A function is a rule that takes a number in and gives exactly one number out. Write f(x) = 3x + 2 and it is a machine: feed it 5, it multiplies by 3 and adds 2, so f(5) = 17. In Extended IGCSE you do two new things with these machines — you chain them together (composite functions) and you run them backwards (inverse functions).
1. Evaluating and reading f(x)
- f(a) means substitute x = a. If f(x) = x² − 1 then f(4) = 16 − 1 = 15.
- You can be asked to solve f(x) = k. If f(x) = 3x + 2 and f(x) = 20, then 3x + 2 = 20, so x = 6.
- The set of allowed inputs is the domain; the set of outputs is the range.
2. Composite functions — inner one first
fg(x) means do g first, then feed the result into f. Read it right-to-left, like peeling from the inside out. The order matters: fg(x) is usually not the same as gf(x).
| Notation | Meaning | Example (f: 2x+3, g: x+1) |
|---|---|---|
| fg(x) | f of g(x) — g first | f(x+1) = 2(x+1)+3 = 2x+5 |
| gf(x) | g of f(x) — f first | g(2x+3) = (2x+3)+1 = 2x+4 |
| ff(x) | f applied twice | f(2x+3) = 2(2x+3)+3 = 4x+9 |
3. Inverse functions — running the machine backwards
The inverse f⁻¹(x) undoes f: if f turns 6 into 15, then f⁻¹ turns 15 back into 6. To find it, follow three steps.
- Step 1: write y = f(x).
- Step 2: swap x and y.
- Step 3: rearrange to make y the subject — that is f⁻¹(x).
ff⁻¹(x) = x: because an inverse undoes the function, applying one straight after the other lands you back where you started. This is a handy check.
Exam technique: when the command word is Find fg(x), state the inner function's output explicitly before substituting — examiners award a method mark for showing g(x) fed into f. For an inverse, always show the swap line: it secures the method even if the algebra slips.