MathematicsExam code: 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
Worked example
Solving a composite equation
f(x) = 4x − 7 and g(x) = x² + 2. Solve fg(x) = 37.
Solution:
- fg(x) = f(g(x)), so g acts first and the whole of its output goes into f
- fg(x) = 4(x² + 2) − 7
- Expand and collect: 4x² + 8 − 7 = 4x² + 1
- Now solve 4x² + 1 = 37
- Subtract 1: 4x² = 36, then divide by 4: x² = 9
- Square root both sides, keeping both signs: x = 3 or x = −3
- Check: g(3) = 11 and f(11) = 44 − 7 = 37, and g(−3) is also 11
- Undoing f first gives the same equation: 4 × g(x) − 7 = 37, so g(x) = 11 and x² + 2 = 11
- If the question asks only for the positive solution, give x = 3 alone