ACT Math Functions: Notation, Graphs & Composition

Functions questions account for roughly 7–9 of the 45 ACT Math questions. Unlike Algebra, Functions questions test your understanding of functions as objects — how f(x) notation works, what f(a) means, how to compose two functions (f(g(x))), how to find inverses, and how to read and transform graphs of functions. The ACT also tests piecewise functions and occasionally asks about the domain and range of composite or restricted functions.

A unique ACT Functions pattern: the test frequently defines a non-standard operation (like "a ★ b = 2a − b") and asks you to evaluate it. Treat these exactly like function notation — substitute the values into the definition and compute.

For f(g(x)) composition, work inside-out: first evaluate g(x) at the given input, then feed that result into f. Never evaluate f first. For example, if f(x) = x + 3 and g(x) = 2x, then f(g(4)) = f(8) = 11.

For inverse functions, remember that f⁻¹ swaps input and output. To find f⁻¹: replace f(x) with y, swap x and y, then solve for y. Verify by checking that f(f⁻¹(x)) = x.

For graph-based function questions, translation rules: f(x) + k shifts up k units; f(x − h) shifts right h units (note: minus in the argument shifts right, plus shifts left — the direction is counterintuitive). Memorize this because it appears regularly.

Reversing composition order: f(g(x)) ≠ g(f(x)) in general. The ACT exploits this by offering both as answer choices. Write out the steps explicitly rather than mentally shortcutting.

Mistaking domain restrictions. If f(x) = √(x − 3), the domain is x ≥ 3 (the expression under the root must be non-negative). If asked for the domain of a composed function, apply both restrictions.

For the non-standard operator questions: students sometimes try to remember a formula. There is none — read the definition provided and substitute. These are almost always one of the easiest questions on the test once you recognize the pattern.

Graph translation rules deserve dedicated drill — make flashcards for the 6 basic transformations (up/down, left/right, vertical stretch/compression, reflection). The ACT tests these predictably and they're pure memorization.

For composition practice, always write f(g(x)) as a two-step process with intermediate work shown. Speed comes after accuracy — don't skip steps to go faster until you're getting these consistently correct.