Function Notation & Interpretation

SECTION 1

Historical Context & Motivation

Long before the modern idea of a function had a precise definition, people were already thinking about relationships between quantities. Ancient astronomers tracked how the position of the Sun depended on the day of the year, and medieval merchants calculated how the cost of goods changed with the number of items purchased. The concept of a function gave mathematicians a compact, reliable way to capture these relationships so they could be analyzed, graphed, and communicated without ambiguity.

1673

Leibniz Coins 'Function'

Gottfried Wilhelm Leibniz was the first mathematician to use the word functio in a mathematical context, referring to quantities that depend on a curve, such as its slope or length at a given point.

1734

Euler Introduces f(x)

Leonhard Euler popularized the notation f(x) in his works, providing a clean and powerful shorthand that is still the universal standard today.

1837

Dirichlet's Modern Definition

Peter Gustav Lejeune Dirichlet proposed that a function is any rule that assigns each input exactly one output — no formula required. This broadened the concept enormously.

1939

Bourbaki Formalizes Domain & Range

The Bourbaki group of French mathematicians formalized set-theoretic language, explicitly defining domain and range as essential parts of a function's specification.

Today, function notation is the backbone of every branch of mathematics. In the IB Applications and Interpretation course, you will use functions to model real-world situations — from population growth to financial projections. The central question this lesson addresses is: How do we write, read, and interpret functions so that we can communicate mathematical relationships precisely?

SECTION 2

Core Principles & Definitions

Before we start evaluating functions or sketching graphs, you need to lock in a handful of foundational ideas. Each one builds on the previous, so take them in order.

Function

A rule that assigns exactly one output to each input. If an input could give two different outputs, the relationship is NOT a function.

Function Notation f(x)

We write f(x) (read "f of x") to name the function f and indicate that x is the input variable. The expression equals the output.

Domain

The complete set of allowable input values. In context, this is often limited by real-world constraints (e.g., time cannot be negative).

Range

The complete set of possible output values that result from substituting every element of the domain into the function.

Interpretation in Context

In the IB, you must explain what f(x) means in the situation described. For example, if C(t) models cost over time, C(5) = 200 means "after 5 hours the cost is $200."

✦ KEY TAKEAWAY

Think of a function like a vending machine. You press one button (the input) and you get exactly one item (the output). The domain is the set of buttons that actually work, and the range is the set of items the machine can actually dispense. If pressing one button could randomly give you a soda or a sandwich, it wouldn't be a proper function machine!

SECTION 3

Visualizing Functions: The Mapping Diagram

One of the clearest ways to see what a function does is through a mapping diagram. This type of diagram shows the domain on the left and the range on the right, with arrows indicating which input maps to which output. The diagram below illustrates a simple function alongside a non-function so you can see the critical difference.

Left: a valid function — every input in the domain maps to exactly one output in the range (two inputs sharing an output is allowed). Right: NOT a function — input 1 maps to both A and B.

Notice that in the valid function on the left, inputs 1 and 2 both map to 5. That is perfectly fine — multiple inputs can share an output. The rule that makes something a function is one-directional: each input must produce exactly one output. On the right, input 1 leads to two different outputs, which violates this rule and disqualifies the relationship from being a function.

SECTION 4

Mathematical Framework

Now let's formalize the notation you'll use throughout the IB course. Understanding each piece of the notation will help you read exam questions quickly and write answers that earn full marks.

FUNCTION NOTATION

f(x) = expression in x

f is the name of the function. x is the input (independent variable). The right side is the rule that produces the output (dependent variable). Other common names include g(x), h(t), or C(n).

EVALUATING A FUNCTION

If f(x) = 3x − 7, then f(4) = 3(4) − 7 = 5

To evaluate, replace every x with the given input value and simplify. The result, 5, is the output.

DOMAIN NOTATION

Domain: {x ∈ ℝ | x ≥ 0} or [0, ∞)

Read: "all real numbers x such that x is greater than or equal to 0." Set-builder notation and interval notation are both accepted in IB exams.

RANGE NOTATION

Range: {y ∈ ℝ | y > −3} or (−3, ∞)

The range is found by considering all possible outputs. A round bracket ( means the endpoint is excluded; a square bracket [ means it is included.

⚠️ Common Domain Restrictions

Watch for these: (1) You cannot divide by zero, so exclude any x that makes a denominator equal to 0. (2) You cannot take the square root of a negative number (in the reals), so the expression inside a square root must be ≥ 0. (3) Real-world contexts often add restrictions — for example, the number of people must be a positive whole number.

SECTION 5

Domain & Range in Context

In the IB Applications and Interpretation course, functions almost always appear inside a real-world scenario. This means you need to determine the domain and range from the context, not just from the algebra. The diagram below shows the graph of a function that models the height of a ball thrown into the air. Study it carefully — it illustrates how context constrains both domain and range.

The graph shows h(t) = −5t² + 20t + 2. The domain is restricted to 0 ≤ t ≤ 4.1 because time starts at the throw and ends when the ball hits the ground. The range is 0 ≤ h ≤ 22 because the ball cannot go below ground level and reaches a maximum height of 22 metres.

Common exam-style questions and how to tackle them

Question	How to Find It	Example Answer
What is f(3)?	Substitute t = 3 into the function.	h(3) = −5(9) + 60 + 2 = 17 metres
What does f(0) represent?	Interpret: the output when the input is 0.	h(0) = 2 → the ball's initial height is 2 m.
What is the domain?	Identify the smallest and largest meaningful input values from the context.	0 ≤ t ≤ 4.1 (throw to landing).
What is the range?	Identify the minimum and maximum output values the function actually reaches.	0 ≤ h ≤ 22 (ground to peak).

SECTION 6

Worked Example

A smartphone app charges a monthly subscription fee. The total cost in dollars after m months is modelled by C(m) = 4.99m + 12, where the 12 represents a one-time setup fee. Let's work through a full set of IB-style questions.

Interpreting C(m) = 4.99m + 12

Step 1 — Evaluate C(6)Replace m with 6: C(6) = 4.99(6) + 12 = 29.94 + 12.

C(6) = 41.94 dollars

Step 2 — Interpret C(6) in contextC(6) = 41.94 means that after 6 months of using the app, the total cost to the user is $41.94, which includes the $12 setup fee plus six monthly payments of $4.99.

After 6 months the total cost is $41.94.

Step 3 — State the domainThe input m represents the number of months of subscription. You cannot subscribe for a negative number of months, so m ≥ 0. If the app offers a maximum 24-month contract, then m ≤ 24. Additionally, months are whole numbers in this context.

Domain: {m ∈ ℤ | 0 ≤ m ≤ 24}

Step 4 — Find the rangeSubstitute the smallest domain value: C(0) = 12. Substitute the largest: C(24) = 4.99(24) + 12 = 119.76 + 12 = 131.76. Since C is increasing, all output values fall between these.

Range: {C ∈ ℝ | 12 ≤ C ≤ 131.76}

Step 5 — Interpret the 4.99 and the 12In context, 4.99 is the rate of change — the cost per month. The 12 is the initial value — the one-time setup fee charged even before any monthly payments.

$4.99/month rate + $12 setup fee

SECTION 7

Common Pitfalls & Tips

Students frequently lose marks on function notation questions not because the math is hard, but because of small misunderstandings. The table below summarizes the most common pitfalls and how to avoid them.

Avoid these mistakes to protect your IB marks

Pitfall	What Students Do Wrong	How to Fix It
Treating f(x) as multiplication	Reading f(x) as "f times x" and trying to distribute.	Remember: f(x) means "f of x" — it names the function and its input, not a product.
Ignoring context for domain	Writing "all real numbers" when the scenario only allows positive integers.	Always re-read the problem. Ask: What values of the input make sense in this situation?
Confusing range with codomain	Stating the range is ℝ without checking which outputs the function actually produces.	The range is only the set of outputs that are actually attained. Sketch or calculate to verify.
Forgetting to interpret	Giving a numeric answer without explaining what it means in context.	IB questions that say "interpret" require a sentence linking the number to the real-world situation.

💡 IB EXAM TIP

Whenever an IB question uses the word "interpret" or "state what … represents", you need a complete sentence that connects the mathematical result to the real-world context. For example, don't just write "f(3) = 17." Instead, write "After 3 seconds, the ball is 17 metres above the ground." This habit alone can be worth multiple marks across a paper.

SECTION 8

Connection to Advanced Topics

Function notation and the ideas of domain and range form the foundation for virtually everything else in the IB Math AI course. The table below shows how these basics connect to more advanced topics you'll encounter later.

SL 2.1 foundations and their advanced extensions

What You Learn Now (SL 2.1)	Where It Leads
Evaluating f(x) at a specific input	Composite functions f(g(x)) — plugging one function into another (SL 2.5)
Domain restrictions	Asymptotes and holes in rational functions; domains of logarithms (SL 2.7)
Range from a graph	Finding the range of transformed functions using translations and reflections (SL 2.4)
Interpretation in context	Modelling with linear, quadratic, exponential, and sinusoidal functions (SL 2.5–2.6)
Understanding inputs and outputs	Inverse functions f⁻¹(x) — reversing input and output (SL 2.3)

Mastering the language of functions now will pay dividends throughout the course. Every modelling question, every graph interpretation, and every optimization problem relies on your ability to read and write function notation fluently. As you move into HL topics or cross-curricular projects, you'll find that the domain-range framework translates directly into constraint analysis in real-world applications.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Explain, in your own words, why the relationship "each student is assigned to the classroom they are currently sitting in" is a function, while "each student is assigned to every teacher they have ever had" might not be a function. Use the terms input, output, and exactly one in your answer.

PROBLEM 2 — BASIC CALCULATION

Given g(x) = 2x² − 3x + 1, find g(−2) and g(0).

PROBLEM 3 — INTERMEDIATE

A function is defined by f(x) = √(10 − 2x). State the domain of f and find the range. Express your answers in interval notation.

PROBLEM 4 — APPLIED

A taxi company charges according to the function C(d) = 3.50 + 1.20d, where d is the distance travelled in kilometres and C is the cost in euros. (a) Find C(15) and interpret your answer. (b) State the domain and range in context. (c) A customer has a budget of €25. What is the maximum distance they can travel?

PROBLEM 5 — CRITICAL THINKING

Two students are arguing. Amir says: "If two different functions have the same domain and the same range, they must be the same function." Priya says: "That's not true — you can have different functions with identical domains and ranges." Who is correct? Support your argument with a specific pair of functions as evidence.

SUMMARY

Lesson Summary

A function is a rule that assigns exactly one output to each input. We write this using function notation f(x), where f names the function and x is the input. To evaluate, substitute the input value and simplify. The domain is the set of all allowable inputs, and the range is the set of all possible outputs. Both can be expressed in set-builder or interval notation.

In the IB Applications and Interpretation course, interpretation in context is essential — you must explain what a function's input, output, and specific values mean within the real-world scenario described. Always check for contextual domain restrictions (negative time, non-integer people, etc.) and remember that the range is only the outputs the function actually achieves, not all theoretically possible values. These fundamentals — notation, domain, range, and interpretation — are the building blocks for every modelling topic you will encounter in the rest of the course.