Functions, mappings, domain and range
Recognise a function as one output per allowed input · Evaluate function notation · Distinguish domain from range
Section C3.1 · Core · Functions, notation, mappings, domain and range
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Substitution
- Ordered pairs
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Recognise a function as one output per allowed input
- Evaluate function notation
- Distinguish domain from range
One output for each input
A function assigns exactly one output to every input in its stated domain. Different inputs may share an output: x ↦ x² is a function even though −2 and 2 both give 4. An input with two different outputs would fail the definition.
f(3) means the value obtained by applying f to input 3; it is not f multiplied by 3. An arrow diagram lists allowed inputs and connects each to its single output. A graph passes the vertical-line test when any vertical line meets it at most once.
Domain and range
The domain is the set of allowed inputs; the range is the set of outputs actually produced. For f(x) = x² with domain {−2, 0, 3}, the range is {0, 4, 9}. For all real x the same rule has range y ≥ 0.
A formula alone may suggest natural restrictions: division by zero is undefined, and a real square root needs a non-negative radicand. A question or context can impose a smaller domain.
Sources for this lesson
Pause and explain
Can two inputs of a function have the same output?
Worked example
Let f(x) = 2x − 1 for x in {0, 2, 5}. Find the mapping and range.
Show the worked solution
- f(0) = −1, f(2) = 3, f(5) = 9.
- Each allowed input has one output. The range is the set of these three outputs.
Answer 0 ↦ −1, 2 ↦ 3, 5 ↦ 9; range {−1, 3, 9}.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For f(x) = x² + 1 on {−1, 0, 2}, find the range.
Give me a hint
Evaluate once for each allowed input.
Compare my reasoning
- f(−1) = 2, f(0) = 1, f(2) = 5.
- Collect distinct outputs.
Range {1, 2, 5}.
Look for these in your work
- State the allowed inputs
- List outputs without repetition
Does {(1, 3), (2, 3), (2, 4)} define a function from inputs {1, 2}? Explain.
Give me a hint
Check each input separately.
Compare my reasoning
- Input 1 has output 3.
- Input 2 has outputs 3 and 4; they are different.
No: input 2 has two outputs.
Look for these in your work
- Identify the offending input
- Do not object merely to repeated outputs
A ticket charge is C(n) = 4n + 2 for n in {1, 2, 3}. What are the domain and range? Is C(1.5) allowed here?
Give me a hint
Use the stated domain, even if substitution is algebraically possible.
Compare my reasoning
- Inputs are the three allowed ticket counts.
- C(1) = 6, C(2) = 10, C(3) = 14.
- 1.5 is not in the stated domain.
Domain {1, 2, 3}; range {6, 10, 14}; C(1.5) is not allowed in this model.
Look for these in your work
- Respect the discrete domain
- Distinguish formula from model
Common mistakes
- Many-to-one does not mean not a function.
- A range is not necessarily the same set as the domain.
What can you explain now?
Can two inputs of a function have the same output?
Compare with the explanation
Yes; each input must have one output, but outputs may repeat.
For example, both −2 and 2 map to 4 under squaring.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Can two inputs of a function have the same output?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.