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Extended Only Algebra and Graphs

Composite Functions for IGCSE Maths

Finding fg(x) and gf(x) by combining two functions. This subtopic is part of Functions in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding composite functions is essen

What You Need to Know

Finding fg(x) and gf(x) by combining two functions. This subtopic is part of Functions in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding composite functions is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Composite Functions

The composite function fg(x) means 'apply g first, then apply f to the result': fg(x) = f(g(x)). Order matters — fg(x) ≠ gf(x) in general. IGCSE Extended tests: finding fg(x) as an algebraic expression, evaluating fg(a) for a specific value, and sometimes solving fg(x) = k. A common exam error is applying f first when fg asks for g first.

Step-by-Step Method

  1. 1

    Identify the order of application

    fg(x): apply g first, then f. Read right to left. gf(x): apply f first, then g.

  2. 2

    Substitute g(x) into f

    Replace every x in f(x) with the entire expression g(x). Use brackets.

  3. 3

    Expand and simplify

    Expand any brackets and collect like terms. Do not leave composite functions un-simplified.

  4. 4

    Evaluate at a specific value if required

    Option 1: substitute into your simplified fg(x) expression. Option 2: work step by step: calculate g(a) first, then substitute that result into f.

  5. 5

    Verify

    Check using a numerical value: compute g(2) then f(result) and compare to your expression evaluated at 2.

Worked Example

Question

f(x) = 3x + 1 and g(x) = x² − 2. (a) Find fg(x). (b) Find gf(2).

Solution

Part (a): fg(x) = f(g(x)) = f(x² − 2) Replace x in f with (x² − 2): fg(x) = 3(x² − 2) + 1 = 3x² − 6 + 1 = 3x² − 5 Part (b): gf(2): apply f first, then g. f(2) = 3(2) + 1 = 7 gf(2) = g(7) = 7² − 2 = 49 − 2 = 47 Answers: (a) fg(x) = 3x² − 5 (b) gf(2) = 47

Exam Tips for Composite Functions

  • fg means g FIRST then f — right to left, like reading Hebrew. Misreading the order is the most common error.
  • When finding fg(x), substitute the entire g(x) expression (with brackets) into f, then expand.
  • To evaluate fg(3): quickest method is g(3) first, then f(result) — you do not need to find fg(x) algebraically first.
  • ff(x) means f applied twice: substitute f(x) back into f.

Practice Questions

Q1: f(x) = x + 4 and g(x) = 2x − 1. Find (a) fg(x) and (b) gf(x). Comment on whether fg(x) = gf(x).

Show hint

fg(x)=f(2x−1)=2x+3. gf(x)=g(x+4)=2x+7. They are not equal.

Q2: h(x) = 1/(2x). Find hh(x) (h applied to h).

Show hint

hh(x) = h(1/(2x)) = 1/(2 × 1/(2x)) = 1/(1/x) = x.

Q3: f(x) = 4x − 3 and g(x) = x² + 1. Solve fg(x) = 37.

Show hint

fg(x) = 4(x²+1)−3 = 4x²+1 = 37. So 4x² = 36, x² = 9, x = ±3.

Frequently Asked Questions

What is composite functions in IGCSE Maths?

Finding fg(x) and gf(x) by combining two functions.

Is composite functions in the Core or Extended syllabus?

Composite Functions is part of the Extended only syllabus for IGCSE Mathematics 0580.

How do I revise composite functions effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on composite functions rather than trying to cover everything at once.

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