If the notation f(x) feels hard, nothing is wrong with you. The notation is only wrapping paper. What is inside is a single promise — put one thing in, and exactly one thing, always the same thing, comes out. Shall we start by pressing a vending machine?
First handle — the promise
That is the question Cliff 2 of the map asks. Here you can answer it with your own finger. First, a machine in working order — press the same button as many times as you like, the same thing comes out.
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A broken machine gives different things for the same button. At that moment it is no longer a vending machine but a lottery machine — and in the language of mathematics, no longer a function. That is the entire definition: a correspondence where each input is assigned exactly one output. Numbers in, numbers out — the same idea as buttons and drinks.
Second handle — three faces
If graphs ever looked like ciphers, the culprit is usually not f(x) but the coordinate plane — the missing feel that one ordered pair is one point. Drag x below and watch the pair in the table, the calculation in the formula, and the point on the plane turn out to be the same one thing.
Same x, same y. A formula is just a table compressed to one line — with room for every number at once.
Where you have been stays marked. A graph is not a picture — it is the footprints of ordered pairs.
Switch to the squaring machine and drag x end to end. The footprints bend. That curve will later be given the name "parabola" — but the name can wait. For now, only this: a different rule leaves a different footprint. Every graph you will ever meet is a relative of these footprints.
Even this promise took two hundred years to name
The position of a planet, the temperature of heating water, the height of a cannonball — to handle quantities that change, people needed a name for the relationship itself: fix one thing, and another is fixed.
function
Late in the 17th century Leibniz first used the word, for quantities attached to a curve. It still meant something close to "formula".
f(x)
In the 18th century Euler spread the notation we still use. "Feed x into the machine called f" — that is still the whole of how to read it.
correspondence
By the 19th century, with Dirichlet, it became clear no formula is required. If each x is assigned exactly one y, it is a function — the vending-machine definition was complete.
Two hundred years from formula to "correspondence". School teaches that story backwards — definition first — so no wonder it feels hard.
The formal names for the vending machine and the footprints.
Browse the full glossary →There are no exercises here. If you once saw the table, the formula and the point move together as you dragged x, this page has done its job.
The first machine left footprints in a dead-straight line. When x grows by 1, how much does y grow — can you tell from the footprints alone? The name for that steepness is the map's next node: slope.