production function
Think of a recipe. Put in so much flour, so many eggs, so much oven time, and out comes a known number of cakes. A production function is exactly this idea written for a whole business: it is the rule that says how much output you get from any given amount of inputs, when those inputs are used as cleverly as current know-how allows.
Economists often write it as Q = f(L, K): the quantity of output Q depends on the amount of labour L and capital K you put in. The little f just means 'is determined by'. So if a print shop with two machines and three workers can turn out 600 flyers an hour, the production function is the relationship that tells you what happens when you add a fourth worker, or a third machine. Crucially it assumes technical efficiency — that you are getting the most output physically possible from those inputs, with no waste, given today's technology. A better recipe (a new technique) shifts the whole function up: the same inputs now yield more.
The production function is the engine room of supply-side economics. Because it ties output to inputs, it is where the cost curves come from: to know what producing a hundred extra units costs, you first need to know how many extra workers and machines those units require, and the production function tells you. It also underlies big-picture ideas like economic growth, where the inputs are a whole nation's labour and capital and the output is its GDP.
A car-wash with one bay and a hose has a production function like this: 1 worker washes 8 cars an hour, 2 workers 18, 3 workers 24, 4 workers 27. Output keeps rising as you add labour, but each extra worker adds a bit less than the one before — because they start crowding the single bay.
A production function is a recipe: this many inputs yield that much output.
The production function describes the best you can do, not the average mess of real life — it assumes no waste and given technology. Real firms often fall short of it (X-inefficiency), and a genuine new technology, not just doing the same thing harder, is what shifts the whole function upward.