IGCSE Maths functions: notation, domain, inverse and composite functions

What each tier expects

The key ideas

Function notation is a compact way of writing a rule. If f(x) = 3x - 5, then f(4) means replace x by 4: f(4) = 12 - 5 = 7. Edexcel also writes the same function as f : x ↦ 3x - 5. You can substitute expressions too: f(2a) = 6a - 5.

The domain is the set of inputs allowed; the range is the set of outputs produced. At IGCSE the domain questions are mostly about what to exclude: you cannot divide by zero, so f(x) = 1/(x - 2) needs x ≠ 2, and you cannot take the square root of a negative, so sqrt(x - 3) needs x ≥ 3. For the range, think about the smallest or largest possible output: h(x) = 2x^2 + 3 can never be less than 3.

A composite function applies one function to the output of another. fg(x) = f(g(x)): work out g(x) first, then put that whole expression into f. In general fg(x) and gf(x) are different.

An inverse function undoes the original: if f(4) = 7, then f^-1(7) = 4. To find it, write y = f(x), rearrange to make x the subject, then rewrite with x in place of y. A quick check is that f(f^-1(x)) should simplify to x.

Worked example 1: f(x) = 3x - 5 and g(x) = x^2 + 1. Find fg(x), gf(x) and fg(2)

1. fg(x) = f(g(x)) = f(x^2 + 1) = 3(x^2 + 1) - 5 = 3x^2 - 2.
2. gf(x) = g(f(x)) = g(3x - 5) = (3x - 5)^2 + 1 = 9x^2 - 30x + 25 + 1 = 9x^2 - 30x + 26.
3. fg(2): either substitute into 3x^2 - 2 to get 12 - 2 = 10, or work in steps: g(2) = 5, then f(5) = 15 - 5 = 10.
4. Check gf(2) both ways: f(2) = 1 and g(1) = 2; the formula gives 36 - 60 + 26 = 2. Both agree, and fg(2) ≠ gf(2), which shows the order matters.

Worked example 2: find the inverse of f(x) = 2x/(x - 3)

1. Write y = 2x/(x - 3).
2. Multiply both sides by (x - 3): y(x - 3) = 2x, so xy - 3y = 2x.
3. Collect the x terms on one side: xy - 2x = 3y.
4. Factorise out x: x(y - 2) = 3y, so x = 3y/(y - 2).
5. Answer: f^-1(x) = 3x/(x - 2). Check: f(4) = 8/1 = 8, and f^-1(8) = 24/6 = 4. Correct.

Worked example 3: solving with functions

1. f(x) = 2x - 3. Solve f(x) = f^-1(x).
2. Find the inverse: y = 2x - 3 gives x = (y + 3)/2, so f^-1(x) = (x + 3)/2.
3. Set them equal: 2x - 3 = (x + 3)/2. Multiply by 2: 4x - 6 = x + 3, so 3x = 9 and x = 3.
4. Check: f(3) = 3 and f^-1(3) = 6/2 = 3. Correct. (The graphs of f and f^-1 meet on the line y = x.)

Common mistakes that cost marks

• Doing composites in the wrong order. fg(x) means g first.
• Writing f^-1(x) as 1/f(x). The -1 means inverse function, not reciprocal.
• Forgetting brackets when substituting an expression: f(x^2 + 1) is 3(x^2 + 1) - 5, not 3x^2 + 1 - 5.
• Stopping the inverse rearrangement too early when x appears twice; you must collect x terms and factorise.
• Leaving the inverse in terms of y instead of x.
• Expanding (3x - 5)^2 as 9x^2 + 25 instead of 9x^2 - 30x + 25.

Exam technique and how a tutor helps

Functions questions on both boards usually come in several short parts: evaluate, find a composite, find an inverse, then solve an equation that uses them. Each part is usually independent, so if you are stuck on the inverse, move on to the next part and come back. Cambridge asks for some answers 'as a fraction in its simplest form', which pulls in algebraic fraction skills.

Check every inverse with one number: pick an input, find f of it, then put the result into your inverse and see whether you get the input back. It takes thirty seconds and catches most errors.

Students often understand functions in a lesson and then freeze on the exam's unfamiliar notation. In one-to-one lessons a tutor can strip a question back to plain substitution, then rebuild it with the notation layer by layer, and give the student mixed questions in both the f(x) and f : x ↦ styles so the format stops being a surprise.

Self-check: can you do these?

• f(x) = 2x + 1 and g(x) = x^2. Find fg(3) and gf(3). (Answers: 19 and 49)
• Find the inverse of f(x) = 4x - 7. (Answer: f^-1(x) = (x + 7)/4)
• Find the inverse of g(x) = 3/(x + 2). (Answer: g^-1(x) = (3 - 2x)/x)
• Which value must be excluded from the domain of f(x) = 1/(x - 2)? (Answer: x = 2)
• Explain why fg(x) and gf(x) are usually different.