ACT-Style: Function Evaluation and Composition

ACT Math · Axiom Academy

EXAMPLE ACT-Style: Function Evaluation and Composition Evaluating a composite function f(g(3)) by working from the innermost function outward If f(x) = 2x + 1 and g(x) = x^2 , what is the value of f(g(3)) ? A composition like f(g(3)) always resolves the same way — work from the inside out, and never mix up which function goes first. Innermost first: in f(g(x)) , evaluate g before f — g is applied to x directly, so it goes first. Order matters: in general. Here f(g(3)) = 19 but g(f(3)) = 49 — swapping the order changes the answer. Keep every term: don't drop a +1 or a coefficient partway through — each piece of f(x) = 2x+1 has to survive into the final substitution. Result: g(3) = 9 , then f(9) = 2(9)+1 = 19 , so — answer choice J. Every "evaluate the composition" ACT question reduces to two clean substitutions done in the right order — inner function first, then plug that number into the outer function.

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