Function concepts, domain, range and notation

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

1. 1. Definition of a Function★☆☆☆☆⏱ 5 min

📘 Definition

Function

$f$ , $f (x)$

A relation between a set of inputs (domain) and a set of outputs (range) where every input maps to exactly one output. Any input that maps to more than one output is not a function.

Example:

$f (x) = x + 3$ maps input $2$ to output $5$ , only one output so it is a function.

The simplest way to check if a graph represents a function is the vertical line test. If any vertical line intersects the graph more than once, the relation is not a function.

📐 Worked Example

Determine if the circle defined by $x^{2} + y^{2} = 16$ is a function.

1
Apply the vertical line test by testing the vertical line $x = 0$ , which intersects the graph at:
2
$(0, 4) \text{ and } (0, -4)$
3
One input $x = 0$ maps to two different outputs $y = 4$ and $y = - 4$ , so:
4
The relation is not a function.

2. 2. Function Notation★★☆☆☆⏱ 5 min

IB exams use two standard forms of function notation, which are interchangeable:

$f (x) = expression$ : The most common form, used for all general contexts
$f : x \mapsto expression$ : Mapping notation, used to explicitly define how inputs map to outputs.

📐 Worked Example

Given $f : x \mapsto 3 x^{2} - 2 x - 8$ , find $f (- 1)$ and all $x$ where $f (x) = 0$ .

1
For $f (- 1)$ , substitute $x = - 1$ :
2
$f(-1) = 3(-1)^2 - 2(-1) - 8 = 3 + 2 - 8 = -3$
3
Set $f (x) = 0$ and factor the quadratic:
4
$3x^2 - 2x - 8 = 0 \implies (3x + 4)(x - 2) = 0$
5
Solve for $x$ to get the solutions:
6
$x = -\frac{4}{3} \text{ or } x = 2$

3. 3. Finding the Domain★★☆☆☆⏱ 6 min

📘 Definition

Domain

The set of all real input values $x$ for which the function is defined. Unless a restricted domain is given, we find the natural domain (all valid inputs).

📐 Worked Example

Find the domain of $f (x) = \frac{1}{x - 3} + x + 1$ .

1
First, apply the denominator restriction:
2
$x - 3 \neq 0 \implies x \neq 3$
3
Next, apply the square root restriction:
4
$x + 1 \geq 0 \implies x \geq -1$
5
Combine the two restrictions to get the domain, written in interval notation:
6
$[-1, 3) \cup (3, \infty)$

4. 4. Finding the Range★★★☆☆⏱ 7 min

📘 Definition

Range

The set of all possible output values $y$ that the function produces when all inputs from the domain are used.

Common methods for finding range: 1) Graph the function and identify all $y$ -values the graph covers, 2) For quadratics, use vertex form to find the minimum or maximum output, 3) Rearrange the function to solve for $x$ and find the domain of the inverse relation.

📐 Worked Example

Find the range of the quadratic function $f (x) = x^{2} + 2 x - 3$ .

1
Rewrite the quadratic in vertex form by completing the square:
2
$f(x) = (x^2 + 2x) - 3 = (x + 1)^2 - 1 - 3 = (x + 1)^2 - 4$
3
A squared term is always non-negative for real $x$ :
4
$(x + 1)^2 \geq 0$
5
Adjust for the constant term to find the minimum output:
6
$(x + 1)^2 - 4 \geq -4$
7
The range is all real numbers greater than or equal to -4:
8
$[-4, \infty)$

5. Common Pitfalls

Wrong move:

Forgetting to exclude values that make denominators zero when finding domain

Why:

Many students only check for square root restrictions and miss denominator restrictions, leading to an incorrect domain

Correct move:

Always check both denominators (not equal to zero) and even roots (non-negative expression) when calculating domain

Wrong move:

Confusing domain (x-values) with range (y-values)

Why:

Mixing up input and output leads to giving the wrong answer in exam questions

Correct move:

Label domain as input (x) and range as output (y) on your working to avoid confusion

Wrong move:

Assuming all quadratics have range $[k, \infty)$ regardless of leading coefficient

Why:

Students remember the result for positive leading coefficients but forget negative leading coefficients flip the parabola

Correct move:

For a quadratic $a (x - h)^{2} + k$ , range is $[k, \infty)$ if $a > 0$ and $(- \infty, k]$ if $a < 0$

Wrong move:

Thinking all relations between x and y are functions

Why:

New students often assume any graph or relation is a function, leading to wrong identification answers

Correct move:

Always apply the vertical line test to confirm a relation is a function before proceeding

6. Quick Reference Cheatsheet

Concept	Key Definition	Checklist
Function	Each input maps to exactly one output	Passes vertical line test
Domain	Set of all valid inputs (x-values)	Check denominator ≠ 0, √(expression) ≥ 0
Range	Set of all possible outputs (y-values)	Read from graph, use vertex form for quadratics
f(a)	Substitute x = a into function, get output	Always substitute correctly, watch negative signs
f(x) = a	Set function equal to a, solve for input x	Check solutions against domain

When this came up on past exams

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

2025 · 1
Find domain of a rational function
2024 · 1
Identify function from set of relations
2023 · 1
Find range of a quadratic function

What's Next

This foundational sub-topic underpins every other function topic in IB AA SL. You will use your understanding of domain, range and function notation to work with more advanced function concepts including composite and inverse functions, and to analyze specific function types like quadratic, exponential, logarithmic and trigonometric functions. These skills are also required for solving equations, modeling real-world scenarios, and sketching graphs of functions, which appear on both Paper 1 and Paper 2 of the IB AA SL exam.