Higher only · inverse and composite functions
Inverse and
composite
functions.
closest to x
goes first
two new pieces of notation —
f−1 · fg
f⁻¹(x) undoes f
fg(x) does g, then f
step 1 of 4
Run it backwards
1. The inverse takes the output back.
2. Reverse the order, opposite operations.
✓ 11 − 3 = 8, then 8 ÷ 2 = 4
f: × 2, then + 3
f⁻¹: − 3, then ÷ 2
step 2 of 4
Inverse with algebra
1. Write y = 2x + 3, then make x the subject.
2. Swap the y for an x at the end.
✓ Same as the machine: − 3, then ÷ 2
y = 2x + 3 write y for f(x)
y − 3 = 2x
x = y − 32 x is the subject
f−1(x) = x − 32
step 3 of 4
Composite: fg(2)
1. fg(x) means do g first, then f.
2. The output of g is the input of f.
✓ fg(2) = 11
f(x) = 2x + 3 g(x) = x²
g(2) = 2² = 4 g first
f(4) = 2 × 4 + 3 then f
fg(2) = 11
step 4 of 4
fg(x) and gf(x)
1. Put the whole of g(x) where x was in f.
2. Swap the order: a different function.
✓ fg(x) is not the same as gf(x)
fg(x) = f(x²)
= 2x² + 3
gf(x) = g(2x + 3)
= (2x + 3)²
inverse and composite —
f−1(x)
reverse order, opposite operations
The function closest to x
always goes first.
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