Composition of Functions

Pre-Calculus · Axiom Academy

LESSON Composition of Functions Chain two functions like machines in series — the output of one becomes the input of the next, and the order you chain them changes the answer. Picture two function machines bolted together. Drop an input x into the first machine g ; whatever comes out is then dropped into the second machine f . The catch that trips everyone up: you always apply the inner function first. In f(g(x)) , the g runs before the f . g runs first — the INNER machine then f runs on g 's output — the OUTER machine Composition is not commutative. Swapping which machine goes first usually changes the result, so in general . Send the same input x=3 down two tracks built from the same two machines — but chained in opposite orders — and they land in different places. g(3) = 4 , then f(4) = 16 . Apply g first, f second. f(3) = 9 , then g(9) = 10 . Apply f first, g second. . Same machines, same input, opposite order — different answer. and are usually different functions. Always respect the order. As formulas, f(g(x)) = (x+1)^2 = x^2 + 2x + 1 while g(f(x)) = x^2 + 1 . Those are genuinely different expressions, so at almost every input they disagree — at x=3 they give 16 and 10 . 3. Building the Composed Formula To get a single formula for f(g(x)) , don't plug in a number — plug in the whole inner function. Take f(x) = x^2 and replace every x with the entire expression for g(x) = x+1 . The x -slot in f is a placeholder, and g(x) drops right into it.

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