GCSE Maths · Topic guide
GCSE Maths functions revision
Function notation appears mainly on Higher tier: evaluating f(a), forming composite functions like fg(x), and finding inverses f⁻¹(x) for one-to-one functions. Examiners penalise sloppy notation — e.g. f(x)² vs f(x²). For composites, apply the inner function first unless the notation tells you otherwise (always check your board's convention in past papers). Domain/range may appear in context with graphs.
Common mistakes in exams
- Substituting into the wrong function in a composite.
- Treating f⁻¹(x) as 1/f(x).
- Reversing the order of composition when the notation is fg(x).
- Forgetting that some functions need domain restrictions for an inverse to exist.
Worked examples & mini quiz
GCSE Maths: Functions
f(a) means substitute x = a. Composite fg(x) usually means f(g(x)) — apply g first. For inverse, swap y and x and rearrange, or use flow diagrams for linear functions.
Worked examples
Example 1
Evaluate f(3) for f(x) = 2x² − 1
f(3) = 2(9) − 1 = 17.
Example 2
Linear inverse
If f(x) = 3x + 2, write y = 3x + 2, swap: x = 3y + 2 → y = (x − 2)/3. So f⁻¹(x) = (x − 2)/3.
Mini quiz (3 questions)
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1. If f(x) = x + 1 and g(x) = 2x, then fg(x) usually means:
- A.f(g(x)) = 2x + 1
- B.g(f(x))
- C.x + 2x
- D.2x²
Show answer
Correct: f(g(x)) = 2x + 1
g(x) = 2x; f(2x) = 2x + 1 (check your spec's convention).
2. The notation f⁻¹(x) typically means:
- A.Inverse function
- B.1/f(x)
- C.f(x²)
- D.Negative f
Show answer
Correct: Inverse function
Inverse reverses the mapping (when it exists).
3. f(0) for f(x) = 5/(x + 2) is:
- A.5/2
- B.0
- C.5
- D.Undefined
Show answer
Correct: 5/2
5/(0+2) = 2.5.
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