What Are Domain and Range?

A function describes a relationship between:

inputs and outputs.

We usually represent the input using:

x

and the output using:

y or f(x).

For example:

y = 2x + 1

We can choose an input value for x and use the function to calculate an output.

If:

x = 3

then:

y = 2(3) + 1

y = 7

So:

Input = 3

Output = 7

Domain and range describe the possible:

inputs and outputs of a function.

---

Domain

The domain of a function is:

the set of all possible input values.

In most functions, the input is represented by:

x.

A useful way to remember this is:

Domain → Inputs → x-values

If a function can accept x-values of:

1, 2, 3, and 4

then its domain is:

Domain = {1, 2, 3, 4}

---

Range

The range of a function is:

the set of all possible output values.

Outputs are usually represented by:

y or f(x).

Remember:

Range → Outputs → y-values

If the outputs of a function are:

3, 5, 7, and 9

then:

Range = {3, 5, 7, 9}

---

Input → Function → Output

We can think of a function as a:

function machine.

An input enters the machine.

The function performs an operation.

An output is produced.

For:

f(x) = 3x + 2

we could have:

Input: 4

↓

Multiply by 3

↓

Add 2

↓

Output: 14

The input belongs to the:

domain.

The output belongs to the:

range.

---

Domain and Range from a Table

Consider:

The domain consists of the:

x-values.

Therefore:

Domain = {-2, -1, 0, 1, 2}

The range consists of the:

y-values.

Therefore:

Range = {1, 3, 5, 7, 9}

---

A Simple Rule

When reading a table:

Domain → look at the x-column

Range → look at the y-column

This works because x represents the:

input

and y represents the:

output.

---

Repeated Values in the Range

Consider:

The domain is:

Domain = {-2, -1, 0, 1, 2}

The outputs are:

4, 1, 0, 1, 4

But when writing the range as a set, we do not need to repeat values.

Therefore:

Range = {0, 1, 4}

Notice:

Different inputs can produce the same output.

---

Domain and Range from Ordered Pairs

A relation might be written as:

(-3, 2), (-1, 4), (0, 5), (2, 4), (5, 8)

The first number in each ordered pair is:

x.

Therefore:

Domain = {-3, -1, 0, 2, 5}

The second number is:

y.

Therefore:

Range = {2, 4, 5, 8}

The value 4 appears twice as an output, but we only list it:

once.

---

Domain and Range from a Mapping Diagram

A mapping diagram can show inputs and outputs visually.

For example:

Domain

1
2
3
4

↓

Range

3
5
7
9

The values on the input side form the:

domain.

The values that actually receive arrows on the output side form the:

range.

---

Domain and Range from a Graph

A graph gives us a visual way to determine:

domain and range.

To determine the domain, ask:

How far does the graph extend horizontally?

To determine the range, ask:

How far does the graph extend vertically?

A useful memory aid is:

Domain → left to right

Range → bottom to top

---

Reading Domain Horizontally

Imagine shining a light on the graph from:

above.

The shadow the graph creates on the x-axis represents its:

domain.

Every x-value touched by that horizontal spread belongs to the:

domain.

---

Reading Range Vertically

Now imagine shining a light from the:

side.

The shadow the graph creates on the y-axis represents its:

range.

Every y-value covered by the vertical spread belongs to the:

range.

---

Example: A Finite Set of Points

Suppose a graph contains the points:

(-3, 2), (-1, 5), (2, 1), (4, 5)

Read the x-coordinates:

-3, -1, 2, 4

Therefore:

Domain = {-3, -1, 2, 4}

Read the y-coordinates:

2, 5, 1, 5

Therefore:

Range = {1, 2, 5}

---

Example: A Line Segment

Suppose a line segment begins at:

(-2, 1)

and ends at:

(4, 7).

If both endpoints are included, then the x-values extend from:

-2 to 4.

Therefore:

Domain: -2 ≤ x ≤ 4

The y-values extend from:

1 to 7.

Therefore:

Range: 1 ≤ y ≤ 7

---

Closed Points

A closed point means that the endpoint is:

included.

For example:

x ≥ 2

includes:

x = 2.

We use:

≥

rather than:

>.

Similarly:

x ≤ 5

includes:

x = 5.

---

Open Points

An open point means the endpoint is:

not included.

For example:

x > 2

does not include:

x = 2.

Similarly:

x < 5

does not include:

x = 5.

---

Inequality Notation

Domain and range are often expressed using:

inequalities.

Examples:

x ≥ 0

means x can be:

0 or any larger value.

x < 5

means x can be:

any value less than 5, but not 5 itself.

-3 ≤ x ≤ 7

means x can be any value from:

-3 through 7, including both endpoints.

---

Interval Notation

Another way to express domain and range is:

interval notation.

For example:

-3 ≤ x ≤ 7

can be written:

[-3, 7]

Square brackets mean:

endpoint included.

For:

-3 < x < 7

we write:

(-3, 7)

Parentheses mean:

endpoint not included.

---

Mixed Endpoints

Suppose:

-3 ≤ x < 7

Then:

-3 is included

but:

7 is not included.

Interval notation:

[-3, 7)

The bracket matches the:

included endpoint.

The parenthesis matches the:

excluded endpoint.

---

Infinity

Suppose the domain is:

x ≥ 2

This means the graph begins at x = 2 and continues indefinitely to the:

right.

We can write:

Domain: x ≥ 2

or:

Domain: [2, ∞)

Infinity is not an actual endpoint.

Therefore, we always use a:

parenthesis with infinity.

---

Domain of an Equation

Sometimes the domain can be determined directly from the:

equation.

Consider:

f(x) = 2x + 5

Can we substitute:

x = 1?

Yes.

Can we substitute:

x = -100?

Yes.

Can we substitute:

x = 0?

Yes.

There is no obvious restriction on x.

Therefore:

Domain = all real numbers

or:

Domain: -∞ < x < ∞

---

Linear Functions

Most ordinary linear functions have:

all real numbers

as their domain.

For example:

f(x) = 4x - 7

Any real value of x can be substituted into the:

equation.

The graph also extends forever to the:

left and right.

For this function:

Domain = all real numbers

and:

Range = all real numbers.

---

Quadratic Functions

Consider:

f(x) = x²

Any real value can be used for:

x.

Therefore:

Domain = all real numbers

However, squaring a real number cannot produce a:

negative result.

So:

Range: y ≥ 0

The graph extends forever:

left and right,

but its lowest y-value is:

0.

---

Another Quadratic Example

Consider:

f(x) = x² + 3

The parabola has been shifted:

3 units upward.

The domain is still:

all real numbers.

But the smallest possible output is:

3.

Therefore:

Range: y ≥ 3

---

Square Root Functions

Consider:

f(x) = √x

Can we substitute:

x = 4?

Yes.

√4 = 2

Can we substitute:

x = 0?

Yes.

√0 = 0

Can we substitute:

x = -4?

Not if we are working with real numbers.

Therefore:

Domain: x ≥ 0

The outputs are also non-negative.

Therefore:

Range: y ≥ 0

---

Why Square Roots Can Restrict the Domain

For real-valued functions:

the quantity inside an even square root cannot be negative.

Consider:

f(x) = √(x - 3)

We need:

x - 3 ≥ 0

Therefore:

x ≥ 3

So:

Domain: x ≥ 3

---

Worked Example: Square Root Restriction

Find the domain of:

f(x) = √(x + 5)

The expression inside the square root must be:

greater than or equal to zero.

Therefore:

x + 5 ≥ 0

Subtract 5:

x ≥ -5

Therefore:

Domain: x ≥ -5

---

Fractions Can Restrict the Domain

Consider:

f(x) = 1 ÷ x

What happens when:

x = 0?

We would have:

1 ÷ 0

Division by zero is:

undefined.

Therefore:

x cannot equal 0.

So:

Domain: x ≠ 0

Notice that the graph never touches:

x = 0.

---

Denominators Cannot Equal Zero

This gives us an important rule:

The denominator of a fraction cannot equal zero.

Consider:

f(x) = 5 ÷ (x - 2)

We cannot allow:

x - 2 = 0

Therefore:

x ≠ 2

So:

Domain = all real numbers except 2

---

Worked Example: Rational Function

Find the domain of:

f(x) = 3 ÷ (x + 4)

The denominator cannot equal zero.

Set:

x + 4 = 0

Therefore:

x = -4

So:

Domain: x ≠ -4

Every other real value of x is allowed.

---

Two Common Domain Restrictions

When finding domain from an equation, two especially important restrictions are:

Division

You cannot divide by:

zero.

Even Square Roots

You cannot take the square root of a negative number when working with:

real numbers.

These restrictions allow us to determine the domain of many functions.

---

Domain Does Not Always Mean All Mathematically Possible Values

Real-world situations can create additional:

restrictions.

Suppose:

C = 5n + 20

represents the cost of renting bicycles, where n is the number of bicycles rented.

Mathematically, we could substitute:

n = -10.

But what would:

-10 bicycles

mean?

It makes no sense in this context.

Therefore, the real-world domain is more restricted than the equation alone might suggest.

---

Domain in Context

Suppose:

C = 5n + 20

where n represents the number of bicycles.

Reasonable values might be:

n = 0, 1, 2, 3, 4, ...

The domain consists of:

non-negative whole numbers.

Why not 2.7?

Because we cannot normally rent:

2.7 bicycles.

Context determines which mathematical values are:

meaningful.

---

Range in Context

Using:

C = 5n + 20

if:

n = 0

then:

C = 20

If:

n = 1

then:

C = 25

If:

n = 2

then:

C = 30

So possible outputs include:

20, 25, 30, 35, ...

These values form the:

range.

---

Discrete and Continuous Domains

A domain can be:

discrete

or:

continuous.

A discrete domain contains separate values.

Example:

number of students

Possible values:

0, 1, 2, 3, ...

You cannot normally have:

14.6 students.

---

Continuous Domains

A continuous domain can contain any value within an:

interval.

Examples include:

• time
• temperature
• distance
• mass
• height

If time can vary from 0 to 10 seconds, then values such as:

2 s

2.5 s

2.537 s

may all be possible.

This creates a:

continuous domain.

---

Worked Example: A Thrown Ball

Suppose the height of a ball is modeled by:

h = -5t² + 20t + 1

where:

t = time in seconds

and:

h = height in metres.

Mathematically, the quadratic equation accepts negative values of t.

But:

negative time before the ball was thrown

may not belong to the situation being modeled.

If the model begins when the ball is thrown, then:

t ≥ 0

until the ball reaches the ground.

The physical situation determines the meaningful:

domain.

---

Domain and Range Have Meaning

Domain and range are not simply mathematical rules.

They answer two useful questions:

Domain

What inputs make sense or are allowed?

Range

What outputs are possible?

This becomes especially important when functions represent:

real situations.

---

Example: Movie Tickets

Suppose:

C = 12n

where:

n = number of tickets

and:

C = total cost in dollars.

If a cinema allows one person to purchase at most 8 tickets, then:

Domain = {0, 1, 2, 3, 4, 5, 6, 7, 8}

The corresponding range is:

Range = {0, 12, 24, 36, 48, 60, 72, 84, 96}

The context creates both:

restrictions and meaning.

---

Example: Filling a Tank

Suppose a tank begins empty and fills at:

5 litres per minute

for 20 minutes.

Let:

V = 5t

where:

t = time in minutes

and:

V = volume of water in litres.

The meaningful domain is:

0 ≤ t ≤ 20

The corresponding range is:

0 ≤ V ≤ 100

Because time and volume can vary continuously, these intervals contain:

all values between the endpoints.

---

Domain from a Graph

When examining a graph, start at its:

leftmost point.

Then move toward the:

rightmost point.

Ask:

Which x-values does the graph contain?

Those values form the:

domain.

If arrows show that the graph continues forever, the domain may extend toward:

infinity.

---

Range from a Graph

Now examine the graph from:

bottom to top.

Ask:

Which y-values does the graph contain?

Those values form the:

range.

Look carefully for:

• highest points
• lowest points
• open endpoints
• closed endpoints
• arrows
• gaps

These features determine whether particular values are:

included.

---

Worked Example: Reading Endpoints

Suppose a graph begins with a closed point at:

(-4, -2)

and ends with an open point at:

(6, 5).

Assume the graph covers all x-values and y-values between these endpoints.

For the domain:

-4 is included

6 is not included

Therefore:

Domain: -4 ≤ x < 6

For the range:

-2 is included

5 is not included

Therefore:

Range: -2 ≤ y < 5

---

Domain and Range of Absolute Value

Consider:

f(x) = |x|

Any real number can be substituted for:

x.

Therefore:

Domain = all real numbers

Absolute value cannot produce a negative output.

Therefore:

Range: y ≥ 0

The graph extends left and right forever but has a minimum value at:

y = 0.

---

Comparing Common Functions

Recognizing these common patterns makes domain and range much easier to:

determine.

---

A Useful Domain Checklist

When given an equation, ask:

Is there a denominator?

If yes:

What values make the denominator zero?

Exclude those values.

Is there an even square root?

If yes:

What values make the expression inside the root negative?

Exclude those values.

Is there a real-world context?

If yes:

Which values actually make sense?

Apply those restrictions.

Are there no restrictions?

Then the domain may be:

all real numbers.

---

A Useful Range Checklist

Finding range from an equation can sometimes be more difficult.

Ask:

Does the function have a minimum?

Example:

y = x²

Minimum:

y = 0

So:

y ≥ 0

Does the function have a maximum?

A downward-opening parabola may have a:

maximum output.

Are some y-values impossible?

Example:

y = 1 ÷ x

The output can never equal:

0.

Does the context restrict outputs?

Example:

A height cannot reasonably be:

negative.

---

Domain and Range from Real Data

Suppose a table records the temperature of water during an experiment.

The recorded domain is:

{0, 2, 4, 6, 8, 10}

The recorded range is:

{20, 29, 38, 47, 56, 65}

But if the function models temperature continuously throughout the experiment, the meaningful domain may instead be:

0 ≤ t ≤ 10

This illustrates an important distinction between:

recorded data points and the underlying situation.

---

Common Mistake: Switching Domain and Range

A common mistake is to reverse:

domain and range.

Remember:

Domain = x = input

Range = y = output

On a graph:

Domain → horizontal

Range → vertical

---

Common Mistake: Listing Repeated Values

Suppose the outputs are:

3, 5, 5, 7, 7, 7

When writing the range as a set:

Range = {3, 5, 7}

Each possible output only needs to be listed:

once.

---

Common Mistake: Ignoring Restrictions

Consider:

f(x) = 4 ÷ (x - 6)

It would be incorrect to say:

Domain = all real numbers

because:

x = 6

would make the denominator:

zero.

The correct domain is:

x ≠ 6.

---

Common Mistake: Ignoring Context

Suppose:

P = 8n

represents the price of n notebooks.

A purely algebraic interpretation might allow:

n = -4

or:

n = 2.7.

But neither represents a sensible number of notebooks.

The mathematical model must be interpreted within:

its context.

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Why Domain and Range Matter

Domain and range help us understand the:

limits of a mathematical model.

They tell us:

• which inputs are possible
• which outputs are possible
• where a graph exists
• where an equation is defined
• which values make sense in context

Without domain and range, we may apply a function to values for which it was never:

intended.

---

Domain and Range in Science

Suppose a scientist models the growth of a plant during a:

30-day experiment.

The time domain might be:

0 ≤ t ≤ 30

Even if the mathematical equation could accept:

t = 1,000,

the model may not be valid that far into the:

future.

Mathematical models have:

limits.

---

Domain and Range in Business

Suppose a company can manufacture at most:

5,000 products per month.

If x represents the number of products manufactured, then the meaningful domain might be:

0 ≤ x ≤ 5000

If only whole products can be manufactured, x would also need to be:

a whole number.

Domain restrictions can therefore represent:

real limitations.

---

Domain and Range in Geometry

Suppose the perimeter of a square is:

P = 4s

where:

s = side length.

Mathematically, the equation works for negative s-values.

But a physical side length cannot be:

negative.

Therefore:

s > 0

for an actual square.

The corresponding perimeter must also satisfy:

P > 0.

---

The DR Method

A simple strategy is the:

DR Method.

D — Domain

Ask:

What x-values or inputs are possible?

Look:

left to right.

R — Range

Ask:

What y-values or outputs are possible?

Look:

bottom to top.

Then check:

Are there mathematical or real-world restrictions?

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Check Your Understanding

1. Define domain.

2. Define range.

3. Which variable usually represents the domain?

4. Which variable usually represents the range?

5. Find the domain of the table:

6. Find the range of the same table.

7. Find the domain of:

(-2, 5), (1, 7), (4, 5), (8, 10)

8. Find the range of the ordered pairs above.

9. Why do repeated output values only need to be listed once in the range?

10. What is the domain of:

f(x) = 5x - 3

11. What is the range of:

f(x) = x²

12. Find the domain of:

f(x) = √x

13. Find the domain of:

f(x) = √(x - 7)

14. Find the domain of:

f(x) = 1 ÷ x

15. Find the domain of:

f(x) = 4 ÷ (x + 3)

16. Explain why division by zero creates a domain restriction.

17. Explain why an even square root can create a domain restriction.

18. A line segment extends from x = -5 to x = 8 with both endpoints included. State its domain.

19. A graph extends from y = 2 upward forever. State its range.

20. Explain the difference between a discrete and continuous domain.

21. A concert venue sells at most 500 tickets. If n represents tickets sold, describe a sensible domain for n.

22. A tank fills for 15 minutes. If t represents time since filling began, describe the domain.

23. Explain why the domain of a mathematical equation may differ from the meaningful domain of a real-world model.

24. What does an open point on a graph tell you about domain or range?

25. Explain how you can determine domain and range visually from a graph.

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Key Terms

• Function: Relationship in which each allowed input has exactly one output.
• Domain: Set of all possible input values of a function.
• Range: Set of all possible output values of a function.
• Input: Value entered into a function.
• Output: Value produced by a function.
• Independent variable: Variable whose value is used as the input.
• Dependent variable: Variable whose value depends on the input.
• Ordered pair: Pair of coordinates written as (x, y).
• Mapping diagram: Diagram showing connections between inputs and outputs.
• Restriction: Condition preventing certain values from belonging to a domain or range.
• Real numbers: Numbers represented on the ordinary number line.
• Discrete: Consisting of separate individual values.
• Continuous: Containing all possible values throughout an interval.
• Endpoint: Value marking the beginning or end of an interval.
• Closed point: Endpoint that is included.
• Open point: Endpoint that is excluded.
• Inequality notation: Method of describing a set of values using symbols such as <, >, ≤, and ≥.
• Interval notation: Method of describing continuous sets of numbers using brackets and parentheses.
• Undefined: Mathematical expression that has no permitted value under the given number system or operation.
• Mathematical model: Function or equation used to represent a real-world situation.

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Key Takeaways

• The domain is the set of possible inputs of a function.
• The range is the set of possible outputs.
• Domain usually corresponds to x-values.
• Range usually corresponds to y-values.
• From a table, domain comes from the x-column and range comes from the y-column.
• From ordered pairs, domain comes from the first coordinates and range from the second coordinates.
• Repeated values only need to appear once when writing a set.
• On a graph, read domain from left to right.
• On a graph, read range from bottom to top.
• Closed endpoints are included.
• Open endpoints are excluded.
• Linear functions such as y = 2x + 3 usually have all real numbers as both their domain and range.
• The function y = x² has all real numbers as its domain but y ≥ 0 as its range.
• The function y = √x has x ≥ 0 as its real-number domain.
• A denominator can never equal zero.
• Therefore, rational functions may have excluded domain values.
• Even square roots cannot contain negative values when working with real numbers.
• Real-world contexts can create additional domain restrictions.
• Counts such as numbers of people or objects usually have discrete domains.
• Measurements such as time, distance, and temperature can have continuous domains.
• The mathematical domain of an equation and the meaningful domain of a model are not always the same.
• Domain asks: What inputs are allowed?
• Range asks: What outputs are possible?
• Understanding domain and range helps us determine both the mathematical and practical limits of a function.