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Representing simple functions

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Sequences and functions

Representing simple functions

A function machine takes a number in and puts a different one out. Run it backwards and you can find what went in.

Forwards with the rule · backwards with the inverse× 21236510inputoutputto find an INPUT, use the inverse: ÷ 2inverse of add is subtract · inverse of multiply is divide
Output 10? Then the input was 10 ÷ 2 = 5.

Function machines

A FUNCTION is a relationship between two sets of numbers. You can draw it as a FUNCTION MACHINE. The numbers you put in are the INPUT. The numbers that come out are the OUTPUT. A machine that adds 3: input 2 → +3 → output 5 input 4 → +3 → output 7 input 5 → +3 → output 8 That is a ONE-STEP function machine, because it does one thing.

Mapping diagrams

You can also draw a function as a MAPPING DIAGRAM: two number lines, with arrows joining each input to its output. For the 'add 3' machine, 2 maps to 5, 4 maps to 7, and 5 maps to 8. 'Maps to' is just another way of saying 'goes in and comes out as'.

Working backwards

To find the OUTPUT, apply the rule. To find the INPUT, work backwards using the INVERSE OPERATION. For a '× 2' machine: input 1 → output 2 input 3 → output 6 output 10 → what was the input? Instead of multiplying by 2, DIVIDE by 2: 10 ÷ 2 = 5. The inverse of adding is subtracting. The inverse of multiplying is dividing.

Worked examples

An 'add 3' machine with input 4

  1. Apply the rule to the input.
  2. 4 + 3 = 7.

Answer: Output 7

A '× 2' machine with output 10 — what went in?

  1. Work backwards with the inverse operation.
  2. Instead of × 2, do ÷ 2.
  3. 10 ÷ 2 = 5.

Answer: Input 5

What does 2 map to on the 'add 3' machine?

  1. 2 + 3 = 5.

Answer: 5

The inverse of adding

  1. Adding is undone by taking away.

Answer: Subtracting

The inverse of multiplying

  1. Multiplying is undone by sharing out.

Answer: Dividing

An 'add 3' machine with output 12

  1. Work backwards: subtract 3.
  2. 12 − 3 = 9.

Answer: Input 9

Where you see this in real life

  • A currency converter is a function machine: pounds in, euros out, one fixed rule.
  • Working out how much you started with after spending £8 is running the machine backwards — the inverse operation.

Key words

Function
a relationship between two sets of numbers
Function machine
a diagram showing a function as input, rule and output
Input
the number that goes in
Output
the number that comes out
Mapping diagram
a diagram with arrows joining inputs to outputs
Inverse operation
the operation that undoes another
One-step function
a machine that does just one thing
Maps to
goes in and comes out as
Rule
what the function machine does to the input
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