ACT Math Functions: The Complete Cheat Sheet + Free Practice
ACT Math Functions: The Complete Cheat Sheet + Free Practice | The School of Mathematics
ACT Math · Cheat Sheet
ACT Math Functions: The Complete Cheat Sheet + Free Practice
9 min read · Full cheat sheet · 7 free quizzes
Functions are everywhere on the ACT Math section — they appear in roughly 10–15% of questions, and they overlap with almost every other topic: quadratics, sequences, transformations, and graphs. The ACT tests functions in very predictable ways, which means this is one of the highest-return topics to study.
This cheat sheet covers every function concept the ACT tests, with clear definitions and examples. Then we link you to 7 free quizzes to practice each skill level.
ACT Functions Cheat Sheet
f(x) notation
f(x) means "the output of f when the input is x." To evaluate f(3), replace every x in the rule with 3.
f(g(x))
Function composition: evaluate g first, then use that result as input to f. Read right-to-left: "f of g of x."
Domain
All valid inputs (x-values). Watch for: division by zero (excluded) and square roots of negatives (excluded when real).
Range
All possible outputs (y-values). For a parabola opening up: range is [vertex y-value, ∞).
f(x) + k
Vertical shift up by k units (graph moves up).
f(x + k)
Horizontal shift LEFT by k units (counterintuitive — the shift is opposite the sign).
−f(x)
Reflection over the x-axis (flip vertically).
f(−x)
Reflection over the y-axis (flip horizontally).
Even function
f(−x) = f(x). Symmetric about the y-axis. Examples: x², cos x.
Odd function
f(−x) = −f(x). Symmetric about the origin. Examples: x³, sin x.
Start Here · Fundamental
ACT Functions — Quiz 1
8 questions on function notation and basic evaluation. Take this before moving to composition or transformations.
The 3 Function Question Types the ACT Always Tests
1. Evaluate f(a) for a given value
The ACT gives you a function rule and asks you to plug in a number. Example: "If f(x) = 2x² − 3x + 1, what is f(−2)?" Substitute −2 for every x: 2(4) − 3(−2) + 1 = 8 + 6 + 1 = 15.
💡 Watch out
When substituting a negative number, put it in parentheses: f(−2) = 2(−2)² not 2−2². The parentheses prevent sign errors, which are the #1 source of mistakes on these questions.
2. Composite functions f(g(x))
Given two functions, evaluate one inside the other. The ACT may give you specific values (find f(g(3))) or ask for the general expression (find f(g(x))). Always evaluate the inside function first.
3. Transformations of graphs
Given a graph of f(x), the ACT asks what the graph of f(x) + 3, f(x − 2), or −f(x) looks like. Memorize the four transformation rules in the cheat sheet above. These questions appear frequently and reward students who know the rules cold.
Practice · Intermediate
ACT Functions — Quizzes 3, 4 & 5
Composite functions, domain/range, and transformations. 10, 10, and 5 questions respectively.
