Functions
ACT function questions usually test whether you can read a rule, connect representations, and predict how outputs change when inputs change.
Core Idea
Treat a function as a machine with a consistent rule: each allowed input gets one output. Most ACT function questions become manageable when you track three things carefully: what inputs are allowed, how the rule changes the input, and what the graph or equation tells you about behavior such as intercepts, growth, turning points, and transformations.
Understanding
Rule: On ACT Mathematics, functions show up as equations, graphs, tables, and short real-world descriptions. The test often changes the presentation, but the job stays the same: identify the rule and read what it implies. That may mean evaluating
A strong shortcut is to connect each question to one of a few predictable tasks:
- read inputs and outputs with function notation
- match a representation to an equation or context
- track domain, range, and key graph features
- recognize the function family and its usual behavior
- decode transformations by separating horizontal and vertical changes
If you keep the original rule in view, the notation stops feeling abstract. You are just asking: what went into the function, what came out, and how did the rule or graph change?
Concept Guides
8Use function notation; evaluate, interpret, and compare function values.
Function notation names outputs, and differences compare two outputs.
Determine domain and range from equations, graphs, or context.
Domain is the allowed inputs, and range is the outputs the function can produce.
Interpret key features of graphs (intercepts, maxima/minima, intervals of increase/decrease).
Read intercepts, turning points, and intervals to understand how a graph behaves.
Work with linear, quadratic, polynomial, exponential, logarithmic, radical, and piecewise functions.
Match a function family to its pattern of change and graph shape.
Translate among representations (equation, graph, table, verbal description).
Translate between words, tables, graphs, and equations by tracking the same rule.
Apply transformations (shifts, reflections, stretches/compressions) to graphs and equations.
Transformations shift, reflect, stretch, or compress a graph or equation.
Interpret rate of change and slope in context; identify proportional vs non-proportional relationships.
Slope is the rate of change, and proportional relationships must pass through the origin.
Solve problems involving inverse relationships or inverse functions when presented.
Inverse functions reverse the original input-output rule.