Functions: inverse and composite (Higher only)
Function notation is Higher tier on Edexcel 1MA1, typically appearing as a 5–7 mark Paper 2H or 3H question that combines composition and inversion.
Function notation
f(x) is read "f of x" — a rule for taking an input x and producing an output. f(3) means substitute x = 3 into the rule.
Example: f(x) = 2x + 5.
- f(3) = 2(3) + 5 = 11.
- f(−1) = −2 + 5 = 3.
- f(a) = 2a + 5.
Inverse functions
f⁻¹(x) is the function that undoes f. To find it:
- Replace f(x) with y: y = 2x + 5.
- Swap x and y: x = 2y + 5.
- Solve for y: y = (x − 5)/2.
- Write as f⁻¹(x) = (x − 5)/2.
Check: f⁻¹(11) = (11 − 5)/2 = 3, which matches f(3) = 11.
Property: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all valid x.
Composite functions
fg(x) means "do g first, then f". So fg(x) = f(g(x)).
Example: f(x) = 2x + 5, g(x) = x²:
- fg(3) = f(g(3)) = f(9) = 23.
- gf(3) = g(f(3)) = g(11) = 121.
- fg(x) = f(x²) = 2x² + 5.
- gf(x) = g(2x + 5) = (2x + 5)².
fg(x) ≠ gf(x) in general — order matters.
Domain considerations (touched on Higher)
The domain is the set of valid inputs. For f(x) = 1/x, x ≠ 0. For f(x) = √x, x ≥ 0. Edexcel asks "state any value of x for which f(x) is undefined".
Worked exam example
f(x) = 3x − 2. g(x) = x + 4.
(a) Find fg(5). f(g(5)) = f(9) = 3(9) − 2 = 25. (b) Find gf(x). g(f(x)) = g(3x − 2) = 3x − 2 + 4 = 3x + 2. (c) Find f⁻¹(x). y = 3x − 2 → x = 3y − 2 → y = (x + 2)/3. So f⁻¹(x) = (x + 2)/3.
Common Edexcel exam tip
On the answer line, write f⁻¹(x) = … (not just the expression). The B1 / A1 typically requires correct notation, not only the simplified expression.
⚠Common mistakes— Common errors
- Computing fg(x) as f(x) × g(x) (multiplication) instead of f(g(x)) (substitution).
- Confusing fg with gf — always do the inner function first.
- Writing f⁻¹(x) = 1/f(x). This is the reciprocal, not the inverse, and it is wrong.
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