Function Arithmetic Visualized

Algebra 2 · Axiom Academy

Function Arithmetic in Business Revenue and cost are just functions. Add, subtract, multiply, or divide them and a business decision falls out the other side. You run Sunny’s Lemonade Stand : revenue is R(x) = 5x ( 5 a cup) and cost is C(x) = 2x + 20 ( 2 a cup plus 20 fixed) — combine the two functions and every business question has an answer. Are you actually making money? The most important question in business: subtract the cost function from the revenue function, output for output, and you get profit — P(x) = R(x) − C(x) . Cost isn’t one thing — it’s Materials M(x) = 2x + 10 (lemons, sugar, cups) plus Labor L(x) = x + 10 (your time). Add the two outputs and you get the total: T(x) = M(x) + L(x) . Discount the price, then find the rate A 10% off sale is a discount function D(x) = 0.9 ; multiply it onto revenue and you get (R · D)(x) = R(x) · D(x) = 4.5x . Divide revenue by cups sold instead and you get the price per cup — R(x) ÷ x (cups sold can never be zero, so this ratio is always defined). Price per cup never moves — R(x) ÷ x = 5x ÷ x = 5 for every x, so the rate stays constant even though revenue scales. New shop, new numbers: a t-shirt stand has R(x) = 25x and C(x) = 10x + 150 . Which operation finds the break-even point — where profit is exactly zero? Subtraction: f(x) − g(x) Profit = Revenue − Cost. Subtract outputs at each input value — this is also how you find a break-even point.

This is the written version of the interactive lesson above. See the full Algebra 2 course.