What domain and range mean
Definition
The domain of a function is the complete set of values you are allowed to put into it – the inputs. The range is the complete set of values the function can produce from that domain – the outputs.
A function is only properly defined once you say what its domain is. Change the domain and you can change the range, the inverse, and whether the inverse even exists.
In exams the domain is often given to you, for example f(x) = 2x + 1, x ≥ 0. When it is not given, you take the largest set of real numbers for which the expression makes sense, excluding values that divide by zero or put a negative inside a square root.
How to find the domain and range: 6 steps
- Check the question for a stated domain. If one is given, use it. Do not widen it.
- If no domain is given, hunt for restrictions. Exclude values that make a denominator zero or take the square root of a negative number.
- Sketch or picture the graph of the function over that domain. A quick sketch is the fastest reliable route to the range.
- Find the range from the sketch. Mark the lowest and highest output values, checking turning points and endpoints, and say whether each end is included.
- Write both answers in correct set notation – for example x ∈ ℝ, x ≠ 2 for a domain, or f(x) ≥ 3 for a range.
- Sanity-check with test values. Substitute one allowed and one excluded input to confirm your answer behaves as claimed.
Worked examples
Example 1 – a restricted domain
f(x) = 1/(x – 2) with no stated domain.
The denominator cannot be zero, so x ≠ 2. Domain: x ∈ ℝ, x ≠ 2.
The graph is a hyperbola which takes every real value except 0, so the range is f(x) ∈ ℝ, f(x) ≠ 0.
Example 2 – range from a sketch
f(x) = x² – 4x + 3, x ∈ ℝ.
Completing the square gives f(x) = (x – 2)² – 1, a parabola with minimum at (2, -1). The domain is all real numbers; the range is f(x) ≥ -1.
The mistake examiners see most
Students find the range by substituting the endpoints of the domain and nothing else. That fails whenever the graph has a turning point inside the domain. In Example 2, testing no values near x = 2 would hide the minimum at -1 entirely. Always sketch, and always look for turning points before writing the range.
Examiner tip
Domain and range questions are worth easy marks because the method never changes: respect the stated domain, sketch, read off the outputs, use correct notation. The marks are lost on notation and on turning points, not on hard algebra. When a question later asks for an inverse, the domain of the inverse is exactly the range you just found – so getting the range right carries marks forward through the whole question.
Exam questions with video solutions
Try each question yourself first, then open the full video solution to check your working. Question 1 is from Edexcel C3 June 2014; Question 2 is from OCR C3 June 2013. The topics these questions draw on are linked under Helpful Tutorials.
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