Composite and inverse functions
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The teaching part (1 min) is free to watch. A free account opens the whole lecture: 4 worked examples, 7 practice questions, 1 real past-paper question with the answer explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
What this lecture covers
- A machine of machines
- Composite functions
- Evaluating a composite
Read the transcript of the teaching part
Last time
Before we start: last time, in The discriminant, a line and a curve, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
A machine of machines
Think of a function as a machine: a number goes in, another comes out. Now plug two machines together, so the output of one feeds the next. And then ask: can I run a machine backwards? Composite and inverse functions are exactly those two ideas.
Composite functions
The composite function f g of x means do g first, then f. We write f of g of x. Read it from the right. The function nearest to x acts first. Take f of x equals two x plus three, and g of x equals x squared. Then f g of x is two x squared plus three. But g f of x is the other way round: two x plus three, all squared. Order matters.
Evaluating a composite
To find f g of two, first find g of two. Squaring two gives four. Now put that four into f. Two times four, plus three, is eleven. So f g of two is eleven. Check g f of two: f of two is seven, and g of seven is forty-nine.
How often it comes up
How often Cambridge has asked each part, counted by ExamYard from every past paper in its question bank: Composite and inverse functions · Inverse of a quadratic function · Inverse of a rational function.
The whole chapter: Functions.
Chapter 1.2: Functions
- Composite and inverse functions 7 min
- Describing combined transformations, equations, ranges and inverses 7 min
- Reversing transformations, sketching transformed graphs and inverses 7 min
Also for A Level Maths 9709 Paper 1
Which topics come up most in A Level Maths 9709 Paper 1 past papers, counted from every paper since 2004.