Study Guide

Function definitions and notation

IB Mathematics: Analysis and Approaches HLΒ· Unit 2: Functions, Topic 1Β· 15 min read

1. Core Definition of a Functionβ˜…β˜†β˜†β˜†β˜†β± 5 min

πŸ“˜ Definition

Function

f:A→Bf: A \to B

A function is a mapping from a domain set (A) (inputs) to a codomain set (B) (possible outputs) where every element in the domain maps to exactly one element in the codomain.

Example:

The rule (f(x) = x^2) mapping real numbers to real numbers is a function.

Functions can be represented in four common ways: as a mapping diagram, an algebraic equation, a table of values, or a graph. The key requirement of one output per input holds in all representations.

πŸ“ Worked Example

Determine which of the following mapping diagrams represents a function: (1) Inputs {1, 2, 3} map to outputs {2, 4, 6}: 1β†’2, 2β†’4, 3β†’4. (2) Inputs {1, 2, 3} map to outputs {2, 4, 6}: 1β†’2, 1β†’4, 2β†’6.

  1. 1

    Recall the core requirement: every input must map to exactly one output for the relation to be a function.

  2. 2

    Check mapping 1: All inputs 1, 2, 3 each have exactly one output. Multiple inputs can map to the same output, this is allowed.

  3. 3

    Conclusion: Mapping 1 is a function.

  4. 4

    Check mapping 2: Input 1 maps to two different outputs (2 and 4). This violates the function definition.

  5. 5

    Conclusion: Mapping 2 is not a function.

Exam tip:

If an input maps to more than one output it is never a function. If multiple inputs map to one output it is always a function.

2. Notation, Domain and Rangeβ˜…β˜…β˜†β˜†β˜†β± 6 min

πŸ“˜ Definition

Domain

The set of all valid input values for the function. If no domain is explicitly stated, we use the implied domain, which is all real numbers for which the function rule is defined.

Example:

For (f(x) = \sqrt{x}), the implied domain is (x \geq 0).

πŸ“˜ Definition

Range

The set of all actual output values produced by the function, which is always a subset of the codomain.

Example:

For (f(x) = \sqrt{x}), the range is also (f(x) \geq 0).

IB uses two standard forms for function notation: explicit form (f: A \to B, f(x) = x^2) that states domain and codomain, and implicit form (f(x) = x^2) that relies on the implied domain convention.

πŸ“ Worked Example

Find the implied domain and range of (f(x) = \frac{1}{x - 2}), where (x \in \mathbb{R}).

  1. 1

    To find domain, identify values that make the function undefined. A fraction is undefined when the denominator is 0:

  2. 2
    xβˆ’2=0β€…β€ŠβŸΉβ€…β€Šx=2x - 2 = 0 \implies x = 2
  3. 3

    The implied domain is all real numbers except 2:

  4. 4
    Domain={x∈R∣xβ‰ 2}\text{Domain} = \{x \in \mathbb{R} \mid x \neq 2\}
  5. 5

    To find range, let (y = \frac{1}{x-2}) and rearrange to solve for x in terms of y:

  6. 6
    y(xβˆ’2)=1β€…β€ŠβŸΉβ€…β€Šx=1y+2y(x - 2) = 1 \implies x = \frac{1}{y} + 2
  7. 7

    x is undefined when (y = 0), so no input produces an output of 0. The range is:

  8. 8
    Range={y∈R∣yβ‰ 0}\text{Range} = \{y \in \mathbb{R} \mid y \neq 0\}

3. The Vertical Line Test for Graphsβ˜…β˜…β˜†β˜†β˜†β± 4 min

When a relation is given as a graph on the (x)-(y) plane, we can quickly test if it is a function of (x) using the vertical line test.

πŸ“ Worked Example

Does the graph of the unit circle (x^2 + y^2 = 1) represent a function of (x)?

  1. 1

    Draw a vertical line at (x = 0). This line intersects the unit circle at two points: ((0, 1)) and ((0, -1)).

  2. 2

    A single input (x = 0) maps to two different outputs, (y = 1) and (y = -1).

  3. 3

    By the vertical line test, the unit circle is not a function of (x).

βœ“ Quick check

Test your understanding: Which of these relations is a function of (x)?

    • Yes

    • No

    • Yes

    • No

    Reveal answer
    1 β€”

    For any (x > 0), there are two values of (y) (positive and negative root), so this is not a function.

4. Common Pitfalls

Wrong move:

Claiming a relation is not a function because multiple inputs share the same output

Why:

The only requirement for a function is one output per input, multiple inputs can map to the same output

Correct move:

Only check that each input has exactly one output, overlapping outputs do not violate the function definition

Wrong move:

Confusing codomain and range, stating the entire codomain as the range

Why:

Codomain is the set of all possible outputs, while range only includes outputs that actually occur

Correct move:

Always calculate the range explicitly from the function rule and domain

Wrong move:

Forgetting to exclude values that make denominators zero or square root arguments negative when finding implied domain

Why:

Implied domain requires all real numbers where the function rule is defined

Correct move:

Always check for division by zero and negative arguments to even roots when calculating domain

Wrong move:

Using the horizontal line test to check if a relation is a function

Why:

The horizontal line test checks if a function is one-to-one, not if a relation is a function

Correct move:

Always use the vertical line test to check if a relation is a function of (x)

5. Quick Reference Cheatsheet

Concept

Key Meaning

Standard Notation

Function

One output per input

,

Domain

Set of all valid inputs

Range

Set of all actual outputs

Implied Domain

Domain when not stated explicitly

All reals where rule is defined

Vertical Line Test

Test if relation is function of

Multiple intersections = not function

6. Frequently Asked

Is every relation a function?

No. A relation only becomes a function when each input in the domain maps to exactly one output. Relations that map an input to multiple outputs are not functions.

Why is domain always required to define a function?

The same function rule can produce entirely different functions if the domain is changed. For example, (f(x) = x^2) with domain (x \geq 0) is a different function to (f(x) = x^2) with domain all real numbers.

When this came up on past exams

AI-estimated based on syllabus patterns β€” cross-check with official past papers for accuracy. Use only as revision-focus signals.

  • 2025 Β· 1

    Find domain of rational function

  • 2024 Β· 1

    Identify function from relations

  • 2023 Β· 1

    State range of square root function

Going deeper

What's Next

Understanding function definitions and notation is the foundation for every other topic in the IB AA HL functions unit, from composite and inverse functions to graph transformations, and all further function topics including polynomials, rational functions, exponentials and logarithms. Mastering the core definition of a function and correctly calculating domain and range will help you avoid costly, easy-to-prevent mistakes in both paper 1 and paper 2 questions, regardless of the topic. Clear understanding of notation is critical for interpreting exam questions correctly and writing solutions that examiners can mark easily. Next, you will build on this foundation to learn more advanced function concepts.