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The domain of a function is the complete set of all possible independent input values (typically x) for which the function is mathematically defined and outputs a real number ( $f(x) \in \mathbb{R}$ ).

If you consider a function as an algebraic processing machine:

\text{Input } x \in \text{Domain} \longrightarrow \fbox{ Function } f \longrightarrow \text{Output } f(x) \in \text{Range}

If an input value produces an impossible mathematical operation—such as division by zero, taking the even root of a negative number, or evaluating the logarithm of a non-positive number—that value cannot belong to the domain and must be excluded.

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Mathematical Domain Notation

UK exam boards accept three standard conventions for expressing domains:

Inequality Notation: Direct algebraic inequalities.
- Example: $x > 2$ , or $-3 \le x \le 5$ .
Set-Builder Notation: Formal set notation stating membership in the real numbers ℝ\mathbb{R}.
- Example: $\{x \in \mathbb{R} : x \neq 4\}$ ("the set of all real numbers x such that x does not equal 4").
- Example: $\{x \in \mathbb{R} : x \ge 0\}$ .
Interval Notation: Bracket notation representing continuous spans on the real number line.
- Round parentheses (a, b) indicate non-inclusive boundaries (< or >).
- Square brackets [a, b] indicate inclusive boundaries ( $\le$ or $\ge$ ).
- Example: $[3, \infty)$ represents $x \ge 3$ .
- Example: $(-\infty, 1) \cup (1, \infty)$ represents $x \neq 1$ .

The Three Fundamental Domain Restrictions

Unless a function involves one of three specific algebraic operations, its natural domain is simply all real numbers ( $x \in \mathbb{R}$ ).

Whenever you analyse a function, check for these three critical conditions:

Restriction 1: Division by Zero (Rational Functions)

Division by zero is undefined in mathematics. For any rational function:

f(x) = \dfrac{P(x)}{Q(x)}

The domain requires that the denominator cannot equal zero:

Q(x) \neq 0

Values of x that make Q(x) = 0 represent vertical asymptotes or points of discontinuity (holes) and must be excluded from the domain.

Restriction 2: Even Roots of Negative Numbers (Radical Functions)

In real analysis, you cannot take an even root (square root $\sqrt(x)$ , fourth root $\sqrt[4](x)$ , etc.) of a negative number:

f(x) = \sqrt{g(x)} \implies g(x) \ge 0

Odd Roots vs. Even Roots:

Odd roots (such as cube roots $\sqrt[3]{x}$ or fifth roots $\sqrt[5]{x}$ ) can accept negative inputs because $(-2)^3 = -8 \implies \sqrt[3]{-8} = -2$ .

Therefore, for $f(x) = \sqrt[3]{g(x)}$ , the radical itself imposes no domain restriction ( $x \in \mathbb{R}$ , provided $g(x)$ is defined).

If an even root appears in a denominator, it cannot equal zero either, turning the weak inequality into a strict inequality:

f(x) = \dfrac{1}{\sqrt{g(x)}} \implies g(x) > 0

Restriction 3: Logarithms of Non-Positive Numbers

The logarithm of zero or a negative number is undefined over the set of real numbers. For any base $a > 0, a \neq 1$ :

f(x) = \ln(g(x)) \quad \text{or} \quad f(x) = \log_a(g(x)) \implies g(x) > 0

The argument of the logarithm must be strictly positive.

Worked Example

Polynomial Functions

Find the domain of each of the following polynomial functions:

(a) $f(x) = 5x^3 - 4x^2 + 11$

(b) $g(x) = 1 - \dfrac{1}{4}x$

Step 1: Check for algebraic restrictions

Are there any denominators containing variables? No.
Are there any even roots? No.
Are there any logarithms or inverse trigonometric expressions? No.

Step 2: Formulate the domain statement

Both functions are polynomials defined for every real number.

\text{Domain of } f(x): x \in \mathbb{R} \quad \text{or} \quad (-\infty, \infty)

\text{Domain of } g(x): x \in \mathbb{R} \quad \text{or} \quad (-\infty, \infty)

Practice Questions & Solutions

Score:

Find the domain of the function:

$\text{[math]}$

Express your answer in set-builder notation.

Solution

Set the denominator equal to zero to find the excluded input value:

$\text{[math]}$

The numerator is a polynomial defined for all real numbers.

The only restriction is that

$\text{[math]}$ .

In set-builder notation:

$\text{[math]}$

Determine the domain of the function:

$\text{[math]}$

Express your answer using inequality notation.

Solution

The expression inside the square root is in the denominator. Therefore, the radicand must be strictly greater than zero:

18 - 6x > 0

Rearrange the inequality:

18 > 6x

Divide by 6:

$\text{[math]}$

The numerator $\text{[math]}$ is defined for all real numbers.

Therefore, the domain is:

x <3

Find the domain of the function:

$\text{[math]}$

Solution

For the natural logarithm to be defined, its argument must be strictly positive:

$\text{[math]}$

A quotient $\text{[math]}$ is strictly positive when either:

Both numerator and denominator are positive:

$\text{[math]}$ and $\text{[math]}$

Intersection: $\text{[math]}$

Both numerator and denominator are negative:

$\text{[math]}$ and $\text{[math]}$

Intersection: $\text{[math]}$

(Alternatively, note that the critical values are $\text{[math]}$ and $\text{[math]}$ . Testing values in the intervals $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ reveals the quotient is positive when $\text{[math]}$ or $\text{[math]}$ ).

Domain:

$\text{[math]}$

The function k(x) is defined by:

$\text{[math]}$

Find the set of values of x for which k(x) is defined, expressing your answer in interval notation.

Solution

The function k(x) is the sum of two square root terms. Both must be defined simultaneously:

First radical condition:

$\text{[math]}$

Second radical condition:

$\text{[math]}$

Find the intersection of both intervals:

$\text{[math]}$

In interval notation:

$\text{[math]}$

Find all values of $\text{[math]}$ in the interval $\text{[math]}$ that must be excluded from the domain of:

$\text{[math]}$

Solution

The function is undefined wherever the denominator is zero:

$\text{[math]}$

Within the specified interval

$\text{[math]}$ :

First quadrant solution:

$\text{[math]}$
Second quadrant solution:

$\text{[math]}$
Excluded values:

$\text{[math]}$

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Gianpiero Placidi

UK-based Chemistry graduate with a passion for education, providing clear explanations and thoughtful guidance to inspire student success.

Theory

Exponent Rules

Composition of Functions

Constant Functions

Convergent and Divergent Sequences

Even and Odd Functions

Exponential Function

Graphing Parabolas

Arithmetic Sequence

Mean Value Theorem

Rolle’s Theorem

Intervals of Increase and Decrease

Periodic Functions

Graphing rational function I

Graphing rational function III

Inflection Points

Increasing and Decreasing Functions

Piecewise Functions

Trigonometric Functions

Linear Function Problems

Domain of a Function

Maxima, Minima and Inflection Points

Radical Functions

Function

Geometric Sequence

Coordinates of X Intercept Explained: GCSE and A Level

Bounded Sequence: Monotonic and Non-Monotic

L’Hôpital’s Rule

Graphing polynomial function I

Types of Functions

Coordinates in the Plane

Optimization

nth Term

Graphing rational function V

Limit of a Sequence

Absolute Value Function

Sequences

Geometric Sequence Problems And Solutions

Arithmetic Sequence Problems with Solutions

Logarithmic Functions

Concave and Convex Functions: Theory and Examples

Rational Functions: Graphs, Hyperbolas and Asymptotes Explained

Bounded Function Examples: Theory and Practice Questions

Maxima and Minima Examples With Solutions

Asymptote Functions: Theory and Practice Problems

Absolute Maxima Explained: Master Max/Min On Intervals

Inverse Function Explained: Simple Steps for A-Level Maths

Graphs of Quadratic Equations: Practice Questions and Answers

Applying Limits in Math: Limit Laws Made Clear

Monotone Sequences: Theory and Exercises

Linear Functions Worksheet With Solutions

Formulas

Sequence Formulas | Arithmetic, Geometric and More

Exercises

Inflection Points Worksheet

Sequences Problems

Concavity and Convexity Worksheet

Increase and Decrease Worksheet

Limit of Sequence Problems

Applications of Derivatives Worksheet

Exponential, Logarithmic and Trigonometric Functions Worksheet

Mean Value Theorem Word Problems

Rational Functions Worksheet

L’Hopital’s Rule Problems

Even and Odd Functions Worksheet

Asymptotes Worksheet

Absolute Value Functions Worksheet

Piecewise Functions Worksheet

Optimization Word Problems

Linear Function Word Problems

Finding Maxima and Minima | Worksheet with Worked Solutions

X and Y Intercepts Worksheet | Exercises with Full Solutions

Finite and Infinite Sequences Worksheet: GCSE and A Level Extension

Composition of Functions Worksheet

Maximum and Minimum Word Problems Worksheet with Answers

Quadratic Function Word Problems

Graphing Worksheet

Finding Inverse Functions Worksheet with Answers

Domain of a Function Worksheet

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Optimization

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Kourtney Sison

August 2026

I get it somehow, there are others who are unfamiliar and I have a hard time with it but it’s okay that I get it.

Vanessa - Editorial Manager Superprof UK

September 2026

Thanks for sharing your thoughts! It’s completely normal to find some parts unfamiliar or more difficult than others. The important thing is that you’re making progress and understanding more as you practise. Keep going! 😊

Kourtney Sison

August 2026

At least I understood some of it, but there are others that I’m confused about and have a hard time understanding, but I get it.

Vanessa - Editorial Manager Superprof UK

September 2026

Thanks for sharing! It’s great that you’ve understood some of it already. Don’t worry if a few parts are still confusing — keep practising and take your time with the trickier sections. You’ll get there! 😊

Normina Mangogawan

August 2026

The lesson was a little difficult for me, especially the word problems, but the examples helped me understand it better.

Vanessa - Editorial Manager Superprof UK

September 2026

Thanks for your feedback! Word problems can definitely be tricky, so we’re glad the examples helped make the lesson clearer. Keep practising — it gets easier with time! 😊

Kourtney Sison /Normina Mangogawan

August 2026

Thankyouuu po

Summarise with AI:

Chapters

Mathematical Domain Notation
The Three Fundamental Domain Restrictions
Worked Example
Practice Questions & Solutions

Domain Worksheet: Learn Domain Rules, Notation and Restrictions

Mathematical Domain Notation

The Three Fundamental Domain Restrictions

Restriction 1: Division by Zero (Rational Functions)

Restriction 2: Even Roots of Negative Numbers (Radical Functions)

Restriction 3: Logarithms of Non-Positive Numbers

Worked Example

Polynomial Functions

Practice Questions & Solutions

Theory

Exponent Rules

Composition of Functions

Constant Functions

Convergent and Divergent Sequences

Even and Odd Functions

Exponential Function

Graphing Parabolas

Arithmetic Sequence

Mean Value Theorem

Rolle’s Theorem

Intervals of Increase and Decrease

Periodic Functions

Graphing rational function I

Graphing rational function III

Inflection Points

Increasing and Decreasing Functions

Piecewise Functions

Trigonometric Functions

Linear Function Problems

Domain of a Function

Maxima, Minima and Inflection Points

Radical Functions

Function

Geometric Sequence

Coordinates of X Intercept Explained: GCSE and A Level

Bounded Sequence: Monotonic and Non-Monotic

L’Hôpital’s Rule

Graphing polynomial function I

Types of Functions

Coordinates in the Plane

Optimization

nth Term

Graphing rational function V

Limit of a Sequence

Absolute Value Function

Sequences

Geometric Sequence Problems And Solutions

Arithmetic Sequence Problems with Solutions

Logarithmic Functions

Concave and Convex Functions: Theory and Examples

Rational Functions: Graphs, Hyperbolas and Asymptotes Explained

Bounded Function Examples: Theory and Practice Questions

Maxima and Minima Examples With Solutions

Asymptote Functions: Theory and Practice Problems

Absolute Maxima Explained: Master Max/Min On Intervals

Inverse Function Explained: Simple Steps for A-Level Maths

Graphs of Quadratic Equations: Practice Questions and Answers

Applying Limits in Math: Limit Laws Made Clear

Monotone Sequences: Theory and Exercises

Linear Functions Worksheet With Solutions

Formulas

Sequence Formulas | Arithmetic, Geometric and More

Exercises

Inflection Points Worksheet

Sequences Problems

Concavity and Convexity Worksheet

Increase and Decrease Worksheet

Limit of Sequence Problems

Applications of Derivatives Worksheet

Exponential, Logarithmic and Trigonometric Functions Worksheet

Mean Value Theorem Word Problems

Rational Functions Worksheet

L’Hopital’s Rule Problems

Even and Odd Functions Worksheet

Asymptotes Worksheet

Absolute Value Functions Worksheet

Piecewise Functions Worksheet

Optimization Word Problems

Linear Function Word Problems

Finding Maxima and Minima | Worksheet with Worked Solutions