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Algebra · Higher

Functions & function notation

Function notation f(x) describes a rule that maps inputs to outputs. You need to evaluate functions, find inputs from outputs, and understand domain and range.

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Key facts to remember

  • 1f(x) = 2x + 3 means "double x then add 3".
  • 2f(4) means substitute x = 4 into the function.
  • 3f(x) = k means find the value of x that gives output k.
  • 4The domain is the set of allowed inputs; the range is the set of possible outputs.
  • 5A function maps each input to exactly one output.
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Worked examples

Example 1

Given f(x) = 3x² − 1, find f(−2) and solve f(x) = 11.

Working

  1. f(−2) = 3(−2)² − 1 = 3(4) − 1 = 12 − 1 = 11
  2. For f(x) = 11: 3x² − 1 = 11 → 3x² = 12 → x² = 4 → x = ±2
Answerf(−2) = 11; x = 2 or x = −2
Example 2

Given g(x) = (x + 1) / (x − 2), find g(5) and state any value excluded from the domain.

Working

  1. g(5) = (5 + 1) / (5 − 2) = 6/3 = 2
  2. Domain exclusion: denominator = 0 when x = 2
  3. x = 2 is excluded from the domain
Answerg(5) = 2; x ≠ 2
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Common mistakes

✗Interpreting f(x) as f × x (multiplication) rather than function notation.
✗Forgetting to include both ± when solving f(x) = k for a quadratic function.
✗Confusing f(a) (evaluate at a) with f(x) = a (solve for x).
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Exam tips

✓Treat f(x) as a substitution instruction — replace every x with the given value.
✓When solving f(x) = k, rearrange the equation and solve just as you would any algebraic equation.

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