Functions & Inverses
What Is a Function?
A function is a rule that takes an input and produces exactly one output. That single-output rule is the foundation for function notation and inverse functions.
One input, one output
If the same input can produce two different outputs, the rule is not a function. A doubling machine is a function because each input has exactly one double.
How to recognize a function
- Identify the input.
- Ask what outputs are possible for that input.
- Accept the rule only if each input has exactly one output.
- Reject any rule that assigns two outputs to the same input.
Worked example
Is “always double the input” a function?
- Input 3 produces 6 only.
- Input −1 produces −2 only.
- No input has more than one output.
Yes — each input has exactly one output, so the rule is a function.
Common mistakes
- Allowing one input to map to two outputs.
- Confusing “many inputs can share an output” with “one input has many outputs.”
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