Knowledge Organiser: Function Notation
This topic summary covers Knowledge Organiser: Function Notation within Function Notation for GCSE Mathematics. Revise Function Notation in Algebra for GCSE Mathematics with 10 exam-style questions and 3 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 5 of 5 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Topic position
Section 5 of 5
Practice
10 questions
Recall
3 flashcards
Knowledge Organiser: Function Notation
Key Terms
- Function: A rule mapping each input to exactly one output
- f(x): Means "the function f applied to input x"
- Domain: The set of allowed input values
- Range: The set of possible output values
- Inverse function f⁻¹(x): Undoes what f does — swaps input and output
Must-Know Facts
- f(3) means substitute x = 3 into f(x)
- f(x) = k means solve the equation for x
- f⁻¹ means inverse (not reciprocal — f⁻¹(x) ≠ 1/f(x))
- To find f⁻¹(x): write y = f(x), swap x and y, rearrange for y
- ff⁻¹(x) = x always (the inverse cancels the function)
- f(x) notation just means "function of x" — it does NOT mean f × x
Key Methods
- Evaluate f(a): replace every x with a and calculate
- Solve f(x) = k: set the expression equal to k and solve for x
- Find f⁻¹(x): write y = f(x), swap x ↔ y, rearrange to make y the subject
- Verify inverse: check f(f⁻¹(x)) = x
Key Formulas
- Evaluate: f(3) means substitute x = 3 into f(x)
- Inverse: swap x and y in y = f(x), then rearrange to y = …
- Composite: gf(x) = g(f(x)) — apply f first, then g
- Domain restriction: state values where f(x) is undefined (e.g. x ≠ 0 for f(x) = 1/x)
Common Mistakes
- f(3) = f × 3: f(x) is a function, not f multiplied by x — substitute x = 3
- gf(x) vs fg(x): Order matters — gf(x) applies f first; fg(x) applies g first
- Inverse: forgetting ±√: If f(x) = x², the inverse gives x = ±√y — consider the domain
- f⁻¹(x) ≠ 1/f(x): The inverse function is NOT the reciprocal
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Practice Questions for Function Notation
If f(x) = x² − 1, what does f(3) equal?
f(x) = x + 3 and g(x) = 2x. Show with an example that fg(x) ≠ gf(x), and explain why the order matters.
Quick Recall Flashcards
10 questions on Function Notation — practise free
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