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Algebra · Higher

Composite & inverse functions

A composite function applies two functions in sequence: fg(x) means apply g first, then f. An inverse function f⁻¹(x) reverses the effect of f, mapping outputs back to inputs.

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Key facts to remember

  • 1fg(x) means f(g(x)) — apply g first, then apply f to the result.
  • 2gf(x) means g(f(x)) — order matters; fg(x) ≠ gf(x) in general.
  • 3f⁻¹(x) is the inverse function: if f(a) = b then f⁻¹(b) = a.
  • 4To find f⁻¹(x): replace f(x) with y, swap x and y, then rearrange for y.
  • 5ff⁻¹(x) = x for all valid inputs.
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Worked examples

Example 1

Given f(x) = 2x + 1 and g(x) = x², find fg(3) and gf(3).

Working

  1. fg(3): first g(3) = 3² = 9, then f(9) = 2(9) + 1 = 19
  2. gf(3): first f(3) = 2(3) + 1 = 7, then g(7) = 7² = 49
Answerfg(3) = 19; gf(3) = 49
Example 2

Find f⁻¹(x) given f(x) = (3x − 2) / 5.

Working

  1. Let y = (3x − 2) / 5
  2. Swap x and y: x = (3y − 2) / 5
  3. Multiply both sides by 5: 5x = 3y − 2
  4. Add 2: 3y = 5x + 2
  5. Divide by 3: y = (5x + 2) / 3
  6. f⁻¹(x) = (5x + 2) / 3
Answerf⁻¹(x) = (5x + 2) / 3
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Common mistakes

✗Reversing the order in fg(x) — always apply the function nearest to x first.
✗Finding f⁻¹(x) as 1/f(x) (the reciprocal) instead of the inverse function.
✗Forgetting to swap x and y when finding the inverse.
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Exam tips

✓For composite functions: work from inside out — the function closest to x acts first.
✓To find inverse: write y =, swap x and y, rearrange for y — then write f⁻¹(x) =.

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