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.

---

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?

---

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.

---

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.

---

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.

One Relationship, Different Representations

A function describes a relationship between:

inputs and outputs.

The same function can be represented in several different ways.

For example, imagine the rule:

Multiply the input by 2, then add 1.

We could represent this using an equation:

f(x) = 2x + 1

We could represent it using a table:

We could also represent it using a:

graph.

Although these representations look different, they all describe exactly the:

same mathematical relationship.

---

The Main Representations of a Function

Functions can commonly be represented using:

• words
• tables
• equations
• graphs
• ordered pairs
• mapping diagrams

In this topic, we will concentrate mainly on:

tables, equations, and graphs.

Being able to move between these representations is an important part of:

understanding functions.

---

Why Use Different Representations?

Each representation helps us see something different.

A table shows:

specific input-output pairs.

An equation shows:

the mathematical rule.

A graph shows:

the visual pattern or shape of the relationship.

Words explain:

what the relationship means.

No single representation is always best.

Different representations help us answer:

different questions.

---

Representing Functions with Tables

A function table lists input values and their corresponding:

outputs.

For example:

f(x) = 3x + 2

Choose some x-values and calculate f(x).

Each row represents one:

input-output pair.

---

Creating a Table from an Equation

Suppose:

f(x) = 2x - 3

We want to create a table.

Choose:

x = -2, -1, 0, 1, 2

Now substitute each value into the equation.

For x = -2:

f(-2) = 2(-2) - 3

f(-2) = -7

For x = -1:

f(-1) = 2(-1) - 3

f(-1) = -5

For x = 0:

f(0) = 2(0) - 3

f(0) = -3

For x = 1:

f(1) = 2(1) - 3

f(1) = -1

For x = 2:

f(2) = 2(2) - 3

f(2) = 1

So:

The table represents the same function as:

f(x) = 2x - 3.

---

Tables Show Specific Values

One advantage of a table is that it gives us:

exact values.

If we want to know f(2), we can simply find:

x = 2

and read the corresponding:

output.

In the previous example:

f(2) = 1

Tables are especially useful when we want to examine:

individual input-output pairs.

---

Limitations of Tables

A table usually shows only:

some values of a function.

Consider:

f(x) = 2x - 3

A table might show x-values from -2 to 2.

But the function may also be defined for:

x = 10

x = 3.5

x = -100

and many other values.

The table does not necessarily show the function's:

entire domain.

---

Representing Functions with Equations

An equation provides a mathematical rule connecting:

inputs and outputs.

For example:

f(x) = 4x + 3

tells us:

multiply the input by 4 and add 3.

An equation allows us to calculate the output for:

any allowed input.

---

Equations Are Compact

Suppose a function contains thousands of possible:

input-output pairs.

A table containing every pair might be impossible to:

write.

But an equation such as:

f(x) = 5x - 2

describes the entire relationship in:

one short statement.

This is one major advantage of:

equations.

---

Equations Reveal Structure

Consider:

f(x) = 3x + 4

The equation immediately tells us that this is a:

linear function.

We can identify:

slope = 3

and:

y-intercept = 4.

The equation reveals mathematical information that may be less obvious from a:

table.

---

Representing Functions with Graphs

A graph represents a function visually.

Each point on the graph has coordinates:

(x, y)

or:

(x, f(x)).

For example, if:

f(2) = 5

then the graph contains the point:

(2, 5).

A graph is created by plotting many input-output pairs.

---

Creating a Graph from a Table

Suppose we have:

Convert each row into an ordered pair:

(-2, -3)

(-1, -1)

(0, 1)

(1, 3)

(2, 5)

Plot these points on a coordinate plane.

Because they come from the linear function:

f(x) = 2x + 1

the points lie on a:

straight line.

---

The Connection Between Tables and Graphs

Every row of a function table corresponds to:

one point on the graph.

For example:

This connection is fundamental.

A table and graph are not separate pieces of mathematics.

They are two ways of displaying the:

same input-output pairs.

---

Graphs Show Overall Behaviour

A major advantage of a graph is that it shows the:

overall pattern.

From a graph we can often quickly see:

• whether the function increases or decreases
• where it crosses the axes
• whether it is linear or curved
• maximum or minimum values
• where the function changes direction
• domain and range
• unusual features

Graphs are especially useful for understanding the:

shape and behaviour of a function.

---

Limitations of Graphs

Graphs may not always provide:

exact values.

If a point appears near:

(4, 7)

it may be difficult to determine whether the actual output is:

7

7.1

or:

6.95.

The accuracy depends partly on the:

scale and quality of the graph.

For exact calculations, an equation or table may sometimes be:

better.

---

From Equation to Table

Consider:

f(x) = x²

Choose several inputs:

x = -3, -2, -1, 0, 1, 2, 3

Calculate the outputs.

Notice the pattern.

Different inputs can produce the:

same output.

For example:

f(-2) = 4

and:

f(2) = 4.

---

From Table to Graph

The previous table produces the points:

(-3, 9)

(-2, 4)

(-1, 1)

(0, 0)

(1, 1)

(2, 4)

(3, 9)

Plotting these points reveals a:

parabola.

The table contains the values.

The equation contains the rule.

The graph reveals the:

shape.

---

From Table to Equation

Sometimes we are given a table and need to determine the:

function rule.

Consider:

Look at how y changes.

Each time x increases by:

1

y increases by:

3.

This suggests a linear function with:

slope = 3.

When:

x = 0

we have:

y = 4.

Therefore, the y-intercept is:

4.

The equation is:

f(x) = 3x + 4

---

Using First Differences

For a linear function, we can examine the:

first differences.

Consider:

The outputs change by:

+4, +4, +4, +4

The constant first difference tells us the function is:

linear.

The slope is:

4.

Since:

f(0) = 2

the equation is:

f(x) = 4x + 2.

---

When x Does Not Increase by 1

Be careful.

Consider:

The y-values increase by:

6

but the x-values increase by:

2.

Therefore:

slope = change in y / change in x

slope = 6 / 2

slope = 3

Since y = 1 when x = 0:

f(x) = 3x + 1

We must compare changes in:

both variables.

---

From Graph to Table

Suppose a graph passes through:

(-2, -5)

(-1, -2)

(0, 1)

(1, 4)

(2, 7)

We can record these points in a table:

The graph has now been converted into a:

table.

---

From Graph to Equation

Suppose a straight-line graph crosses the y-axis at:

2

and rises:

3 units

for every:

1 unit to the right.

Then:

slope = 3

and:

y-intercept = 2.

Using:

y = mx + b

we obtain:

y = 3x + 2

or:

f(x) = 3x + 2.

---

Finding Slope from Two Points

Suppose a graph contains:

(1, 4)

and:

(3, 10).

Calculate the change in y:

10 - 4 = 6

Calculate the change in x:

3 - 1 = 2

Therefore:

slope = 6 / 2

slope = 3

If the graph also crosses the y-axis at 1, then:

f(x) = 3x + 1.

---

From Words to an Equation

Functions can also begin with a:

verbal description.

Suppose:

A taxi charges $4 initially and $2 for every kilometre travelled.

Let:

d = distance travelled

and:

C(d) = total cost.

The starting cost is:

4.

The cost increases by:

2 for every kilometre.

Therefore:

C(d) = 2d + 4

---

From Words to a Table

Using:

C(d) = 2d + 4

we can create:

The table shows specific examples of the relationship described in:

words.

---

From Words to a Graph

We can plot:

(0, 4)

(1, 6)

(2, 8)

(3, 10)

and so on.

The resulting graph shows that the total cost:

increases at a constant rate.

Now the same relationship has four representations:

Words: $4 starting fee plus $2/km

Equation: C(d) = 2d + 4

Table: specific distances and costs

Graph: visual representation of cost against distance

---

Each Representation Answers Different Questions

Consider the taxi example.

If we ask:

What is the exact cost for 5 km?

The table may answer this quickly:

$14.

If we ask:

What rule calculates the cost for any distance?

The equation is useful:

C(d) = 2d + 4.

If we ask:

How does cost change as distance increases?

The graph provides an immediate:

visual picture.

---

Comparing the Representations

Each representation provides a different:

view of the same relationship.

---

Representing a Nonlinear Function

Consider:

f(x) = x² + 1

The equation tells us the:

rule.

A table gives:

When plotted, the points create a:

curved parabola.

This demonstrates that multiple representations are useful for:

nonlinear functions as well as linear functions.

---

Recognizing Linear Relationships from Tables

Consider:

The output increases by:

3

every time the input increases by:

1.

This constant rate of change suggests a:

linear relationship.

---

Recognizing Linear Relationships from Graphs

A linear relationship produces a:

straight-line graph.

The constant slope shows that the output changes at a:

constant rate.

If the graph curves, the rate of change is generally:

not constant.

Therefore, graphs make it easy to distinguish between:

linear and nonlinear relationships.

---

Recognizing Nonlinear Relationships from Tables

Consider:

The changes in y are:

+1, +3, +5, +7

The first differences are:

not constant.

Therefore, the relationship is:

not linear.

In fact:

y = x².

---

Ordered Pairs as a Bridge

Ordered pairs provide a useful connection between:

tables and graphs.

A row such as:

becomes:

(3, 8).

That ordered pair becomes:

a point on the graph.

So we can think of the process as:

Table row → Ordered pair → Graph point

---

Equations Generate Ordered Pairs

An equation can generate as many ordered pairs as we:

need.

For:

f(x) = 2x + 1

choose x = 4.

Then:

f(4) = 9

This creates the ordered pair:

(4, 9).

Choose x = 10.

Then:

f(10) = 21

This creates:

(10, 21).

Every allowed input creates a corresponding:

point on the function's graph.

---

Graphs Represent More Than the Points We Plot

When drawing a continuous function such as:

f(x) = 2x + 1

we might calculate only five points.

But the function contains infinitely many points between:

those calculated points.

For example:

x = 0.5

gives:

f(0.5) = 2

So:

(0.5, 2)

also lies on the graph.

The table helps us construct the graph, but it does not contain:

every possible point.

---

Discrete Functions

Not every function should be drawn as a:

continuous line.

Suppose:

C = 5n

represents the cost of buying n notebooks.

Possible values of n might be:

0, 1, 2, 3, 4, ...

Values such as:

2.6 notebooks

do not make sense.

The graph should therefore contain:

separate points.

This is a:

discrete function.

---

Continuous Functions

Suppose:

d(t) = 60t

represents distance travelled at 60 km/h.

Time could be:

1 hour

1.5 hours

1.57 hours

or many other values.

The inputs can vary continuously.

Therefore, the graph can normally be drawn as a:

continuous line.

---

Converting Between Representations

A useful skill is being able to move freely between:

words ↔ equation ↔ table ↔ graph

For example:

Words

A quantity begins at 3 and increases by 2 for each unit of x.

↓

Equation

f(x) = 2x + 3

↓

Table

↓

Graph

Plot:

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

and connect appropriately.

All four representations describe:

the same relationship.

---

Worked Example 1: Equation to Table

Given:

f(x) = -2x + 5

complete the table.

For x = -1:

f(-1) = -2(-1) + 5 = 7

For x = 0:

f(0) = 5

For x = 1:

f(1) = 3

For x = 2:

f(2) = 1

For x = 3:

f(3) = -1

Therefore:

---

Worked Example 2: Table to Equation

Consider:

The y-values increase by:

4

when x increases by 1.

Therefore:

slope = 4.

When x = 0:

y = -2.

Therefore:

y-intercept = -2.

The equation is:

f(x) = 4x - 2.

---

Worked Example 3: Graph Information to Equation

A straight-line graph passes through:

(0, 6)

and:

(2, 10).

Calculate the slope:

slope = (10 - 6) / (2 - 0)

slope = 4 / 2

slope = 2

The y-intercept is:

6.

Therefore:

f(x) = 2x + 6.

---

Worked Example 4: Words to Multiple Representations

A streaming service charges:

$8 per month plus a one-time $12 registration fee.

Let:

m = number of months

and:

C(m) = total cost.

Equation:

C(m) = 8m + 12

Table:

Graph:

Plot the points:

(0, 12)

(1, 20)

(2, 28)

(3, 36)

(4, 44)

Because months are normally counted in whole numbers for this model, the context may make the representation:

discrete.

---

Checking That Representations Match

Suppose an equation is:

f(x) = 3x - 1

but a table contains:

Does the table represent the equation?

Check x = 2:

f(2) = 3(2) - 1

f(2) = 5

But the table says:

6.

Therefore, the table does:

not match the equation.

---

Checking a Point Against an Equation

Suppose the graph of:

f(x) = 2x + 3

appears to contain the point:

(4, 11).

Check:

f(4) = 2(4) + 3

f(4) = 11

Therefore:

(4, 11)

does lie on the graph.

This is a useful way to verify whether different representations:

agree.

---

Comparing Two Functions

Multiple representations also help us compare:

different functions.

Suppose:

f(x) = 2x + 1

and:

g(x) = x + 4.

A table can compare their outputs:

We can immediately see:

f(3) = g(3) = 7.

---

Comparing Functions Graphically

If we graph:

f(x) = 2x + 1

and:

g(x) = x + 4

the lines intersect at:

(3, 7).

This shows visually that both functions produce the same output when:

x = 3.

The table and graph communicate the same:

intersection.

---

Choosing the Best Representation

Different situations may favour different representations.

Use a table when:

• exact individual values matter
• the dataset is small
• you want to compare selected inputs and outputs

Use an equation when:

• you need the mathematical rule
• you want to calculate new values
• you want a compact representation

Use a graph when:

• you want to see the overall pattern
• you want to identify trends
• you want to compare functions visually
• you want to identify intersections or turning points

Use words when:

• the context and meaning need to be explained

---

Representations Work Together

Strong mathematical understanding means being able to look at one representation and imagine:

the others.

When you see:

f(x) = 3x + 2

you should begin to imagine:

• a table increasing by 3
• a straight-line graph
• a y-intercept of 2
• an input-output rule
• a constant rate of change

These are not separate facts.

They are different features of:

one function.

---

Multiple Representations in Science

Suppose an object moves at constant speed.

A scientist might describe the relationship in words:

The object moves at 5 m/s.

Equation:

d(t) = 5t

Table:

Graph:

A straight line through:

(0, 0)

with a slope of:

5.

Each representation provides useful information about the:

same physical motion.

---

Multiple Representations in Business

Suppose a company charges:

$25 fixed fee plus $10 per item.

Words describe the:

pricing system.

Equation:

C = 10n + 25

A table can show costs for selected quantities.

A graph can show how cost changes as the number of items:

increases.

This is why functions are so useful for:

mathematical modelling.

---

Multiple Representations in Everyday Life

The same idea appears in:

• phone plans
• taxi fares
• wages
• electricity costs
• distance and time
• temperature changes
• population growth
• savings
• scientific experiments
• business profits

In each situation, multiple representations help us understand the relationship from:

different perspectives.

---

Common Mistake: Treating Representations as Separate

A table, equation, and graph are not:

different functions

simply because they look different.

If they represent the same input-output rule, they describe:

the same function.

---

Common Mistake: Connecting Discrete Points

Suppose a graph represents:

number of concert tickets purchased.

Values such as:

2.4 tickets

do not make sense.

Therefore, the plotted points should not automatically be connected with:

a continuous line.

Always consider the:

context.

---

Common Mistake: Reading a Graph Inaccurately

When converting a graph to a table:

• check the axis scale
• identify coordinates carefully
• include negative signs
• distinguish x from y
• do not estimate when an exact value is clearly shown

Graph reading requires:

precision.

---

Common Mistake: Assuming Every Pattern Is Linear

A table may contain a clear pattern without being:

linear.

For example:

1, 4, 9, 16, 25

has a strong pattern.

But the differences are:

3, 5, 7, 9.

They are not constant.

The relationship is:

quadratic, not linear.

---

The REPRESENT Strategy

When working with multiple representations, ask:

R — Relationship

What relationship is being described?

E — Equation

Can I write a mathematical rule?

P — Points

What input-output pairs are known?

R — Rate or Pattern

How do the values change?

E — Examine the Graph

What shape should the relationship have?

S — Same Relationship?

Do all representations agree?

E — Explain

What does the relationship mean?

N — Note Restrictions

Is the function discrete, continuous, or otherwise restricted?

T — Test

Choose an input and check that every representation gives the same output.

---

Check Your Understanding

1. Name three common ways of representing a function.

2. What information does a table show particularly well?

3. What information does an equation show particularly well?

4. What information does a graph show particularly well?

5. Explain how a row in a table corresponds to a point on a graph.

6. Create a table for f(x) = 2x + 3 using x = 0, 1, 2, 3, 4.

7. Write the ordered pairs produced by your table.

8. Create a table for g(x) = x² - 1 using x = -2, -1, 0, 1, 2.

9. A table has outputs 5, 8, 11, 14 as x increases by 1. What is the constant first difference?

10. A table contains (0, 4), (1, 6), (2, 8), and (3, 10). Determine the equation.

11. A straight line has slope 5 and y-intercept -2. Write its equation.

12. Explain how you can convert an equation into a table.

13. Explain how you can convert a table into a graph.

14. Explain how you can convert a linear graph into an equation.

15. What does a constant first difference tell you about a function?

16. Why might a table fail to show the entire function?

17. Give one advantage and one limitation of a graph.

18. Give one advantage and one limitation of an equation.

19. Give one advantage and one limitation of a table.

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

21. A taxi charges $6 initially plus $3 per kilometre. Write a function for the total cost.

22. Create a table for the taxi function for distances of 0, 1, 2, 3, and 4 km.

23. Explain what the slope of the taxi graph represents.

24. Explain what the y-intercept of the taxi graph represents.

25. Explain how a table, equation, graph, and verbal description can all represent the same function.

---

Key Terms

• Function: Relationship assigning each allowed input exactly one output.
• Representation: A way of displaying or describing mathematical information.
• Table: Arrangement showing selected input-output pairs.
• Equation: Mathematical statement describing the rule connecting variables.
• Graph: Visual representation of the relationship between variables.
• Ordered pair: Pair of coordinates written as (x, y).
• Input: Value supplied to a function.
• Output: Value produced by a function.
• Function notation: Notation such as f(x) used to describe a function's output.
• Linear function: Function with a constant rate of change and a straight-line graph.
• Nonlinear function: Function whose graph is not a straight line.
• Slope: Rate of change of a linear function.
• Y-intercept: Point where a graph crosses the y-axis.
• First difference: Difference between consecutive output values.
• Discrete function: Function represented by separate input values.
• Continuous function: Function that can take all allowed values throughout intervals.
• Mathematical model: Mathematical representation of a real-world relationship.
• Rate of change: Amount by which one variable changes compared with another.
• Independent variable: Input variable.
• Dependent variable: Output variable.

---

Key Takeaways

• A function can be represented using words, tables, equations, graphs, ordered pairs, and mapping diagrams.
• Different representations can describe exactly the same mathematical relationship.
• A table shows specific input-output pairs clearly.
• An equation gives a compact mathematical rule for the function.
• A graph shows the overall shape and behaviour of a function.
• Words help explain the meaning and context of a relationship.
• Every row of a function table corresponds to an ordered pair.
• Every ordered pair corresponds to a point on the graph.
• An equation can be converted into a table by choosing inputs and calculating outputs.
• A table can be converted into a graph by plotting its input-output pairs.
• A linear table can often be converted into an equation by identifying the slope and y-intercept.
• Constant first differences suggest a linear relationship when the x-values increase by equal amounts.
• If x-values do not increase by 1, calculate slope using change in y / change in x.
• Straight-line graphs represent linear relationships.
• Curved graphs generally represent nonlinear relationships.
• Tables usually show only selected values rather than every possible value of a function.
• Graphs reveal patterns clearly but may not always provide exact values.
• Equations allow outputs to be calculated for any allowed input.
• Discrete functions should not automatically be drawn as continuous lines.
• Context helps determine whether a function should be discrete or continuous.
• Different representations are useful for answering different types of questions.
• Checking the same input in several representations can verify that they describe the same function.
• Strong mathematical understanding means being able to move confidently between tables, equations, graphs, and words.
• Multiple representations provide different views of one underlying relationship, helping us understand functions more completely.

Functions Are Everywhere

Functions are not just mathematical equations.

They describe relationships that occur throughout:

everyday life, science, technology, business, and nature.

Whenever one quantity depends on another quantity, there may be a function.

For example:

The cost of fuel depends on how many litres you buy.

The distance travelled depends on how long you travel.

The amount you earn depends on how many hours you work.

The area of a circle depends on its radius.

In each situation, one quantity acts as an:

input

and another quantity is the resulting:

output.

---

Input → Relationship → Output

A useful way to recognize a function is:

Input → Rule or Relationship → Output

For example:

Hours worked → Pay rate → Total pay

If you earn $15 per hour:

1 hour → $15

2 hours → $30

5 hours → $75

The number of hours determines the:

total pay.

We can model this relationship using:

P = 15h

where:

h = hours worked

and:

P = total pay.

---

Independent and Dependent Variables

In real-world functions, we often describe the two quantities as:

independent and dependent variables.

The independent variable is the input.

The dependent variable is the output.

The dependent variable changes because of, or is determined by, the:

independent variable.

---

Example: Distance and Time

Imagine a cyclist travelling at a constant speed of:

12 km/h.

The distance travelled depends on:

time.

Therefore:

Time = independent variable

Distance = dependent variable

We can write:

d(t) = 12t

where:

t = time in hours

and:

d(t) = distance travelled in kilometres.

After 3 hours:

d(3) = 12(3)

d(3) = 36

The cyclist has travelled:

36 km.

---

How to Identify the Variables

Ask:

Which quantity determines or helps determine the other?

That quantity is usually the:

independent variable.

Then ask:

Which quantity changes in response?

That quantity is usually the:

dependent variable.

For example:

The amount of water in a tank increases as filling time increases.

Time is the:

independent variable.

Amount of water is the:

dependent variable.

---

Independent Does Not Mean Unrelated

The word independent can sometimes be confusing.

It does not necessarily mean the variable has nothing to do with other quantities.

Instead, in a function model it usually means that we treat it as the:

input variable.

The dependent variable is then calculated or observed based on that:

input.

---

Identifying Functions in Everyday Life

Consider a parking garage that charges:

$3 per hour.

The total cost depends on the:

number of hours parked.

We could write:

C = 3h

where:

h = hours parked

and:

C = parking cost.

This is a real-world:

function.

---

Why Is It a Function?

For each allowed number of hours parked, there is exactly:

one corresponding cost.

For example:

1 hour → $3

2 hours → $6

3 hours → $9

An input cannot produce two different outputs under the same:

pricing rule.

That is why the relationship is a:

function.

---

Real-World Relationships That Are Functions

Many familiar relationships can be modeled as functions.

Examples include:

• wages depending on hours worked
• cost depending on quantity purchased
• distance depending on travel time
• temperature depending on time
• population depending on year
• area depending on dimensions
• electricity cost depending on energy used
• fuel used depending on distance travelled
• height of a projectile depending on time
• account balance depending on time

The important question is:

Does each allowed input correspond to exactly one output?

---

A Relationship That Is Not a Function

Suppose we try to describe:

people → phone numbers

A single person might have:

• a personal phone number
• a work number
• another mobile number

Therefore, one input could correspond to:

several outputs.

Under that particular definition, the relationship would not be a:

function.

The way we define the input and output matters.

---

Context Matters

Consider:

Student → height

At a specific moment in time, each student has one:

height.

This can be treated as a function.

But:

Student → friends

would not normally be a function because one student may have:

many friends.

When deciding whether a real-world relationship is a function, carefully identify:

the input and the output.

---

Functions and Units

Real-world functions usually involve:

units.

Suppose:

C = 4n

represents the cost of buying notebooks.

Then:

n = number of notebooks

and:

C = cost in dollars.

If:

C(5) = 20

we should interpret the result as:

5 notebooks cost $20.

Not simply:

20.

Units give mathematical answers their:

real-world meaning.

---

Interpreting Function Notation in Context

Suppose:

T(t)

represents the temperature t hours after sunrise.

Then:

T(4) = 28

means:

The temperature 4 hours after sunrise is 28°C.

It does not simply mean:

4 becomes 28.

Function notation should always be interpreted using the:

context of the situation.

---

Another Context Example

Suppose:

P(t)

represents the population of a town t years after 2025.

If:

P(10) = 75,000

then this means:

10 years after 2025, the model gives a population of 75,000.

Since:

2025 + 10 = 2035

we could also say:

The model gives a population of 75,000 in 2035.

---

Describing How Quantities Change

Functions allow us to describe how one quantity changes as:

another quantity changes.

For example:

C = 5n

As the number of items increases, the total cost:

increases.

More specifically:

For every additional item, the cost increases by:

$5.

This is called a:

rate of change.

---

Rate of Change

The rate of change tells us how much the dependent variable changes when the independent variable:

changes.

Suppose:

d(t) = 60t

where d is distance in kilometres and t is time in hours.

The rate of change is:

60 km/h.

This means:

For every additional hour, distance increases by 60 km.

---

Rates Have Units

A real-world rate of change often combines:

two units.

Examples include:

60 km/h

$15/hour

5 L/min

$3/kg

2°C/hour

The units help explain what the rate:

means.

---

Positive Relationships

Sometimes, as one variable increases, the other variable also:

increases.

For example:

More hours worked → more money earned

More items purchased → greater total cost

More time travelling → greater distance travelled

These are examples of:

increasing relationships.

---

Decreasing Relationships

Sometimes, as one quantity increases, another:

decreases.

For example:

A car begins with 50 litres of fuel and uses:

5 litres per hour.

We could model the remaining fuel as:

F(t) = 50 - 5t

As time increases, the amount of fuel:

decreases.

The rate of change is:

-5 L/h.

The negative sign tells us that fuel is being:

used up.

---

Constant Relationships

Sometimes the output remains unchanged even when the input:

changes.

Suppose a parking garage charges a flat daily fee of:

$20.

The cost might be modeled as:

C = 20

for the allowed parking period.

Whether you park for 2 hours or 6 hours, the cost remains:

$20.

This is a:

constant function.

---

Linear Real-World Functions

Many simple real-world situations can be modeled using:

linear functions.

A common form is:

y = mx + b

or:

f(x) = mx + b

where:

m = rate of change

and:

b = starting value.

This structure appears frequently in:

real-world models.

---

Understanding the Starting Value

Consider:

C(d) = 2d + 5

Suppose this represents a taxi fare.

The taxi charges:

$5 before any distance is travelled

plus:

$2 per kilometre.

Therefore:

5 = starting fee

and:

2 = rate per kilometre.

---

Starting Value + Rate × Input

Many real-world linear functions can be understood using:

Output = Starting Value + (Rate × Input)

For example:

A gym charges:

$30 membership fee

plus:

$8 per visit.

Let:

v = number of visits

and:

C(v) = total cost.

Then:

C(v) = 30 + 8v

or:

C(v) = 8v + 30.

---

Creating a Function from a Situation

To create a simple function model:

Step 1: Identify the independent variable.

Step 2: Identify the dependent variable.

Step 3: Determine the starting value.

Step 4: Determine the rate of change.

Step 5: Write the function.

Step 6: Check that the function makes sense in context.

---

Worked Example: Movie Tickets

Movie tickets cost:

$12 each.

Let:

n = number of tickets

and:

C = total cost.

There is no additional starting fee.

Therefore:

C = 12n

For 4 tickets:

C(4) = 12(4)

C(4) = 48

So:

4 tickets cost $48.

---

Representing the Movie Ticket Function

We can represent:

C = 12n

using a table:

Each row represents:

one input-output pair.

---

Is the Movie Ticket Function Continuous?

Can someone normally purchase:

2.7 movie tickets?

No.

The number of tickets must normally be:

whole numbers.

Therefore, this real-world function has a:

discrete domain.

Its graph should normally contain:

separate points rather than a continuous line.

---

Worked Example: Filling a Swimming Pool

A pool already contains:

500 litres

of water.

Water enters at:

40 litres per minute.

Let:

t = time in minutes

and:

V(t) = volume of water in litres.

Starting amount:

500 L

Rate:

40 L/min

Therefore:

V(t) = 40t + 500

---

Interpreting the Pool Function

For:

V(t) = 40t + 500

the number:

500

represents the:

initial volume of water.

The number:

40

represents the:

rate at which water enters the pool.

If:

t = 10

then:

V(10) = 40(10) + 500

V(10) = 900

After 10 minutes, the pool contains:

900 litres.

---

Continuous Real-World Functions

Unlike movie tickets, time can take values such as:

2 minutes

2.5 minutes

2.57 minutes

and so on.

Therefore, the pool example can usually be modeled as a:

continuous function.

This is an important example of how:

context affects the domain of a function.

---

Worked Example: Cooling

Suppose a drink begins at:

80°C

and cools by approximately:

3°C per minute

over a short period.

A simple model might be:

T(t) = 80 - 3t

where:

t = time in minutes

and:

T(t) = temperature in °C.

The negative rate indicates that temperature:

decreases over time.

---

Interpreting the Cooling Function

Calculate:

T(5).

T(5) = 80 - 3(5)

T(5) = 65

Interpretation:

After 5 minutes, the model predicts a temperature of 65°C.

The phrase:

"the model predicts"

is important because mathematical models are often:

approximations of reality.

---

Models Have Limits

The cooling function:

T(t) = 80 - 3t

cannot remain accurate forever.

After 30 minutes, it predicts:

T(30) = -10°C.

That is unlikely to describe a drink cooling naturally in an ordinary room.

The model may work reasonably well only over a:

limited time interval.

Real-world functions often have restricted:

domains.

---

Mathematical Models

A mathematical model uses mathematics to represent a:

real situation.

Models can help us:

• describe patterns
• calculate unknown values
• make predictions
• compare situations
• understand relationships
• make decisions

But models are usually:

simplifications of reality.

---

A Model Is Not Reality

Suppose:

P(t) = 1000 + 50t

models a population.

The function assumes the population increases by exactly:

50 individuals per year.

Real populations are affected by:

• births
• deaths
• migration
• resources
• disease
• environmental conditions

Therefore, the mathematical function is a:

model, not a perfect description of reality.

---

Functions in Shopping

Suppose apples cost:

$4 per kilogram.

Let:

m = mass of apples in kilograms

and:

C(m) = cost.

Then:

C(m) = 4m

For:

m = 2.5 kg

we have:

C(2.5) = 4(2.5)

C(2.5) = 10

So:

2.5 kg of apples costs $10.

Because mass can take decimal values, this function can be:

continuous.

---

Functions in Employment

Suppose a worker earns:

$18 per hour.

Then:

P = 18h

where:

h = hours worked

and:

P = pay.

If the worker works 7.5 hours:

P(7.5) = 18(7.5)

P(7.5) = 135

The worker earns:

$135.

---

Functions with a Fixed Fee

Suppose a plumber charges:

$50 call-out fee

plus:

$40 per hour.

Let h represent hours worked.

Then:

C = 40h + 50

The number:

40

represents the:

hourly rate.

The number:

50

represents the:

fixed starting charge.

---

Comparing Two Real-World Functions

Suppose two taxi companies charge:

Company A: A(d) = 2d + 5

Company B: B(d) = 3d + 2

For a 4 km trip:

A(4) = 2(4) + 5 = 13

B(4) = 3(4) + 2 = 14

The functions allow us to compare:

two real-world pricing systems.

---

Functions in Science

Science is full of relationships between:

variables.

Examples include:

Distance = speed × time

Density = mass / volume

Force = mass × acceleration

Voltage = current × resistance

Many scientific equations can be interpreted as:

functions.

One quantity depends mathematically on:

one or more other quantities.

---

Functions in Biology

Suppose scientists measure plant height each week.

They might define:

H(t)

where:

t = time

and:

H(t) = plant height.

A table, graph, or equation could then describe how plant height:

changes over time.

Time is the:

independent variable.

Plant height is the:

dependent variable.

---

Functions in Chemistry

A chemist may investigate how temperature affects:

reaction rate.

Temperature could be treated as the:

independent variable.

Reaction rate could be treated as the:

dependent variable.

The relationship could be represented using:

a graph or mathematical model.

Functions therefore connect mathematics directly with:

experimental science.

---

Functions in Physics

Suppose an object moves at a constant velocity of:

5 m/s.

Its position might be modeled as:

d(t) = 5t

A table could show:

The graph would be:

a straight line through the origin.

The slope represents:

speed.

---

Functions in Geometry

Suppose we want to describe the area of a square.

If:

s = side length

then:

A(s) = s²

The area depends on:

side length.

Side length is the:

independent variable.

Area is the:

dependent variable.

This function is nonlinear because its graph is:

curved.

---

Not All Real-World Functions Are Linear

Many introductory examples are linear because they are:

easy to model.

But real-world functions can also be:

• quadratic
• exponential
• periodic
• piecewise
• inverse
• logarithmic

For example, the area of a circle is:

A(r) = πr²

This is not a linear relationship.

Doubling the radius does not simply:

double the area.

---

Example: Area of a Circle

Consider:

A(r) = πr²

If:

r = 2

then:

A(2) = 4π

If:

r = 4

then:

A(4) = 16π

The radius doubled from:

2 to 4.

But the area became:

four times as large.

This illustrates a:

nonlinear relationship.

---

Reading Real-World Graphs

Graphs are particularly useful for real-world functions because they show:

how quantities change.

When interpreting a real-world graph, ask:

• What does the x-axis represent?
• What does the y-axis represent?
• What are the units?
• Is the graph increasing or decreasing?
• Is the relationship linear?
• Is the rate of change constant?
• What do intercepts mean?
• Are there maximum or minimum values?
• What domain makes sense?

These questions help transform a graph into a:

meaningful explanation.

---

Interpreting a Distance-Time Graph

Suppose a distance-time graph rises steadily.

This tells us:

distance increases as time increases.

If the graph is a straight line, the object is moving at a:

constant speed.

A steeper line represents:

a greater speed.

A horizontal section means the distance is not changing, so the object is:

stationary.

---

Interpreting an Intercept

Suppose:

C(x) = 5x + 20

represents a service charge.

The graph crosses the y-axis at:

20.

This means that when:

x = 0

the cost is still:

$20.

In context, the y-intercept represents the:

fixed starting charge.

Intercepts often have important:

real-world meanings.

---

Interpreting Slope

For:

C(x) = 5x + 20

the slope is:

5.

If x represents hours, the slope might mean:

$5 per hour.

If x represents kilograms, it might mean:

$5 per kilogram.

The number alone is not enough.

We must interpret the slope using:

the variables and units.

---

Building a Model from a Table

Suppose:

The cost increases by:

$4 per hour.

Therefore:

rate = 4

When:

h = 0

the cost is:

$10.

Therefore:

starting value = 10

The function is:

C = 4h + 10.

---

Interpreting the Model

For:

C = 4h + 10

we can say:

There is a $10 starting charge, and the cost increases by $4 for each additional hour.

This verbal explanation is often just as important as:

writing the equation.

A good mathematical model should be understandable in:

context.

---

Building a Model from Words

Suppose:

A water tank begins with 200 L and loses 8 L every minute.

Identify the variables.

Independent variable:

time, t

Dependent variable:

volume, V

Starting value:

200 L

Rate:

-8 L/min

Therefore:

V(t) = 200 - 8t

---

Interpreting the Tank Model

For:

V(t) = 200 - 8t

calculate:

V(10).

V(10) = 200 - 8(10)

V(10) = 120

Interpretation:

After 10 minutes, 120 L of water remains.

---

Finding a Meaningful Domain

For:

V(t) = 200 - 8t

can t equal:

100 minutes?

The equation would produce:

V(100) = -600 L.

A tank cannot contain:

negative water.

The tank reaches zero when:

200 - 8t = 0

8t = 200

t = 25

Therefore, a meaningful domain is:

0 ≤ t ≤ 25.

---

Real-World Restrictions

Mathematical equations may allow values that do not make sense in:

real situations.

Examples:

Number of people cannot be negative.

Number of cars usually must be a whole number.

Time may have a starting and ending point.

A container cannot hold more than its capacity.

Distance cannot always continue increasing forever.

Always interpret the:

context.

---

Prediction and Interpolation

Suppose we have data from:

0 to 10 hours.

Using a model to estimate a value at:

6 hours

is called interpolation because the value lies within the observed range.

Interpolation is often reasonably:

reliable, assuming the model fits the data well.

---

Extrapolation

Using the same model to predict a value at:

100 hours

is called extrapolation.

This extends the model beyond the observed:

data range.

Extrapolation can be much less reliable because the relationship may:

change outside the known interval.

---

Models and Predictions

Functions allow us to make predictions.

But predictions depend on:

how appropriate the model is.

A model may work well:

within a certain domain

but poorly:

outside that domain.

This is why mathematical modelling requires both:

calculation and judgment.

---

A Complete Real-World Function Example

A bicycle rental shop charges:

$10 starting fee plus $6 per hour.

Let:

h = number of hours

and:

C = total cost.

The function is:

C = 6h + 10

The independent variable is:

hours rented.

The dependent variable is:

total cost.

The rate of change is:

$6/hour.

The starting value is:

$10.

For 5 hours:

C(5) = 6(5) + 10

C(5) = 40

Interpretation:

Renting the bicycle for 5 hours costs $40.

This single example connects:

variables, function notation, equations, rates, context, and interpretation.

---

A Strategy for Real-World Functions

When you encounter a real-world function, use these questions:

1. What quantities are changing?

Identify the:

variables.

2. Which quantity is the input?

This is usually the:

independent variable.

3. Which quantity depends on the input?

This is the:

dependent variable.

4. What is the starting value?

Look for the value when the input is:

zero.

5. What is the rate of change?

Determine how quickly the output:

increases or decreases.

6. Can I write a function?

For a simple linear model:

Output = Rate × Input + Starting Value

7. Does the answer make sense?

Check:

units, domain, and context.

---

Common Mistake: Reversing the Variables

Suppose:

Total cost depends on the number of tickets purchased.

Correct:

Number of tickets = independent variable

Total cost = dependent variable

The cost depends on the number of tickets, not:

the other way around.

---

Common Mistake: Ignoring Units

Suppose:

d(4) = 240

If d represents kilometres and the input represents hours, the interpretation should be:

After 4 hours, the distance travelled is 240 km.

Units are essential to:

meaningful interpretation.

---

Common Mistake: Ignoring the Starting Value

Suppose a taxi charges:

$4 starting fee plus $2/km.

Incorrect:

C(d) = 2d

Correct:

C(d) = 2d + 4

The starting fee must be included even when:

d = 0.

---

Common Mistake: Ignoring Negative Rates

Suppose a tank loses:

3 L/min.

If it begins with 100 L, the model is:

V(t) = 100 - 3t

not:

V(t) = 100 + 3t.

The negative rate represents a quantity that is:

decreasing.

---

Common Mistake: Extending Models Forever

A function may be mathematically defined for many values but only make sense over a:

limited domain.

Always ask:

Does this input make sense in the real situation?

A good mathematical model includes:

reasonable restrictions.

---

Check Your Understanding

1. What is a real-world function?

2. Define independent variable.

3. Define dependent variable.

4. If total pay depends on hours worked, which variable is independent?

5. If plant height is measured over time, which variable is dependent?

6. A worker earns $20 per hour. Write a function for total pay P.

7. Use your function to calculate P(7).

8. Interpret P(7) in context.

9. A taxi charges $5 initially plus $3/km. Write a function for total cost C(d).

10. What does the 5 represent in your function?

11. What does the 3 represent?

12. Calculate the cost of a 10 km journey.

13. A tank contains 600 L and loses 20 L/min. Write a function for its volume.

14. Which variable is independent in the tank model?

15. Which variable is dependent?

16. Why is the rate of change negative?

17. When will the tank become empty?

18. What is a reasonable domain for the tank function?

19. Explain the difference between a discrete and continuous real-world function.

20. Is the number of movie tickets purchased usually discrete or continuous? Explain.

21. Is the mass of fruit purchased usually discrete or continuous? Explain.

22. A table begins at $15 and increases by $4 for every additional hour. Write a function describing the relationship.

23. Explain what the slope represents in your function.

24. Explain what the y-intercept represents.

25. Why should mathematical models not always be extrapolated far beyond their original data?

26. Give an example of a real-world function from science.

27. Give an example of a real-world function from everyday life.

28. Explain why units are important when interpreting functions.

29. Describe how you would determine whether a real-world relationship is a function.

30. Explain how functions can help us understand and predict real-world situations.

---

Key Terms

• Function: Relationship in which each allowed input has exactly one output.
• Real-world function: Function used to represent a relationship between quantities in a real situation.
• Independent variable: Input variable used to determine or predict another quantity.
• Dependent variable: Output variable whose value depends on the input.
• Input: Value supplied to a function.
• Output: Value produced by a function.
• Variable: Quantity that can change.
• Function notation: Notation such as f(x) used to represent the output of a function.
• Rate of change: Amount the dependent variable changes compared with the independent variable.
• Starting value: Value of the dependent variable when the input is zero.
• Linear function: Function with a constant rate of change.
• Constant function: Function whose output remains unchanged.
• Mathematical model: Mathematical representation of a real-world situation.
• Discrete: Consisting of separate individual values.
• Continuous: Able to take all values throughout an interval.
• Domain: Set of meaningful or allowed input values.
• Range: Set of possible output values.
• Interpolation: Estimating a value within the range of known data.
• Extrapolation: Predicting beyond the range of known data.
• Slope: Rate of change of a linear function.
• Y-intercept: Output value when the input is zero.

---

Key Takeaways

• Functions can describe relationships found throughout everyday life, science, business, and technology.
• A function connects an input to exactly one output.
• The independent variable is usually the input.
• The dependent variable is usually the output.
• The dependent variable changes in relation to the independent variable.
• Function notation can give real-world quantities meaningful names, such as C for cost or d(t) for distance.
• Function values should always be interpreted using their context and units.
• Real-world functions can be represented using words, equations, tables, and graphs.
• The rate of change describes how one quantity changes compared with another.
• Rates should be interpreted with appropriate units such as km/h, $/hour, or L/min.
• Positive rates describe increasing relationships.
• Negative rates describe decreasing relationships.
• A constant function has an output that does not change as the input changes.
• Many simple real-world situations can be modeled using linear functions.
• In a linear model, the slope represents the rate of change.
• The y-intercept often represents the starting value.
• A useful linear model is Output = Rate × Input + Starting Value.
• Real-world models may have restricted domains.
• Counts such as people, tickets, or cars are usually discrete.
• Measurements such as time, distance, mass, and temperature can often be continuous.
• Not every real-world function is linear.
• Mathematical models are simplifications and may not remain accurate forever.
• Interpolation estimates values within known data, while extrapolation predicts beyond it.
• Predictions become less reliable when a model is used far outside the conditions for which it was developed.
• A good real-world model should be mathematically correct and meaningful in context.
• Functions allow us to describe, calculate, compare, and predict relationships between changing quantities.