Function Machines

Pre-Calculus · Axiom Academy

A function is a machine: drop a number in, get one out. The real power shows up when you start wiring machines together. One machine: a number in, a number out A function isn't a formula to memorize — it's a machine . Feed it a number, it follows one fixed rule, and a single number comes out. Once you see a function that way, everything in this unit becomes a question about plumbing : what can you build by connecting machines? Watch a number drop into the machine f(x) = x² . The machine squares whatever it's given and drops the result out the bottom. The input cycles, but the rule never changes. That's the whole idea of a function: one input, one rule, one output — every single time. Chain them: send one machine's output into the next Here are two machines bolted in a row. Drag the dial to choose an input. It runs through f(x) = x + 1 first; whatever drops out is fed straight into g(x) = x² . Connecting machines like this is called composition — and the combined machine is just g(f(x)) = (x + 1)² . Order matters: feed the SAME input the other way (square first, then add one) and you'd usually get a different number. Combine them: two machines, one input, added results Chaining isn't the only move. Split one input so it runs through both machines at the same time, then add the two results. Drag the dial: the same number feeds f(x) = x + 1 and g(x) = x² , and the totals meet at the bottom. This new combined machine is (f + g)(x) = x² + x + 1 .

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