Composition of functions

Composing two functions means feeding the output of one into the other: (f o g)(x) = f(g(x)), where g is applied first. If f(x) = x + 3 and g(x) = 2x, then f(g(x)) = 2x + 3, but g(f(x)) = 2(x + 3) = 2x + 6, so composition is usually not commutative: order matters. To evaluate f(g(4)), work from the inside out: g(4) = 8, then f(8) = 11. The domain of a composition must respect both functions: if f(x) = square root of x and g(x) = x - 5, then f(g(x)) = square root of (x - 5), which requires x >= 5. You can also decompose a function into simpler pieces, such as writing h(x) = (3x + 1)^2 as f(g(x)) with g(x) = 3x + 1 and f(x) = x^2. Composition models chained relationships, like a cost that depends on the number of items, which depends on time, and a function and its inverse undo each other: f(f^-1(x)) = x.

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