0580
Functions
Algebra and Sequences
Reading the order
- A composite function feeds the output of one function straight into another
- gf(x) is shorthand for g(f(x)), and the function written next to the bracket acts first
- So gf means f then g, and fg means g then f, which is the reverse of the order most people read it in
- Rewriting gf(x) as g(f(x)) before doing anything is the surest way to avoid reversing it
- ff(x) applies the same function twice, and it is not the same as squaring the output
Working one out
- With a number, work out the inner function first and put that answer into the outer one
- With algebra, substitute the whole inner expression into the outer rule and then expand
- The substitution line scores a mark before any expanding is done, so write it out
Worked example
Numbers and algebra
f(x) = 4x − 7 and g(x) = x² + 2. Find gf(2), fg(2) and gf(x).
Solution:
- For gf(2), f acts first: f(2) = 8 − 7 = 1
- Then g acts on that: g(1) = 1 + 2 = 3, so gf(2) = 3
- For fg(2), g acts first: g(2) = 4 + 2 = 6
- Then f acts on that: f(6) = 24 − 7 = 17, so fg(2) = 17
- For gf(x), substitute the whole of f(x) into g: g(4x − 7) = (4x − 7)² + 2
- Expand the bracket: 16x² − 56x + 49 + 2
- So gf(x) = 16x² − 56x + 51