Non-commutativity of composition
SAT glossary · Course Foundations
Definition
f(g(x)) ≠ g(f(x)): the innermost machine runs first, which is the opposite of the reading order and is exactly why the error is so common. With f(x)=x^2-3 and g(x)=2x+1, f(g(3))=46 while g(f(3))=13 — and 13 is always on the page as the reversed-composition distractor.
Non-commutativity of composition in plain English
When you plug one function into another — f(g(x)) — you run the INNERMOST function first (the one closest to x), which is actually the opposite order from how you'd normally read left to right, and that backwards feel is exactly why mixing up the order is such a common mistake. With f(x)=x^2-3 and g(x)=2x+1, computing f(g(3)) means: first do g(3)=2(3)+1=7, then plug THAT into f: f(7)=7^2-3=46. But computing g(f(3)) reverses it: first f(3)=3^2-3=6, then g(6)=2(6)+1=13 — a totally different number. That 13 isn't a coincidence — it's specifically included as the 'you did the composition backwards' trap answer.
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