Use function notation, domains, inverses and composites
Evaluate function rules and mappings, describe domains and ranges, and reverse or compose functions in the correct order.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
In gf(x), which function acts first?
What you will learn
- Use function notation and mapping diagrams
- Determine domain and range for Extended functions
- Find inverse and composite functions for Extended
A function maps each permitted input to one output
In f(x) = 2x − 3, f(4) means substitute 4 for x: f(4) = 5. The input set is the domain; outputs produced form the range. Mapping diagrams show the same input-output rule.
Core candidates use function notation and mapping diagrams. For Extended functions, identify restrictions: a denominator cannot be zero and a real square root needs a non-negative radicand.
Reverse or compose the rule carefully
To find an inverse, write y = f(x), swap x and y, then solve for y. An inverse reverses inputs and outputs on the stated domain.
The composite gf(x) means g(f(x)): apply f first, then g. Cambridge does not expect candidates to find the domain and range of a composite function, so keep that exclusion explicit.
Pause and explain
Find the inverse of f(x) = 2x − 6.
Worked example
Let f(x) = 3x + 4 and g(x) = x². Find f(−2), f⁻¹(x) and gf(x).
Show the worked solution
- f(−2) = 3(−2) + 4 = −2.
- From y = 3x + 4, swap x and y and solve: f⁻¹(x) = (x − 4)/3.
- gf(x) = g(3x + 4) = (3x + 4)².
Answer f(−2) = −2; f⁻¹(x) = (x − 4)/3; gf(x) = (3x + 4)².
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) = 2x − 3, g(x) = x² + 1, find f(5), fg(x) and gf(x).
Give me a hint
Use fg(x) = f(g(x)) and gf(x) = g(f(x)); do not commute the order.
Compare my reasoning
- f(5) = 7.
- fg(x) = f(x² + 1) = 2x² − 1.
- gf(x) = g(2x − 3) = (2x − 3)² + 1.
7; fg(x) = 2x² − 1; gf(x) = (2x − 3)² + 1.
Look for these in your work
- Substitution correct
- First composition expanded
- Second order kept distinct
For h(x) = 1/(x − 2), state its domain and range. Find the inverse of p(x) = 3x + 5.
Give me a hint
Exclude the input that makes the denominator zero and the output the reciprocal cannot reach.
Compare my reasoning
- Domain: x ≠ 2.
- Range: h(x) ≠ 0 because a reciprocal cannot equal zero.
- For y = 3x + 5, swap and solve: p⁻¹(x) = (x − 5)/3.
Domain x ≠ 2; range h(x) ≠ 0; p⁻¹(x) = (x − 5)/3.
Look for these in your work
- Denominator restriction found
- Range restriction justified
- Inverse solved after swapping
A converter maps Celsius c to Fahrenheit f(c) = 9c/5 + 32. Find the inverse and check a conversion round trip for 20°C.
Give me a hint
Swap input and output and solve for c; then apply both rules to a sample.
Compare my reasoning
- Set F = 9c/5 + 32.
- Subtract 32 and multiply by 5/9: c = 5(F − 32)/9.
- f(20) = 68; the inverse gives 5(68 − 32)/9 = 20.
Inverse c = 5(F − 32)/9; the round trip returns 20°C.
Look for these in your work
- Formula rearranged
- Sample conversion calculated
- Inverse round trip verified
Common mistakes
- Reading f(4) as f multiplied by 4
- Swapping variables without solving for the inverse
- Applying g before f in gf
- Claiming a composite domain/range requirement beyond the syllabus
What can you explain now?
If f(x) = 2x + 1, find f(3).
Compare with the explanation
7
Substitute 3: 2(3) + 1.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do you find an inverse, and what order does gf(x) apply the two rules?
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.