ACT Mathematics — function families, composition and transformations

On the enhanced ACT, Functions is one of the five Preparing for Higher Math subcategories, and ACT names the families it draws on: linear, radical, piecewise, polynomial and logarithmic functions, along with manipulating and translating functions and reading important features of their graphs. This material works through those families beyond the plain evaluation of f(x), with questions that take about a minute each, the pace the 50-minute section asks for.

You evaluate a piecewise-defined function at inputs that fall in different pieces, and compose two functions in the right order — the trap being that f(g(3)) and g(f(3)) are different numbers. Two questions cover graph transformations: translating y = f(x) left and down, and following one point through a reflection across the y-axis followed by a vertical shift. Exponential functions appear as models, a used car losing a fixed percent of its value each year and a bacteria culture that doubles every three hours, so you practice choosing between a factor of 1 − r and 1 + r and counting doubling periods.

The polynomial questions ask for all the zeros of a cubic after factoring out a common x, for a missing coefficient given one zero (the factor theorem), and for a remainder without long division (the remainder theorem). The last group covers a logarithmic equation rewritten in exponential form, the inverse of a linear function, and the range of a translated square-root function, where the domain and the range are easy to confuse.

The material offers two formats. The quiz has twelve multiple-choice questions with four answer choices each, as on the enhanced ACT, and every explanation names the slip behind each wrong choice, so a miss tells you what to fix. The flashcards collect the rules and formulas worth knowing by heart.

The content is based on ACT's published description of the mathematics section (reporting categories and the topics listed for each). It is independent practice and is not produced or endorsed by ACT.

• Evaluate piecewise-defined functions and compositions of functions in the correct order
• Describe and apply translations and reflections of the graph of y = f(x)
• Write and use exponential growth and decay models
• Find zeros of polynomials and use the factor and remainder theorems
• Solve a logarithmic equation, find an inverse function and state the range of a radical function

Practice material written by Zestly, based on ACT's published description of the ACT mathematics section (act.org, Mathematics Test Description: Preparing for Higher Math — Functions). Formulas beyond the basics are given in the questions, as ACT states that recall of complex formulas is not required.