Function Notation

Pre-Calculus · Axiom Academy

A function is a named machine: feed it an input, it runs one rule, and out comes the output. The notation f(x) says exactly that. Every function written in notation has three labeled parts. The name identifies the function, the input in parentheses names what goes in, and the rule on the right says what to do with it. The parentheses mean "of" — they hold the input, they are NOT multiplication. 2. Running the Machine: Evaluating f(x) To evaluate a function at a number, drop that number in for x everywhere it appears in the rule, then simplify. The animation feeds through f(x)=x+3 one at a time — watch x get replaced and the output drop out the bottom. f(2) does not mean " f times 2 ." It means "run the machine f on the input 2 " — plug 2 into the rule. f(2) is "the output of f when the input is 2 ." For f(x)=x+3 that output is 5 . The name and the input variable are just labels . The machine's behavior lives in the rule, not in which letters we picked. The animation keeps one machine and relabels its nameplate and input slot — f(x) , then g(t) , then h(n) , then A(r) — while the structure "name of input equals rule" stays fixed. Name g , input t — squares the input. g(3) = 9 . Name h , input n — subtracts 5 . h(9) = 4 . Name A (area), input r (radius) — area of a circle. Name f , input x — doubles the input. f(7) = 14 . Reading notation aloud keeps it clear:

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