What is a Function?

Pre-Calculus · Axiom Academy

A function is a machine with a promise: feed it an input, and it hands back exactly one output — every single time. Picture a little machine: a number goes in, the machine does one fixed thing to it, and a number comes out. That is the whole idea of a function. The rule never changes its mind — the same input always produces the same single output. Watch a number ride into the machine f(x) = x + 3 . The machine adds 3 , and exactly one result drops out the far end. The name f labels the machine; the x stands for whatever input you feed it. In goes x , out comes f(x) = x + 3 — one input, one output, no surprises. Slide the input x left and right. The machine f(x) = x + 3 sends it to one height, and that pair drops a point onto the grid. Move along and the dots trace out the graph — the picture of the function is just every input paired with its one output. f(x) = x + 3 : feed it x , read off the one height f(x) — that pair is a single point on the graph. One input, one output — the golden rule Here is the rule that makes a function a function: each input gets exactly one output. Drag the vertical line across each graph. Where it crosses tells you how many outputs that input has. If any vertical line ever hits a graph more than once, that graph is not a function. A circle fails: at most inputs the line hits it twice, so one x would need two outputs — and a function never allows that.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.