Building Functions for Cost, Profit and Distance Settings (AF 602, AF 702)

A single number like "revenue" or "cost" tells you nothing by itself. You need both as functions of the same variable, usually the number of units n, before you can ask anything useful. Revenue is what comes in: price per unit times n. Cost is what goes out: fixed overhead plus a per-unit cost times n. Profit only means something once you have both rules lined up, because profit is revenue minus cost, written as one new function P(n) = R(n) - C(n). You subtract the entire cost expression, fixed cost and all, not just the per-unit part.

Break-even is the n where R(n) = C(n), the point where profit is exactly zero. Set revenue equal to cost, solve for n, and that's how many units clear the fixed overhead. Raise the fixed cost and that break-even point moves up, because there's more overhead to recover before profit turns positive; the per-unit profit margin stays the same, so you're just dividing a bigger fixed cost by the same number. This is also why the revenue-maximizing quantity and the profit-maximizing quantity are usually different n values: revenue only grows as you sell more, but cost grows too, and profit cares about the gap between them, not the size of revenue alone.

The same subtract-and-set-up habit builds distance functions. A point moving along y = x² has coordinates (x, x²), so its distance from the origin is √(x² + (x²)²), built straight from the distance formula with the curve's own rule plugged in for y. Both ideas, cost settings and curve settings, are the same skill: turn a situation into an algebraic rule in one variable, then work with that rule.