This guide covers the Functions portion of ACT Math, which makes up about 12–15% of the test (roughly 7–9 out of 60 questions). You'll need to work with several function types, including linear, radical, piecewise, polynomial, exponential, logarithmic, and trigonometric functions. The questions test everything from evaluating functions algebraically to reading and manipulating their graphs.

🤓 Functions Subtopics

The ACT organizes Functions into two main skill areas:

🔢 Definitions, Notations, and Applications of Functions

This is the algebraic side. You'll calculate input and output values, manipulate a function's algebraic form, interpret what parts of a function mean in context, and sometimes build a function from a word problem.

📈 Representing Functions as Graphs

This is the visual side. You'll graph functions on a coordinate plane, identify equations from graphs, and work with transformations (shifts, stretches, reflections). You also need to understand how changing a graph affects the underlying equation, and vice versa.

🔢 Definitions, Notations, and Applications of Functions

🧠 What You Need to Know

What is a function? A function describes a relationship between inputs and outputs where each input produces exactly one output. If you plug in a value for $x$ , you get back one and only one value for $f(x)$ .

Types of functions you should recognize:

Linear (e.g., $f(x) = 2x + 5$ )
Polynomial (e.g., $f(x) = x^3 - 4x + 1$ )
Radical (e.g., $f(x) = \sqrt{x + 3}$ )
Exponential (e.g., $f(x) = 3 \cdot 2^x$ )
Logarithmic (e.g., $f(x) = \log_2(x)$ )
Piecewise (different formulas for different intervals of $x$ )

For a deeper review of these, check out the ACT Math: Algebra & Functions guide.

Key skills the ACT tests:

Evaluating a function (finding the output given an input):

Identify the input value from the problem.
Replace the input variable (usually $x$ ) with that value in the function's formula.
Compute the result. You should get exactly one output.

Solving a function (finding the input given an output):

Set the given output value equal to the function's formula.
Solve for the input variable. You may get more than one answer, since multiple inputs can produce the same output.

Interpreting a function in context: The ACT often wraps functions inside word problems. Ask yourself: what does the input represent? The output? What does the y-intercept mean in this scenario? What about the slope or rate of change?

Formulating a function: Some problems describe a situation in words and ask you to write the matching equation. Translate the relationship piece by piece into math.

🤓 Applying Your Knowledge

Interpreting and Evaluating Functions Practice

The number of fish, $f$ , in Skipper's Pond at the beginning of each year can be modeled by the equation $f(x) = 3(2^x)$ , where $x$ represents the number of years after the beginning of the year 2000. For example, $x = 0$ represents the beginning of the year 2000, $x = 1$ represents the beginning of the year 2001, and so forth. According to the model, how many fish were in Skipper's Pond at the beginning of the year 2006?

A) 96 B) 192 C) 384 D) 1,458 E) 46,656

Credits: ACT, Inc - Question 7 from Preparing for the ACT Test Guide | 2023-2024

The answer is B) 192.

This problem tests both function evaluation and reading a function in context. The trap here is plugging in $x = 2006$ directly. But the problem defines $x$ as the number of years after 2000, so:

$x = 2006 - 2000 = 6$

Now plug $x = 6$ into the function:

$f(6) = 3(2^6) = 3 \times 64 = 192$

The number of fish at the beginning of 2006 is 192.

Always read carefully how the input variable is defined. On the ACT, the most common mistake on problems like this is using the raw year instead of calculating the correct input.

Formulating Functions Practice

If a publisher charges $15 for the first copy of a book that is ordered and $12 for each additional copy, which of the following expressions represents the cost of $y$ books?

F) $12y + 3$ G) $12y + 15$ H) $15y - 3$

J) $15y + 3$ K) $15y + 12$

Credits: ACT, Inc - Question 60 from Preparing for the ACT Test Guide | 2023-2024

The answer is F) 12y + 3.

The key is recognizing that not every book costs the same amount:

The first book costs $15.
The remaining $y - 1$ books each cost $12.

So the total cost is:

$15 + 12(y - 1)$

Simplify:

$15 + 12y - 12 = 12y + 3$

A common mistake is writing $12y$ for the additional books, forgetting that $y$ is the total number of books, not the number of additional books. You need to subtract 1 from $y$ to account for the first copy.

📈 Representing Functions as Graphs

🧠 What You Need to Know

Graphing functions: You should be familiar with the general shapes of common function types. A linear function is a straight line, a quadratic is a parabola, an exponential curves steeply upward (or downward), and trig functions oscillate in waves. If you're unsure about a graph, plug in a few values of $x$ , calculate $f(x)$ , and plot the points. If you have a graphing calculator, use it.

Determining an equation from a graph: Look at key features of the graph: where does it cross the axes? What's its shape? What's its amplitude or period (for trig functions)? Match those features to the answer choices.

Transformations of graphs: This is a big one. You need to know how algebraic changes to a function affect its graph:

Vertical shift: $f(x) + k$ shifts the graph up by $k$ units (down if $k$ is negative)
Horizontal shift: $f(x - h)$ shifts the graph right by $h$ units (left if $h$ is negative)
Vertical stretch/compression: $A \cdot f(x)$ stretches the graph vertically by a factor of $A$
Horizontal stretch/compression: $f(Bx)$ compresses the graph horizontally by a factor of $B$
Reflection: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis

🤓 Applying Your Knowledge

Manipulating Graphs Practice

Credits: ACT, Inc - Question 58 from Preparing for the ACT Test Guide | 2023-2024

The answer is K) 3cos(2x).

Start by recalling what the basic cosine function $\cos(x)$ looks like:

Image Courtesy of Study.com

The standard $\cos(x)$ has an amplitude of 1 and a period of $2\pi$ . Now look at the graph in the problem: it has an amplitude of 3 (the wave reaches up to 3 and down to -3) and a period of $\pi$ (one full cycle completes in $\pi$ units).

Here's how those transformations work algebraically:

Amplitude: Multiplying the function by a constant $A$ changes the amplitude to $|A|$ . So $3\cos(x)$ has an amplitude of 3.
Period: Multiplying $x$ by a constant $B$ changes the period to $\frac{2\pi}{|B|}$ . For a period of $\pi$ , you need $B = 2$ , since $\frac{2\pi}{2} = \pi$ . So $\cos(2x)$ has a period of $\pi$ .

Image Courtesy of Onlinemathlearning.com

Combining both: $3\cos(2x)$ gives you amplitude 3 and period $\pi$ , which matches the graph.

When you're stuck on a trig graph problem, identify the amplitude and period from the graph first, then match them to the general form $A\cos(Bx)$ or $A\sin(Bx)$ . You can also plug in a specific $x$ -value from the graph into each answer choice to see which one gives the correct $y$ -value.

Conclusion

That covers the two main areas of ACT Functions: working with functions algebraically (evaluating, solving, interpreting, and formulating) and working with them graphically (graphing, identifying equations, and understanding transformations). With 7–9 questions in this category, it's worth practicing both sides thoroughly.

For more practice, check out the other ACT Math Guides.

tldr; The ACT Math section is 60 minutes, 60 questions, calculators allowed throughout. Functions make up about 7–9 questions. You need to handle functions both algebraically (evaluate, solve, interpret, formulate) and graphically (graph, identify equations, apply transformations).

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