

{x in R | x >= 0}
{x in R | x > 0}
{x in R | x <= 0}
{x in R | x = 0}

Show answer and explanation

{x in R | x >= 0}

Set notation identifies the variable, the set it belongs to, and the condition.

Key terms

Domain: The set of all possible input values for a function.
Range: The set of all possible output values for a function.

Continue through MCR3U

View the complete MCR3U Ontario Grade 11 Mathematics curriculum and lessons

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation A1.3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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