Domain and Range

IB Mathematics AA HL· 10 min read

1. Finding Domain Algebraically★★☆☆☆⏱ 3 min

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📘 Definition

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 $f (x) = \frac{1}{4 - x ^{2}}$

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.

2. Finding Range of Functions★★★☆☆⏱ 4 min

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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 $f (x) = x^{2} - 4 x + 3$ 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
$[-1, 3]$

3. Domain and Range of Composite Functions★★★★☆⏱ 3 min

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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.

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📐 Worked Example

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

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

4. Domain and Range of Inverse Functions★★★☆☆⏱ 2 min

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A key relationship for inverse functions swaps domain and range: $dom (f^{- 1}) = range (f)$ and $range (f^{- 1}) = dom (f)$ . 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 $(- \infty, \infty)$ .

5. Common Pitfalls

Wrong move:

Allowing the expression under an even root in the denominator to equal zero.

Why:

Zero under the root gives zero in the denominator, which is undefined.

Correct move:

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 move:

For a quadratic on a restricted interval, only using the vertex to find range.

Why:

The vertex gives the global min/max, but endpoints can give a larger/smaller value on the restricted domain.

Correct move:

Always evaluate the quadratic at the endpoints of the domain interval to find the full range.

Wrong move:

For composite $f (g (x))$ , only checking that $g (x)$ is in the domain of $f$ , and forgetting restrictions on $g$ .

Why:

Restrictions on the inner function do not disappear when composing.

Correct move:

Always intersect the domain of $g$ with the set of valid $x$ for the outer function $f$ .

Wrong move:

Assuming a function automatically has domain of all real numbers unless stated otherwise.

Why:

Rational, root, and log functions have inherent domain restrictions built into their definition.

Correct move:

Always check for denominators, even roots, and log arguments when finding domain of any function.

6. Quick Reference Cheatsheet

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

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 composite function
2023 · 2
Determine range of restricted quadratic

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.