Writing an Expression for the Composite of Two Functions (F 708)

Think of f and g as two machines wired in series. A number goes into g first, g spits out a value, and that value goes straight into f. The composite f(g(x)) is the single formula that describes the whole pipeline without ever stopping to compute a number in the middle. That's the intuition: instead of plugging in one number at a time, you plug in the entire expression g(x) at once, so the formula works for every input simultaneously.

Mechanically, composing is just substitution with expressions instead of numbers. To find f(g(x)), take the formula for f(x), and everywhere you see x, write g(x) in its place, parentheses and all, then simplify. For f(x) = 2x + 5 and g(x) = x² - 3, replace x in f with the whole expression x² - 3: f(g(x)) = 2(x² - 3) + 5 = 2x² - 6 + 5 = 2x² - 1. Order matters completely: g(f(x)) sends f's output into g instead, giving a different formula almost every time.

This substitution habit connects directly to the rest of the unit. Transformations are really composition with simple outer functions, like replacing x with x - 3 to shift a graph, or multiplying by -1 to flip it. Inverse functions are the special case where composing f with its own inverse undoes everything, leaving f(f⁻¹(x)) = x. Once substitution-of-expressions feels automatic here, both of those ideas become the same skill applied to a new situation.