Read function domains, ranges and inverse conditions
Describe what a function does, find its allowed inputs and outputs, and decide whether it can have an inverse.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which statement defines a function?
What you will learn
- Distinguish a function from a relation that gives one input several outputs
- Find a range from a stated domain
- Use one-to-one behaviour to decide whether an inverse exists
A function assigns one output to each allowed input
The domain is the set of permitted inputs. The range (image set) is the set of outputs that those inputs actually produce. A function may be many-to-one: different inputs can share an output, but one permitted input cannot have two outputs.
A function is one-to-one on a domain when different inputs always give different outputs. An inverse function can undo the original function only when that one-to-one condition holds on the chosen domain.
Check the domain before claiming an inverse
For f(x) = x² on −2 ≤ x ≤ 3, the range is 0 ≤ f(x) ≤ 9, but f(−2) = f(2). The function is many-to-one, so it has no inverse function on that whole domain.
Restricting the domain to 0 ≤ x ≤ 3 makes f one-to-one. Its range is then 0 ≤ y ≤ 9 and the inverse is f⁻¹(x) = √x. The restriction is part of the definition, not an optional afterthought.
Pause and explain
What restriction makes f(x) = x² invertible?
Worked example
For f(x) = (x − 1)² with 1 ≤ x ≤ 4, state the range and whether f has an inverse.
Show the worked solution
- The domain begins at the turning point x = 1, so f is increasing on the stated domain.
- The minimum value is f(1) = 0 and the maximum is f(4) = 9.
- Every output occurs once, so f is one-to-one on this domain.
Answer Range 0 ≤ f(x) ≤ 9; an inverse exists on this restricted domain.
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 ≤ x ≤ 3, find the range and decide whether f is one-to-one.
Give me a hint
Find the least and greatest outputs, then compare two inputs with the same square.
Compare my reasoning
- The least output is 0 at x = 0; the greatest is 9 at x = 3.
- The range is 0 ≤ f(x) ≤ 9.
- f(−1) = f(1) = 1, so two inputs share an output.
Range [0, 9]; not one-to-one, so no inverse function on the full domain.
Look for these in your work
- Range matches endpoints and turning point
- Repeated output identified
- Inverse decision justified
A model f maps a positive side length x to the square's area x². State a suitable domain, range and inverse.
Give me a hint
A physical side length is positive; the area cannot be negative.
Compare my reasoning
- Use x > 0 for a positive side length.
- Then f(x) = x² has range f(x) > 0 and is one-to-one.
- Solving y = x² with x > 0 gives x = √y.
Domain x > 0, range f(x) > 0, and f⁻¹(x) = √x for x > 0.
Look for these in your work
- Context supports the domain
- Range follows from the domain
- Positive square root chosen
A student says a function has an inverse because every output in their table has a value. What additional check is needed?
Give me a hint
Focus on whether an output belongs to one input or several.
Compare my reasoning
- An inverse must return each output to exactly one original input.
- Check whether distinct domain inputs ever produce the same output.
- If they do, restrict the domain or explain that no inverse function exists as stated.
The function must be one-to-one on its domain; a table can show repeated outputs that prevent an inverse.
Look for these in your work
- Inverse condition stated
- Repeated outputs considered
- Possible domain restriction named
Common mistakes
- Using the full real-line range when the question gives a restricted domain
- Assuming every function is one-to-one
- Giving an inverse without stating or checking the domain
What can you explain now?
Why does x² on −3 ≤ x ≤ 3 not have an inverse function?
Compare with the explanation
The inputs 2 and −2 both give 4
Two different inputs have the same output, so the function is not one-to-one on that domain.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Define domain, range and one-to-one, then state the test for an inverse function.
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.