Use function notation and composition in the right order
Evaluate functions, build composite functions and explain why changing their order can change the result.
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
In fg(x), which function acts first?
What you will learn
- Read and use f(x), f⁻¹(x), fg(x) and f²(x)
- Form composite functions by substitution
- Compare fg and gf without assuming they are equal
Notation records which input is being used
f(3) means substitute 3 for x in f(x). fg(x) means f(g(x)): apply g first and feed its output into f. The notation f²(x) means f(f(x)), not [f(x)]².
Keep brackets until substitution is complete. They protect the inner expression and make the order visible.
Composition order is part of the operation
With f(x) = 2x − 1 and g(x) = x² + 3, fg(x) = 2(x² + 3) − 1 = 2x² + 5.
In the other order, gf(x) = (2x − 1)² + 3 = 4x² − 4x + 4. These are different functions, so writing the order is essential.
Pause and explain
Does f²(x) usually mean [f(x)]²?
Worked example
Given f(x) = 3x + 2 and g(x) = x² − 1, find fg(x), gf(x) and f(2).
Show the worked solution
- Substitute g(x) into f: fg(x) = 3(x² − 1) + 2.
- Substitute f(x) into g: gf(x) = (3x + 2)² − 1.
- Evaluate f(2) directly: 3(2) + 2 = 8.
Answer fg(x) = 3x² − 1; gf(x) = 9x² + 12x + 3; f(2) = 8.
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 + 1 and g(x) = x² − 3, find fg(x).
Give me a hint
Replace the input of f with the whole expression g(x).
Compare my reasoning
- Write f(g(x)) = 2(x² − 3) + 1.
- Expand the outer function.
- Simplify the constant terms.
fg(x) = 2x² − 5.
Look for these in your work
- Correct order
- Substitution bracket retained
- Simplification correct
For the same f and g, find gf(x) and state whether it equals fg(x).
Give me a hint
Now place f(x) inside g(x).
Compare my reasoning
- Write g(f(x)) = (2x + 1)² − 3.
- Expand: 4x² + 4x + 1 − 3.
- Compare with 2x² − 5.
gf(x) = 4x² + 4x − 2, so gf is not equal to fg.
Look for these in your work
- Order reversed correctly
- Square expanded correctly
- Comparison made
A data process first doubles a reading and then adds 1. Define functions for the two stages and write the composite in words and notation.
Give me a hint
The first process should be the inner function.
Compare my reasoning
- Let g(x) = 2x represent doubling.
- Let f(x) = x + 1 represent adding 1.
- Apply g, then f: fg(x) = f(g(x)).
fg(x) = 2x + 1: double the reading, then add 1.
Look for these in your work
- Each stage modelled
- Order matches the description
- Composite notation correct
Common mistakes
- Reading fg(x) as g(f(x))
- Dropping brackets when substituting a negative or multi-term expression
- Confusing f²(x) with f(x)²
What can you explain now?
If f(x) = x + 4 and g(x) = 2x, what does fg(x) mean?
Compare with the explanation
f(g(x)) = 2x + 4
The right-hand function g acts first, then f acts on its output.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Explain the difference between fg(x), gf(x) and f²(x).
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.