Functions (intro)
The idea
Picture a machine with an input slot and an output tray: feed it a number, and a fixed rule decides exactly what comes out. That is a function — a rule that assigns to each input exactly one output. You already work with patterns and plot pairs on the coordinate plane; a function organizes a pattern into input-output pairs you can list in a table, draw as a graph, or state as a rule like 'triple the input, then subtract 2'.
The defining promise is reliability: the same input must always produce the same single output. Feeding in 9 today and getting 25, then feeding in 9 tomorrow and getting 31, would mean the rule is not a function. The misconception to drop is that a function must be a formula — a table of game scores or a graph of temperature over a day qualifies, as long as each input has exactly one output. Different inputs sharing an output is perfectly fine; one input owning two outputs is not.
Worked example
A number machine uses the rule: triple the input, then subtract 2. What output does the input 9 produce, and which input produces the output 19?
- Run the rule forward on 9: triple gives 3 × 9 = 27, then subtracting 2 gives 27 − 2 = 25.
- To find the input behind 19, run the rule backwards in reverse order: undo the subtraction first, 19 + 2 = 21, then undo the tripling, 21 ÷ 3 = 7.
- Confirm the backwards trip by running 7 forward: 3 × 7 − 2 = 21 − 2 = 19, exactly the target output.
- Notice why the backwards trip was so clean: this particular rule never gives two different inputs the same output, so 19 traces back to a single input. Being a function only promises one output per input — a rule like squaring sends both 4 and −4 to 16, so its backwards question would have two answers.
Answer. The input 9 produces the output 25, and the input 7 is the one that produces 19.
Check your understanding
- What makes a rule fail to be a function, and can you invent an everyday rule that fails this way?
- Why is it acceptable for two different inputs to share an output, but not for one input to have two outputs?
- How can you decide from a table of pairs whether it could represent a function?
- When does running a rule backwards get complicated, and what feature of the rule causes the trouble?
Build the foundations first
Functions (intro) builds on these concepts. If any feel shaky, start there.