Functions & Graphing
Feed 20 into the rule `C × 9/5 + 32` and exactly one number falls out: 68. Plot every input beside its output and the rule stops being arithmetic, it becomes a shape you can read at a glance.
- ▸A rule that returns two answers for one input can't be plotted as a single readable curve.
- ▸A straight plot lifts the output equally per step; a bending plot lifts it differently as you move.
- ▸A model's training progress hides inside a plot's shape, the final number alone can lie about whether it worked.
Feed a number into a rule and get one number back. Put in a Celsius reading of 20, the rule C × 9/5 + 32 hands back 68 Fahrenheit. Put in 20 again tomorrow and it still hands back 68, same input, same answer, every time. That is the whole contract: one input in, exactly one number out.
Now do it for every input at once. Slide the input left and right and mark where each answer lands. String those marks together and you get a curve. The curve isn't a separate picture drawn next to the rule, it *is* the rule. Reading the height of the curve above an input is the same act as running the rule on that input.
The one-answer contract has a visible signature. Drop a vertical line anywhere and it should touch the curve exactly once, because one input has exactly one answer. A shape that a vertical line crosses twice (a sideways U, a full circle) breaks the contract: one input would owe two answers. That is the line between a plottable rule and a relation that isn't one.