# Function concepts, domain, range and notation

> IB Mathematics: Analysis and Approaches SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u2-function-concepts-domain-range-and/

This sub-topic introduces the core definition of a function, standard IB notation, and methods to find domain and range for common functions. It is the foundational building block for all further work on functions in IB AA SL.

**Prerequisites:** Basic set and interval notation; Coordinate geometry and graphing of equations

## Learning objectives

- Distinguish between functions and non-function relations using the vertical line test
- Use standard IB function notation correctly to evaluate and solve for inputs/outputs
- Calculate the domain of a function by identifying all invalid input values
- Find the range of common functions including quadratics and rational functions

## 1. Definition of a Function

**Function** — 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.

*Notation:* $f$, $f(x)$

*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.

> **mnemonic**
>
> One input → one output is the rule. If a vertical line crosses twice, it's not a function for you.

**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. Function Notation

IB exams use two standard forms of function notation, which are interchangeable:
1. $f(x) = \text{expression}$: The most common form, used for all general contexts
2. $f:x \mapsto \text{expression}$: Mapping notation, used to explicitly define how inputs map to outputs.

**Exam command terms**

Common exam question phrasing for function notation has consistent meanings:

- **Find $f(a)$** — Substitute $x=a$ into the function to get the output *($f(2)$ means replace all $x$ with $2$)*

- **Find $x$ where $f(x) = a$** — Set the function equal to $a$ and solve for input $x$ *($f(x)=5$ means solve for $x$ when output equals 5)*

**Worked example:** Given $f:x \mapsto 3x^2 - 2x - 8$, 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. Finding the Domain

**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).

> **tip**
>
> Always check for two common domain restrictions: 1) Denominators can never equal zero, 2) Expressions under even roots (like square roots) can never be negative.

**Worked example:** Find the domain of $f(x) = \frac{1}{x - 3} + \sqrt{x + 1}$.

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. Finding the Range

**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 $f(x) = x^2 + 2x - 3$.

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. $$[-4, \infty)$$

## Common pitfalls

- **Wrong:** Forgetting to exclude values that make denominators zero when finding domain
  - Why it fails: Many students only check for square root restrictions and miss denominator restrictions, leading to an incorrect domain
  - Correct: Always check both denominators (not equal to zero) and even roots (non-negative expression) when calculating domain
- **Wrong:** Confusing domain (x-values) with range (y-values)
  - Why it fails: Mixing up input and output leads to giving the wrong answer in exam questions
  - Correct: Label domain as input (x) and range as output (y) on your working to avoid confusion
- **Wrong:** Assuming all quadratics have range $[k, \infty)$ regardless of leading coefficient
  - Why it fails: Students remember the result for positive leading coefficients but forget negative leading coefficients flip the parabola
  - Correct: For a quadratic $a(x-h)^2 +k$, range is $[k, \infty)$ if $a>0$ and $(-\infty, k]$ if $a<0$
- **Wrong:** Thinking all relations between x and y are functions
  - Why it fails: New students often assume any graph or relation is a function, leading to wrong identification answers
  - Correct: Always apply the vertical line test to confirm a relation is a function before proceeding

## 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 |

## 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.

- [Composite and inverse functions](https://www.owlsprep.com/study/ib-math-aa-sl-u2-composite-and-inverse-functions/)
- [Transformations of graphs of functions](https://www.owlsprep.com/study/ib-math-aa-sl-u2-transformations-of-graphs-of-functions/)
- [Linear functions and their graphs](https://www.owlsprep.com/study/ib-math-aa-sl-u2-linear-functions-and-their-graphs/)

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-aa-sl-u2-function-concepts-domain-range-and/