Evaluating a Composite Function

Evaluating a composite function means running the inner function on the input, then running the outer function on that result — inside out, one machine at a time.

For f(x) = x² and g(x) = x + 1, evaluate f(g(2)). Inner first: g(2) = 2 + 1 = 3. Outer next: f(3) = 9. Now the reverse order, g(f(2)): inner f(2) = 4, outer g(4) = 5. Same machines, opposite wiring, different answers — 9 against 5 — and neither matches the product f(2) · g(2) = 12. Writing the intermediate value down is not a beginner's crutch; it is the method, and it is where wiring mistakes surface before they compound. The habit scales: three chained machines run the same way, innermost first.