# Domain and Range

> IB Mathematics AA HL · Unit 2: Functions
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u2-domain-and-range/

This module teaches you to find domain and range for algebraic, composite, and inverse functions, a foundational skill that appears in almost every functions question on IB AA HL exams.

**Prerequisites:** [Basic function concepts and notation](https://www.owlsprep.com/study/ib-math-aa-hl-u2-introduction-to-functions/)

## Learning objectives

- Identify domain restrictions for functions from algebra, graphs, and context
- Calculate range of common and restricted-domain functions
- Find domain and range of composite and inverse functions
- Avoid common exam traps involving domain and range

## Finding Domain Algebraically

**Domain Restriction** — A value of x that makes a function undefined is excluded from the domain. Common restrictions include division by zero, negative values under even roots, and non-positive arguments for logarithms.

To find the domain, start with all real numbers, then exclude any x that violate the common restrictions above. Always check all parts of the function for restrictions.

**Worked example:** Find the domain of $f(x) = \frac{1}{\sqrt{4 - x^2}}$

1. First, identify restrictions: the expression under the square root must be positive (it is in the denominator, so cannot be zero):
2. $$4 - x^2 > 0$$
3. Rearrange the inequality:
4. $$x^2 < 4 \implies -2 < x < 2$$
5. There are no other restrictions, so the domain is:
6. $$(-2, 2)$$

> **Exam tip:** Always check if a restricted value is open or closed on the number line for interval notation.

*Calculator:* forbidden

## Finding Range of Functions

Two common methods for finding range: (1) rearrange the function to solve for x in terms of y, then find the domain of this inverse relation, or (2) use known properties of the function (e.g., vertex of a quadratic) to find minimum/maximum values.

**Worked example:** Find the range of $f(x) = x^2 - 4x + 3$ for $x \in [1, 4]$

1. Start by completing the square to find the vertex:
2. $$f(x) = (x - 2)^2 - 1$$
3. This parabola opens upward, so the minimum output occurs at the vertex $x=2$, which gives $y=-1$.
4. Next check the endpoints of the domain interval for the maximum:
5. $f(1) = (1-2)^2 -1 = 0$, $f(4) = (4-2)^2 -1 = 3$
6. The maximum output is 3, so the range is:
7. $$[-1, 3]$$

*Calculator:* forbidden

## Domain and Range of Composite Functions

For a composite function $f(g(x))$, the domain is the set of all $x$ such that: (1) $x$ is in the domain of $g$, and (2) $g(x)$ is in the domain of $f$. You must check both conditions.

The domain of $f(g(x))$ is the intersection of the domain of $g$ and the set of $x$ where $g(x)$ is valid for $f$.

> tip

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

1. First find domain of $g$: $x - 1 \neq 0 \implies x \neq 1$.
2. Next, $g(x)$ must be non-negative for $f$: $\frac{1}{x - 1} \geq 0 \implies x - 1 > 0 \implies x > 1$.
3. Intersect both conditions: $x > 1$ satisfies both $x \neq 1$ and the non-negative requirement.
4. Domain is $(1, \infty)$.

*Calculator:* forbidden

## Domain and Range of Inverse Functions

A key relationship for inverse functions swaps domain and range: $\text{dom}(f^{-1}) = \text{range}(f)$ and $\text{range}(f^{-1}) = \text{dom}(f)$. This saves time on exam questions that don't require you to find the full inverse expression.

**Worked example:** If $f(x) = e^{x + 3}$ with domain $(-\infty, \infty)$, what is the range of $f^{-1}(x)$?

1. Apply the inverse property: range of $f^{-1}$ equals domain of $f$.
2. Domain of $f$ is all real numbers $(-\infty, \infty)$.
3. So range of $f^{-1}(x)$ is $(-\infty, \infty)$.

*Calculator:* forbidden

## Common pitfalls

- **Wrong:** Allowing the expression under an even root in the denominator to equal zero.
  - Why it fails: Zero under the root gives zero in the denominator, which is undefined.
  - Correct: Use a strict inequality ($>0$) for expressions under even roots in denominators, non-strict ($\geq 0$) only if the root is not in the denominator.
- **Wrong:** For a quadratic on a restricted interval, only using the vertex to find range.
  - Why it fails: The vertex gives the global min/max, but endpoints can give a larger/smaller value on the restricted domain.
  - Correct: Always evaluate the quadratic at the endpoints of the domain interval to find the full range.
- **Wrong:** For composite $f(g(x))$, only checking that $g(x)$ is in the domain of $f$, and forgetting restrictions on $g$.
  - Why it fails: Restrictions on the inner function do not disappear when composing.
  - Correct: Always intersect the domain of $g$ with the set of valid $x$ for the outer function $f$.
- **Wrong:** Assuming a function automatically has domain of all real numbers unless stated otherwise.
  - Why it fails: Rational, root, and log functions have inherent domain restrictions built into their definition.
  - Correct: Always check for denominators, even roots, and log arguments when finding domain of any function.

## Cheatsheet

| Function Type | Common Domain | Common Range |
| --- | --- | --- |
| Polynomial | $(-\infty, \infty)$ | Odd degree: $(-\infty, \infty)$; Even degree: bounded on one side |
| Rational $\frac{1}{x-a}$ | $x \neq a$ | $y \neq 0$ |
| $\sqrt{ax + b}$ | $ax + b \geq 0$ | $[0, \infty)$ |
| $\log_k(ax + b)$ | $ax + b > 0$ | $(-\infty, \infty)$ |
| $e^{ax + b}$ | $(-\infty, \infty)$ | $(0, \infty)$ |
| Inverse $f^{-1}(x)$ | Range of original $f$ | Domain of original $f$ |

## What's next

Domain and range are foundational to every topic in functions for IB AA HL. You will use this skill to find valid solutions to equations, sketch graphs, evaluate composite and inverse functions, and solve optimization problems. Getting domain and range right is almost always required to earn full marks on longer multi-part questions, so mastering the common restrictions and methods is critical. Next, you will build on this knowledge to study composite and inverse functions in depth, then learn to sketch and transform graphs of common function types.

- [Composite and Inverse Functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-composite-and-inverse-functions/)
- [Transformations of functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-transformations-of-functions/)
- [Linear and quadratic functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-linear-and-quadratic-functions/)

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