ACT Math Study Notes: Understanding and Using Functions

Function Definition and Notation

What a function is (and why it matters)

A function is a rule that assigns exactly one output to each allowed input. You can think of it like a reliable machine: you feed in an input, and the machine returns one specific output every time. This “one output per input” idea is the whole point—functions are useful because they are predictable, and that predictability lets you model relationships (distance vs. time, cost vs. number of items, population vs. years, etc.) and analyze them with algebra and graphs.

A relationship is not a function if an input can lead to two different outputs. For instance, if one input value produces two different y-values on a graph, it fails the function rule.

Function notation

The most common way to write a function is function notation, like f(x), which means “the output of function f when the input is x.” This does not mean f times x—a very common misunderstanding.

If you are given an equation like

f(x) = 2x + 3

then f(5) means substitute 5 for x:

f(5) = 2(5) + 3 = 13

On the ACT, function notation shows up constantly because it packages an input-output relationship neatly, especially when you compare multiple functions (like f and g).

Domain and range

The domain is the set of all allowed inputs (often all real numbers, but not always). The range is the set of all possible outputs.

Domain restrictions usually come from operations that are not allowed in real-number algebra:

• Division by zero is not allowed.
• Even roots of negative numbers (like square roots) are not real.
• Logarithms of non-positive numbers are not real.

For example, if

f(x) = \frac{1}{x - 2}

then x - 2 cannot be 0, so x \ne 2.

How to tell if a graph is a function

A graph represents a function if it passes the vertical line test: every vertical line intersects the graph at most once. If a vertical line hits the graph twice, that means one x value corresponds to two different outputs, so it is not a function.

Common function notations you may see

Examples

Example 1 (Evaluate): If f(x) = x^2 - 4x + 1, find f(-2).

Substitute -2:

f(-2) = (-2)^2 - 4(-2) + 1 = 4 + 8 + 1 = 13

Example 2 (Domain): Find the domain of

g(x) = \sqrt{x - 5}

Inside the square root must be nonnegative:

x - 5 \ge 0

So x \ge 5. The domain is all real numbers at least 5.

Exam Focus

• Typical question patterns:
  • Evaluate expressions like f(3), f(a+1), or f(x-2) by substitution.
  • Identify domain/range from formulas or graphs.
  • Decide whether a graph/relation is a function (often via the vertical line test).
• Common mistakes:
  • Treating f(x) as multiplication instead of “function output.”
  • Forgetting domain restrictions (especially square roots and denominators).
  • Mixing up domain (inputs) and range (outputs) when reading a graph.

Linear Functions

What linear functions are

A linear function is a function whose graph is a straight line. Linear functions model constant rate-of-change situations: each time you increase the input by 1, the output changes by the same amount.

The most common form is slope-intercept form:

y = mx + b

where:

• m is the slope (rate of change)
• b is the y-intercept (the output when x = 0)

You will also see linear functions written as f(x) = mx + b.

Why slope matters

Slope tells you how steep a line is and in which direction it goes.

• If m > 0, the line rises as x increases.
• If m < 0, the line falls as x increases.
• If m = 0, the line is horizontal.

Given two points (x1, y1) and (x2, y2) , slope is

m = \frac{y2 - y1}{x2 - x1}

This formula is especially important on the ACT because problems often give you two points or a table and ask for the equation.

Building a line from information

From slope and a point (point-slope idea): If you know slope m and a point (x1, y1) , you can write

y - y1 = m(x - x1)

Then you can convert to y = mx + b if needed.

From two points: Find slope with the slope formula, then plug into point-slope form using either point.

Examples

Example 1 (Equation from two points): Find the equation of the line through (2, 5) and (6, 1) .

Slope:

m = \frac{1 - 5}{6 - 2} = \frac{-4}{4} = -1

Use point-slope with (2, 5) :

y - 5 = -1(x - 2)

Simplify:

y - 5 = -x + 2

y = -x + 7

Example 2 (Interpret slope): If C(t) = 20t + 50 models cost C in dollars for t hours, then 20 means the cost increases by 20 dollars per hour, and 50 is the starting cost when t = 0.

Exam Focus

• Typical question patterns:
  • Find slope from points, tables, or graphs.
  • Write a linear function given two points or slope and intercept.
  • Interpret slope and intercept in word problems.
• Common mistakes:
  • Reversing differences in the slope formula inconsistently (mixing numerator order and denominator order).
  • Confusing slope with y-intercept.
  • Reading the y-intercept from a graph incorrectly (it occurs where x = 0).

Polynomial Functions

What polynomial functions are

A polynomial function is built from powers of the input using nonnegative integer exponents. A general polynomial looks like

f(x) = an x^n + a{n-1} x^{n-1} + \dots + a1 x + a0

where:

• n is a nonnegative integer called the degree
• coefficients an, \dots, a0 are real numbers
• a_n \ne 0

Polynomials matter because they are smooth, continuous (no breaks), and they model many real situations approximately. On the ACT, polynomials appear in graph analysis, factoring, finding zeros, and understanding end behavior.

Degree and shape (big-picture behavior)

The degree strongly influences the graph:

• Degree 1: line
• Degree 2: parabola (quadratic)
• Degree 3 and higher: can have multiple turns

A key idea is end behavior, which depends mainly on the leading term a_n x^n:

• If n is even and a_n > 0, both ends go up.
• If n is even and a_n < 0, both ends go down.
• If n is odd and a_n > 0, left end down and right end up.
• If n is odd and a_n < 0, left end up and right end down.

You don’t usually need a formal proof on the ACT; you just need to recognize the pattern.

Zeros (roots) and factors

A zero (or root) of f is an input that makes the output zero:

f(x) = 0

If r is a zero, then (x - r) is a factor. This is one of the most testable links between algebra and graphs: x-intercepts occur where y = 0.

For quadratics, factoring or the quadratic formula may be used. If

ax^2 + bx + c = 0

then the solutions are

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The expression b^2 - 4ac is the discriminant, which tells you how many real solutions exist.

Examples

Example 1 (Zeros from factoring): Solve x^2 - 5x + 6 = 0.

Factor:

x^2 - 5x + 6 = (x - 2)(x - 3)

Set each factor to zero:

x - 2 = 0 \Rightarrow x = 2

x - 3 = 0 \Rightarrow x = 3

So the zeros are 2 and 3, and the graph crosses the x-axis at (2, 0) and (3, 0) .

Example 2 (End behavior): For

f(x) = -2x^4 + x - 1

the leading term is -2x^4. Even degree with negative leading coefficient means both ends go down.

Exam Focus

• Typical question patterns:
  • Identify zeros/x-intercepts from a factored polynomial.
  • Determine end behavior from the leading term.
  • Match a polynomial equation to a graph using intercepts and general shape.
• Common mistakes:
  • Forgetting that zeros correspond to x-intercepts where y = 0.
  • Losing a negative sign when factoring or applying the quadratic formula.
  • Assuming the graph always crosses at a zero; sometimes it touches and turns (even multiplicity), though ACT questions often keep this conceptual.

Radical Functions

What radical functions are

A radical function contains a root of the variable, most commonly a square root. A typical form is

f(x) = \sqrt{x}

or a transformed version like

f(x) = \sqrt{x - h} + k

Radical functions matter because they introduce domain restrictions and have distinctive graph shapes. They also model real constraints—like distance from an equation involving squaring, or physical quantities that cannot be negative.

Domain restrictions (the key skill)

For an even root (square root, fourth root, etc.), the expression inside must be nonnegative.

If

f(x) = \sqrt{g(x)}

then you must have

g(x) \ge 0

For an odd root (like cube root), negative inputs are allowed.

Graph shape intuition

The parent function f(x) = \sqrt{x} starts at (0,0) and increases to the right, but it increases more slowly over time (it “flattens out”). Transformations shift, stretch, or reflect this shape.

Examples

Example 1 (Domain): Find the domain of

f(x) = \sqrt{2x + 6}

Require inside nonnegative:

2x + 6 \ge 0

2x \ge -6

x \ge -3

So the domain is all real x where x \ge -3.

Example 2 (Solve an equation with a radical): Solve

\sqrt{x - 1} = 4

Square both sides:

(\sqrt{x - 1})^2 = 4^2

x - 1 = 16

x = 17

Check (important because squaring can create extraneous solutions):

\sqrt{17 - 1} = \sqrt{16} = 4

It works.

Exam Focus

• Typical question patterns:
  • Determine domain from a radical expression.
  • Match a square-root graph to an equation using starting point and shape.
  • Solve simple radical equations (often one radical).
• Common mistakes:
  • Forgetting to restrict the domain for even roots.
  • Not checking solutions after squaring.
  • Mixing up square root and absolute value behavior (they are not the same).

Piecewise Functions

What piecewise functions are

A piecewise function is defined by different rules on different parts of the domain. This is how you model real situations where the rule changes after a threshold—like shipping costs (different rates after a weight limit), tax brackets, or parking fees.

A piecewise function might look like

f(x) = \begin{cases} x + 2 & x < 0 \ x^2 & x \ge 0 \end{cases}

Even if you don’t write it in the brace format on the ACT, the idea is the same: you must pay attention to which rule applies for the given input.

How to evaluate piecewise functions

To find f(a):

1. Look at the conditions (like x < 0 or x \ge 0).
2. Decide which interval contains the input a.
3. Use only that formula.

Graphing and endpoints (open vs. closed circles)

When graphing, the boundary point is crucial:

• If the condition includes equality (like x \le 2 or x \ge 2), the point at x = 2 is included: use a closed dot.
• If it does not include equality (like x < 2 or x > 2), the point is excluded: use an open dot.

This affects continuity and whether the function has a jump.

Examples

Example 1 (Evaluate): Suppose

f(x) = \begin{cases} 3x & x \le 1 \ x + 4 & x > 1 \end{cases}

Find f(1) and f(2).

For x = 1, use the first rule (because 1 \le 1):

f(1) = 3(1) = 3

For x = 2, use the second rule (because 2 > 1):

f(2) = 2 + 4 = 6

Example 2 (Understand a jump): If the two pieces give different outputs at the boundary, the graph has a jump discontinuity. For the function above, the first piece gives 3 at x=1, while the second piece would give 5 at x=1 (but it’s not used there). That mismatch produces a visible jump.

Exam Focus

• Typical question patterns:
  • Evaluate f(a) for a given piecewise definition.
  • Interpret a graph made of two different “chunks” with different rules.
  • Identify whether a boundary point is included (open vs. closed dot).
• Common mistakes:
  • Using the wrong formula because you ignore the inequality condition.
  • Treating x = 1 as if it belongs to both pieces when only one includes equality.
  • Confusing a hole (open circle) with an actual defined value.

Exponential and Logarithmic Functions

Exponential functions: what they are

An exponential function has the variable in the exponent. A common form is

f(x) = a \cdot b^x

where:

• a is the initial value (often the output when x = 0)
• b is the growth/decay factor

If b > 1, the function shows exponential growth. If 0 < b < 1, it shows exponential decay. Exponential models appear whenever a quantity changes by a constant percentage over equal time steps (interest, population growth, radioactive decay).

A key anchor point is when x = 0:

f(0) = a \cdot b^0 = a

Logarithmic functions: what they mean

A logarithmic function is the inverse of an exponential function. A logarithm answers the question:

“Which exponent do I need to put on the base to get this number?”

Definition:

y = \log_b(x)

means exactly the same as

b^y = x

This equivalence is the heart of nearly every ACT log problem: you switch between log form and exponential form to solve.

Domain restrictions and key properties

For logs, the input must be positive:

x > 0

Also, the base must be positive and not equal to 1 (these conditions are usually assumed on standardized tests).

Useful properties that show up:

\log_b(1) = 0

because b^0 = 1.

\log_b(b) = 1

because b^1 = b.

Examples

Example 1 (Exponential evaluation): If f(x) = 3 \cdot 2^x, then

f(4) = 3 \cdot 2^4 = 3 \cdot 16 = 48

Example 2 (Convert log to exponential): Solve

\log_2(x) = 5

Convert to exponential form:

2^5 = x

x = 32

Example 3 (Convert exponential to log): Rewrite 10^y = 3 in log form.

y = \log_{10}(3)

Exam Focus

• Typical question patterns:
  • Evaluate exponential expressions and recognize growth vs. decay.
  • Convert between y = \log_b(x) and b^y = x.
  • Interpret exponential graphs (y-intercept and rapid growth/decay).
• Common mistakes:
  • Thinking logarithms can take negative or zero inputs (they cannot in real numbers).
  • Mixing up what is the base and what is the exponent when converting forms.
  • Treating b^x like bx (confusing exponential with linear behavior).

Function Transformations and Translations

Why transformations matter

Transformations let you take a “parent” function (a basic shape) and shift, stretch, compress, or reflect it to match a new equation. On the ACT, this is powerful because you can interpret graphs quickly without plotting lots of points.

A common transformed form is

y = a \cdot f(b(x - h)) + k

Each parameter changes the graph in a predictable way.

Translations (shifts)

• Horizontal shift: y = f(x - h) shifts the graph of f right by h.
• Vertical shift: y = f(x) + k shifts the graph up by k.

The horizontal shift is the one students most often reverse. A good mental check is to test a key point: if the original graph had a special point at x = 0, then f(x - 3) moves that feature to x = 3.

Reflections

• Over the x-axis: y = -f(x)
• Over the y-axis: y = f(-x)

Reflection over the y-axis flips left and right, which is why the negative is inside the function input.

Stretches and compressions

• Vertical stretch by factor a (if |a| > 1) or compression (if 0 < |a| < 1):

y = a f(x)

• Horizontal stretch/compression affects the input:

y = f(bx)

If |b| > 1, the graph is compressed horizontally; if 0 < |b| < 1, it is stretched.

Examples

Example 1 (Transform a parabola): Start with f(x) = x^2. Describe

g(x) = (x - 2)^2 + 3

The x - 2 shifts right 2. The +3 shifts up 3. So the vertex moves from (0,0) to (2,3) .

Example 2 (Reflection and vertical stretch): For f(x) = \sqrt{x}, describe

g(x) = -2\sqrt{x}

The negative reflects across the x-axis, and the factor 2 stretches vertically (outputs are doubled in magnitude).

Exam Focus

• Typical question patterns:
  • Identify shifts and reflections from an equation like f(x - 4) + 1.
  • Match a transformed equation to one of several graphs.
  • Find a key point (like vertex or starting point) after a transformation.
• Common mistakes:
  • Reversing the direction of horizontal shifts (misreading x - h).
  • Confusing vertical scaling a f(x) with horizontal scaling f(bx).
  • Forgetting that a negative inside the input, f(-x), reflects across the y-axis.

Analyzing Graphs and Key Features of Functions

Why graph analysis is a core ACT skill

ACT function questions often don’t require heavy algebra—they require you to read meaning from a graph quickly and accurately. A graph encodes multiple ideas at once: intercepts, intervals where the function increases/decreases, maximum/minimum points, and domain/range.

Intercepts

• x-intercepts occur where the graph crosses the x-axis, meaning y = 0. In function notation, these solve

f(x) = 0

• y-intercept occurs where the graph crosses the y-axis, meaning x = 0. That point is

f(0)

Intercepts are frequently used to match an equation to a graph or to interpret a context (like “starting value”).

Increasing, decreasing, and constant behavior

A function is:

• increasing on an interval if outputs rise as inputs increase
• decreasing on an interval if outputs fall as inputs increase
• constant on an interval if outputs stay the same

On the ACT, you’re often asked about these intervals from a graph without calculus—just follow the direction of the curve as you move left to right.

Maximum and minimum (extrema)

A maximum is a highest output value (globally or on a given interval). A minimum is a lowest output value.

On a graph, these show up as peaks and valleys. Quadratics are especially common: a parabola opening upward has a minimum at its vertex; opening downward has a maximum.

Domain and range from a graph

To find domain from a graph, ask: “Which x values appear?” Project the graph down to the x-axis.

To find range, ask: “Which y values appear?” Project the graph across to the y-axis.

Be careful with endpoints:

• A closed dot means included.
• An open dot means excluded.

Average rate of change

The average rate of change from x = a to x = b is the slope of the secant line through the points (a, f(a)) and (b, f(b)) :

\frac{f(b) - f(a)}{b - a}

This connects function thinking to linear thinking: it’s “change in output over change in input.” Many ACT questions phrase this as “average speed,” “average rate,” or “slope between two points on the curve.”

Examples

Example 1 (Average rate of change from values): If f(2) = 7 and f(6) = 19, then the average rate of change from 2 to 6 is

\frac{19 - 7}{6 - 2} = \frac{12}{4} = 3

So over that interval, the function increases by 3 output units per 1 input unit on average.

Example 2 (Intercept interpretation): If a graph of cost vs. items has y-intercept (0, 12) , that means even with 0 items you still pay 12 units of cost (a fixed fee).

Exam Focus

• Typical question patterns:
  • Identify intercepts, maxima/minima, domain/range, and intervals of increase/decrease from a graph.
  • Compute average rate of change using two points or given values like f(a) and f(b).
  • Match a story (context) to a graph by interpreting intercepts and overall shape.
• Common mistakes:
  • Confusing x-intercepts with y-intercepts (always check whether y = 0 or x = 0).
  • Ignoring open vs. closed endpoints when stating domain or range.
  • Reading the graph right-to-left (interval behavior must be described as x increases).