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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.