Compose functions in the stated order
Evaluate f(g(x)) with a linear inner function · Form the composite rule · Preserve any restrictions on inner and outer inputs
Exam scope and assessment notes
Extended content; not a Core requirement. 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
- Function substitution
- Expanding brackets and domain restrictions
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
- Evaluate f(g(x)) with a linear inner function
- Form the composite rule
- Preserve any restrictions on inner and outer inputs
Apply the inside first
f(g(x)) means apply g to x, then use that output as the input to f. Replace every input occurrence in f by the entire expression g(x), using brackets. This 0444 requirement uses a linear inner function.
If f(x) = x² and g(x) = 2x − 1, then f(g(x)) = (2x − 1)², whereas g(f(x)) = 2x² − 1. Composition is generally order-dependent, and it is not multiplication of outputs.
Both stages must be defined
An allowed composite input must belong to g's domain and must produce an output in f's domain. For f(u) = 1/u and g(x) = x − 3, the composite excludes x = 3.
When expanding, retain restrictions from the original stages; a simplified expression does not remove them.
Sources for this lesson
Pause and explain
Which stage happens first in f(g(x))?
Worked example
f(u) = u² + 2 and g(x) = 3x − 1. Find f(g(x)) and f(g(2)).
Show the worked solution
- Replace u with the whole linear expression: (3x − 1)² + 2.
- Expand to 9x² − 6x + 3.
- g(2) = 5 and f(5) = 27.
Answer f(g(x)) = 9x² − 6x + 3; f(g(2)) = 27.
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.
f(u) = u² and g(x) = x + 4. Find f(g(2)) and f(g(x)).
Give me a hint
Insert the full result of g.
Compare my reasoning
- g(2) = 6 and f(6) = 36.
- f(g(x)) = (x + 4)² = x² + 8x + 16.
36; composite x² + 8x + 16.
Look for these in your work
- Keep the brackets
- Check the numerical result separately
f(u) = 1/u and g(x) = 2x + 1. Find f(g(x)) and its excluded input.
Give me a hint
The outer denominator cannot be zero.
Compare my reasoning
- Composite is 1/(2x + 1).
- 2x + 1 = 0 gives x = −1/2.
1/(2x + 1), with x ≠ −1/2.
Look for these in your work
- State the restriction
- Use the inner output as the denominator
A sensor maps x to g(x) = 2x + 3, and software squares that result using f(u) = u². Compare the real pipeline with swapping the two stages at x = 1.
Give me a hint
Write both orders before evaluating.
Compare my reasoning
- Actual f(g(1)) = 5² = 25.
- Swapped g(f(1)) = 2×1² + 3 = 5.
Actual output 25; swapped output 5. Order matters.
Look for these in your work
- Model the actual sequence of operations
- Do not multiply the two outputs
Common mistakes
- Do not apply the outer function first.
- Square the full bracket, not just its first term.
What can you explain now?
Which stage happens first in f(g(x))?
Compare with the explanation
g is applied first.
Its output becomes the input of f.
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Make it stick
Which stage happens first in f(g(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.