Introduction to Domain and Range Revision Notes | GCSE AQA Higher Further Maths

1. What Are Domain and Range?

Every function is a machine: you feed in a number, it produces a number. But not every machine accepts every input — and not every output is possible. That is exactly what domain and range describe.

The domain is the set of all values you are allowed to put into the function. The range is the set of all values the function can actually produce as outputs. Get these two ideas clear and the rest of the topic falls into place.

Think of the domain as the list of ingredients a recipe will accept. If a recipe says 'no nuts', then nuts are excluded from the domain of that recipe. In the same way, a function might say 'no zero inputs' because dividing by zero is undefined.

The range answers the question: what values can actually come out? For the function $f(x) = x^2$, the output is always zero or positive — you can never square a real number and get a negative result. So the range is $f(x) \geq 0$.

2. Expressing the Domain

The domain can be written in several ways depending on the restriction involved. The two most common forms you will meet at GCSE Further Maths are inequality notation and exclusion statements.

Notice that the phrasing 'all x, except $x = 0$' is perfectly acceptable exam notation — you do not have to force everything into a single inequality. Use whichever form describes the restriction most clearly.

3. Finding the Domain — Worked Examples

Two situations create domain restrictions at GCSE level: division by zero and square roots of negative numbers. Spotting which one applies is the first step in every domain question.

Worked example: Finding the Domain of a Rational Function

Find the domain of $f(x) = \dfrac{5}{x - 3}$.

1. The function has a denominator: $x - 3$ — Division by zero is undefined, so we must find when the denominator equals zero.
2. $x - 3 = 0 \Rightarrow x = 3$ — At x = 3 the function is undefined — this value must be excluded.
3. Domain: all $x$, except $x = 3$ — Every other real number is a valid input. This is the standard exclusion-statement form.

Answer: Domain: all $x$, except $x = 3$

Worked example: Finding the Domain of a Square Root Function

Find the domain of $g(x) = \sqrt{x - 2}$.

1. The function contains $\sqrt{x - 2}$ — You cannot take the square root of a negative number (in real-number maths), so the expression inside must be zero or positive.
2. $x - 2 \geq 0$ — The expression under the root must be greater than or equal to zero.
3. $x \geq 2$ — Add 2 to both sides. This is the domain in inequality notation.
4. Domain: $x \geq 2$ — Any input of 2 or above produces a valid (real) output.

Answer: Domain: $x \geq 2$

4. Finding the Range

Once you know the domain, you can work out the range by asking: given the allowed inputs, what outputs are actually achievable? The graph of the function is your most reliable guide here.

For many GCSE functions, the range comes down to identifying a minimum or maximum output value. The key is to think about what happens to $f(x)$ as $x$ varies across its domain.

Worked example: Finding the Range of a Quadratic Function

Find the range of $f(x) = x^2 - 1$ for all real $x$.

1. Domain: all real $x$ (no restrictions) — There is no division or square root, so every real number is a valid input.
2. $x^2 \geq 0$ for all real $x$, so $x^2 - 1 \geq 0 - 1 = -1$ — Squaring any real number gives zero or a positive result. The smallest value of x² is 0, achieved when x = 0.
3. When $x = 0$: $f(0) = 0^2 - 1 = 0 - 1 = -1$ — The output -1 is actually reached, so we use ≥ (inclusive), not > (strict).
4. Range: $f(x) \geq -1$ — The function produces every value from -1 upwards. There is no upper limit.

Answer: Range: $f(x) \geq -1$

The graph confirms the range: the curve has a minimum point at $(0, -1)$ and rises without bound in both directions. Every $y$-value at or above $-1$ is hit by the curve, so $f(x) \geq -1$ is correct.

Worked example: Finding the Range of a Rational Function

Find the range of $f(x) = \dfrac{5}{x - 3}$ (domain: all $x$, except $x = 3$).

1. $f(x) = \dfrac{5}{x - 3}$ — As x varies across its domain, the denominator (x - 3) takes every non-zero real value.
2. If $\dfrac{5}{x - 3} = 0$, then $5 = 0$, which is impossible. — A fraction with a non-zero numerator can never equal zero. So f(x) = 0 is not in the range.
3. For any target $k eq 0$: set $\dfrac{5}{x-3} = k$, so $x = \dfrac{5}{k} + 3$, which is always a valid $x$. — Every non-zero output value k corresponds to a valid input, so all non-zero values are in the range.
4. Range: all $f(x)$, except $f(x) = 0$ — This mirrors the domain exclusion idea — here it is the output value 0 that is excluded.

Answer: Range: all $f(x)$, except $f(x) = 0$

5. Reading Domain and Range from a Graph

Graphs give you a visual shortcut. The domain is read from the horizontal extent of the graph (the x-axis spread), and the range is read from the vertical extent (the y-axis spread).

• Scan left to right along the x-axis: the domain covers all x-values where the graph exists.
• Scan up and down along the y-axis: the range covers all y-values (f(x) values) that the graph reaches.
• A filled circle on a graph means that endpoint is included (use ≤ or ≥); an open circle means it is excluded (use < or >).
• A vertical asymptote at x = a means x = a is excluded from the domain.
• A horizontal asymptote at y = b means f(x) = b is excluded from the range.

The graph of $f(x) = \dfrac{1}{x-3}$ has a vertical asymptote at $x = 3$ (the curve never touches that line) and a horizontal asymptote at $y = 0$ (the curve never reaches zero). Reading directly from the graph: domain is all $x$ except $x = 3$, and range is all $f(x)$ except $f(x) = 0$.

6. Practice Problems

The best way to consolidate domain and range is to work through a variety of function types. Try each question before checking the solution — the method is always the same: identify the restriction, then state the domain or range using correct notation.

7. Exam Technique and Common Mistakes

Domain and range questions are reliable mark-winners once you know the two traps examiners set most often. Both come down to notation precision and sign awareness.

One final point worth remembering: the domain and range are linked. A change in the domain can change the range. If you restrict the domain of $f(x) = x^2$ to $x > 0$ only, the range becomes $f(x) > 0$ — the value $f(x) = 0$ is no longer achievable because $x = 0$ is excluded from the domain.

Mastering domain and range means you can describe any function completely — not just what it does, but where it works and what it produces. That precision is exactly what exam mark schemes reward.