Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who can evaluate f(x) and g(x) separately but lose marks when questions ask for fg(x), gf(x) or f⁻¹(x).
What query it owns: how to find composite and inverse functions in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the revision-guide angle, while Tutopiya’s Composite and Inverse of Functions subtopic page owns the learning resource and the free Composite and Inverse quiz owns the practice.

Composite and inverse functions sit near the end of the Functions unit in Cambridge IGCSE Mathematics (0580/0607), and they reward students who understand order and notation. A composite function applies one function to the output of another; an inverse function undoes the original. This guide explains what each notation means, how to work them step by step, and how to answer the exact command words examiners use.

Key takeaways

• fg(x) means f(g(x)) — apply g first, then f. Order matters; fg(x) is usually not equal to gf(x).
• f⁻¹(x) is the inverse function: if f(a) = b, then f⁻¹(b) = a.
• To find an inverse, write y = f(x), swap x and y, then rearrange to make y the subject.
• Always check that the input to the inner function lies in its domain before evaluating a composite.

What are composite and inverse functions in Cambridge IGCSE Maths?

A composite function combines two functions so the output of one becomes the input of the other. If f(x) = 2x + 1 and g(x) = x², then fg(x) = f(g(x)) = 2(x²) + 1 = 2x² + 1. An inverse function f⁻¹(x) reverses f: whatever f does to x, f⁻¹ undoes. Cambridge IGCSE Extended papers test both ideas with algebraic functions and sometimes with domain restrictions.

Read the full notes and worked examples on Tutopiya’s Composite and Inverse of Functions subtopic page before you attempt questions.

The core notation you must master

Notation	Meaning	Order of operations
fg(x)	f(g(x))	g first, then f
gf(x)	g(f(x))	f first, then g
f⁻¹(x)	Inverse of f	Undoes f
f⁻¹(a)	Input that gives output a	Solve f(x) = a

If you are unsure about which values are allowed as inputs, revise Domain and Range of Functions first — domain errors are the most common reason composite questions go wrong.

How to find a composite function — step by step

1. Identify the inner function from the notation. In fg(x), g is inner; in gf(x), f is inner.
2. Substitute the inner function into the outer one. Replace every x in the outer rule with the entire inner expression.
3. Simplify the result — expand brackets and collect like terms.
4. Check the domain if the question gives restrictions (e.g. x > 0).

Example: f(x) = 3x − 2, g(x) = x + 4. Find fg(2).

• g(2) = 6, then f(6) = 16. So fg(2) = 16.

Test whether the method has stuck with the free Composite and Inverse quiz.

How to find an inverse function — step by step

1. Write y = f(x) using the function rule.
2. Swap x and y (this is the standard IGCSE method).
3. Rearrange to make y the subject — that expression is f⁻¹(x).
4. Verify by checking f(f⁻¹(x)) = x.

Example: f(x) = (x + 3)/2. Write y = (x + 3)/2, swap → x = (y + 3)/2, rearrange → y = 2x − 3. So f⁻¹(x) = 2x − 3.

Composite and inverse in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Find fg(x)	Composite with g inside f	”f(x) = 2x + 1, g(x) = x². Find fg(x).”
Find gf(x)	Composite with f inside g	”Find gf(x) in its simplest form.”
Find f⁻¹(x)	Algebraic inverse	”Find f⁻¹(x), the inverse of f.”
Show that	Prove a given result	”Show that ff⁻¹(x) = x.”
Work out	Evaluate at a value	”Work out fg(3).”
State	Give a value with no working	”State the value of f⁻¹(7).”

Worked exam-style stems (how to answer the wording)

1. “f(x) = 2x − 5, g(x) = x² + 1. Find fg(x) in its simplest form.” fg(x) = f(g(x)) = 2(x² + 1) − 5 = 2x² − 3. Mark-scheme reward: correct substitution, then simplified expression.

2. “f(x) = (3x − 1)/2. Find f⁻¹(x).” y = (3x − 1)/2 → swap → x = (3y − 1)/2 → 2x = 3y − 1 → y = (2x + 1)/3. Answer: f⁻¹(x) = (2x + 1)/3.

3. “Show that ff⁻¹(x) = x.” “Show that” means the answer is given — earn marks by substituting your inverse back into f and simplifying to x with clear working.

When you can recognise the wording instantly, work the full set on the Functions topical past papers and the Composite and Inverse quiz.

How composite and inverse connect to the rest of Functions

Composite functions build directly on Domain and Range of Functions and feed into Graphs of Functions, where you may sketch f and f⁻¹ as reflections in y = x. When you are ready to mix topics, the Cambridge IGCSE Maths resource hub lets you move from a weak subtopic straight into the next.

Common mistakes students make

• Treating fg(x) as f(x) × g(x) instead of f(g(x)).
• Assuming fg(x) = gf(x) without checking — they are usually different.
• Forgetting to swap x and y when finding an inverse.
• Evaluating a composite when the inner output is outside the domain of the outer function.

When you need more support

If composite or inverse questions keep costing marks, work through the Domain and Range quiz and the Functions topical past papers to pinpoint the gap, then get focused help from a Cambridge IGCSE Maths tutor.

Frequently asked questions

What is the difference between fg(x) and gf(x)? fg(x) means apply g first, then f. gf(x) means apply f first, then g. Unless the functions are specially related, the two composites give different results.

How do I find f⁻¹(x) in IGCSE Maths? Write y = f(x), swap x and y, then rearrange to make y the subject. That expression is f⁻¹(x). Always check by substituting back.

Is composite functions hard in Cambridge IGCSE? The algebra is straightforward if you know the order rule. Most lost marks come from wrong order, poor substitution, or domain slips — all fixable with focused practice.

How should I revise composite and inverse functions? Read the subtopic notes, work three examples by hand, then take the Composite and Inverse quiz. Revisit any inverse rearrangements you got wrong before moving on.

Ready to master Cambridge IGCSE Maths composite and inverse functions?

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