Concept of function, domain, range, and graph

IB Mathematics: Applications and Interpretation SL· 40 min read

1. Defining a Function: Relations vs Functions★☆☆☆☆⏱ 10 min

📘 Definition

Function

$f (x)$

A relation between inputs and outputs where every input maps to exactly one output. Multiple inputs can map to the same output, but one input cannot map to multiple outputs.

Example:

$f (x) = 2 x + 3$ is a function, as every $x$ gives exactly one $f (x)$

For graphical relations, we use the vertical line test to check if a relation is a function: if any vertical line intersects the graph more than once, the relation is not a function.

📐 Worked Example

Determine if the graph of $x^{2} + y^{2} = 16$ (a circle of radius 4 centered at the origin) is a function.

1
Recall the vertical line test rule: any vertical line crossing the graph more than once means the relation is not a function.
2
Draw a vertical line at $x = 0$ . This line intersects the circle at two points:
3
$y = 4 \text{ and } y = -4$
4
One input $x = 0$ maps to two outputs, so the relation is not a function.

Exam tip:

If an exam question asks "is this relation a function", always justify your answer with the vertical line test or the definition.

2. Finding the Domain of a Function★★☆☆☆⏱ 15 min

The domain is the set of all valid input ( $x$ ) values. We exclude values that create undefined expressions: division by zero, or square (even) roots of negative numbers. For real-world functions, domain is also restricted by context (e.g. time cannot be negative).

📐 Worked Example

Find the domain of $f (x) = 2 x - 6$ , write your answer in interval notation.

1
Identify the restriction: the expression under a square root must be non-negative for real-valued functions.
2
Set up and solve the inequality:
3
$2x - 6 \geq 0 \\ 2x \geq 6 \\ x \geq 3$
4
Write the domain in interval notation, using a closed bracket for the included endpoint $x = 3$ .
5
$\text{Domain: } [3, \infty)$

3. Calculating the Range of a Function★★☆☆☆⏱ 15 min

The range is the set of all possible output ( $y$ ) values a function can produce, given its domain. The range is often restricted by the shape of the function: for example, quadratics have a minimum or maximum that bounds the range.

📐 Worked Example

Find the range of $f (x) = (x - 2)^{2} + 1$ , where the domain is all real numbers $x$ .

1
All squared real numbers are non-negative, so:
2
$(x - 2)^2 \geq 0$
3
Add 1 to both sides to get the inequality for $f (x)$ :
4
$(x - 2)^2 + 1 \geq 1$
5
The minimum output is 1, and $f (x)$ increases to infinity. Write the range in interval notation:
6
$\text{Range: } [1, \infty)$

✓ Quick check

Test your understanding:

What is the range of $f (x) = 5 - x^{2}$ for all real $x$ ?
- $(- \infty, 5]$
- $[5, \infty)$
- $(- \infty, 5)$
- All real numbers
Reveal answer
$(-\infty, 5]$ —
Correct! $- x^{2} \leq 0$ , so $5 - x^{2} \leq 5$ , with maximum value 5.

4. Domain and Range from Graphs★★☆☆☆⏱ 10 min

When reading domain and range from a graph: domain is the spread of the graph along the horizontal $x$ -axis, and range is the spread along the vertical $y$ -axis. Closed circles mean the endpoint is included, open circles mean it is not included.

📐 Worked Example

A function has a graph starting at a closed circle $(- 2, 1)$ , and ending at an open circle $(3, 6)$ . Find the domain and range.

1
Read the $x$ -values covered: from $x = - 2$ (included) to $x = 3$ (not included). So domain is:
2
$[-2, 3)$
3
Read the $y$ -values covered: from $y = 1$ (included) to $y = 6$ (not included). So range is:
4
$[1, 6)$

Exam tip:

Marks are awarded for correct notation, so make sure you use the correct bracket type for included/excluded endpoints.

5. Common Pitfalls

Wrong move:

Assuming all relations are functions, skipping the vertical line test.

Why:

Many common relations like circles, horizontal parabolas, and ellipses are not functions.

Correct move:

Always check if the relation meets the function definition, or apply the vertical line test for graphs.

Wrong move:

Confusing domain and range, swapping input and output values.

Why:

Domain describes inputs (x) and range describes outputs (y), swapping them gives the entirely wrong answer.

Correct move:

Remember: Domain is Horizontal (x-axis), Range is Vertical (y-axis).

Wrong move:

Including values that cause division by zero in the domain.

Why:

Division by zero is undefined, so these inputs are not valid.

Correct move:

Set the denominator equal to zero, solve for x, and exclude those values from your domain.

Wrong move:

Including negative values under a square root in the domain.

Why:

IB AI SL only works with real-valued functions, so square roots of negative numbers are undefined.

Correct move:

Always set the expression under a square root to be greater than or equal to zero.

Wrong move:

Treating open circle endpoints as included in domain/range.

Why:

An open circle means the point is not part of the graph, so the value is not included.

Correct move:

Use closed brackets for closed circles, open parentheses for open circles in interval notation.

6. Quick Reference Cheatsheet

Concept	Definition	Key Rule
Function	Each input maps to exactly 1 output	Use vertical line test for graphs
Domain	Set of all valid input (x) values	Exclude division by zero, negative roots
Range	Set of all valid output (y) values	Check minimum/maximum values of function
Vertical Line Test	Check if graph is a function	If crosses > once, not a function

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
2023 · 2
State range from given graph

What's Next

Now that you have mastered the core concepts of functions, domain, and range, you are ready to explore more complex function topics that build on this foundation. This core definition is used in every function topic you will encounter for IB AI SL, from linear functions to exponential models, and it is essential for solving the real-world application problems that are the focus of this syllabus. Understanding how to restrict domain and interpret range will also help you when you work with composite and inverse functions later in the course. You will apply these skills constantly when drawing graphs, solving equations, and interpreting results in exam questions.