Function Notation for Simple Functions of Two Variables (F 511)

Some real quantities depend on more than one thing. Shipping cost depends on both how heavy a package is and how far it travels — one number alone can't name a price. Function notation handles this by giving the machine two slots instead of one: f(x, y) means feed in a value for x and a value for y, and the formula tells you exactly what to do with each before combining them into a single output.

The mechanism is just substitution, done twice. For f(x, y) = 3x - 2y, the x-slot gets multiplied by 3 and the y-slot gets multiplied by -2, then the two results add together. Order matters: the first number listed always fills x, the second always fills y. If you freeze x and only let y move, the x-term never changes — it just sits there contributing the same fixed amount each time. Whatever change happens to the output comes entirely from the y-term, scaled by its coefficient. Increase y by 1 in f(x, y) = 5x - 2y, and the output drops by 2, because -2y falls by 2 while 5x stays put.

This sits right next to single-variable function notation and the four representations of a function. The same substitution habit — replace the letter, respect the order, compute carefully — carries over whether a function takes one input or two. It also sets up the made-up-operation notation the test likes to spring, where a symbol like x star y is just another two-input machine wearing a disguise.