Model Comparison: Functions

═══════════════════════════════════════

Gemini 3 Pro

═══════════════════════════════════════

What You Need to Know

• Functions are the backbone of ACT Algebra: You must be comfortable treating f(x) exactly like a variable y and understand that inputs (x) map to specific outputs (f(x)).

• Graphs and Equations are interchangeable: You will often need to match a graph to its equation or look at an equation and predict the shape, intercepts, and asymptotes of the graph.

• Composite Functions appear frequently: Mastering the "inside-out" method for f(g(x)) is essential for medium-to-hard difficulty questions.

• Domain restrictions are distinct: Always check denominators (cannot be zero) and even roots (cannot be negative) when asked for the domain.

Function Definition and Notation

The Definition of a Function

A relation is a function if every input (x-value) has exactly one output (y-value). Visually, this is tested using the Vertical Line Test: if a vertical line can pass through more than one point on the graph, it is not a function.

Function Notation

The notation f(x) represents the output value (the y-coordinate) associated with the input value x.

• Evaluation: If given f(x) = 2x + 1, to find f(3), you substitute 3 wherever you see x.
  f(3) = 2(3) + 1 = 7

• Composite Functions: This involves plugging one function into another. The notation f(g(x)) or (f \circ g)(x) means you apply g first, then apply f to the result.

  • Example: If f(x) = x^2 and g(x) = x + 1, then f(g(3)) means finding g(3) first (3+1=4), then finding f(4) (4^2 = 16).

Domain and Range

• Domain: All possible valid input values (x).

• Range: All resulting output values (y).

Exam Focus

• Why it matters: This is foundational. You will see 3-5 questions purely on evaluating or interpreting function notation.

• Typical question patterns:

  • "If f(x) = 3x^2 - x, what is f(-2)?" (Watch your signs!)

  • "For which value of x is the function f(x) = \frac{1}{x-4} undefined?"

  • "Given the graph of f(x), what is f(2)?" (Look for the y-value where x=2).

• Common mistakes: Confusing f(2) (plug in 2 for x) with f(x) = 2 (set the equation equal to 2 and solve for x).

Linear Functions

Linear functions form straight lines and typically appear as f(x) = mx + b.

• m (Slope): The rate of change, or \frac{\text{rise}}{\text{run}}. Formula: m = \frac{y2 - y1}{x2 - x1}.

• b (y-intercept): The starting value where the line crosses the vertical axis (x=0).

Real-World Modeling

The ACT loves word problems modeled by linear functions.

• Variable Cost: Corresponds to the slope (m).

• Fixed Cost: Corresponds to the y-intercept (b).

Exam Focus

• Why it matters: Linear models are the most common word problem type on the ACT Math section.

• Typical question patterns:

  • "A cell phone plan costs \$20 per month plus \$0.10 per minute. Write a function C(m) for the cost."

  • Comparing the slopes of two lines (parallel lines have equal slopes; perpendicular lines have negative reciprocal slopes).

• Common mistakes: Mixing up the independent variable (x) and dependent variable (y) in word problems.

Polynomial Functions

Polynomials include quadratic (x^2), cubic (x^3), and higher-degree functions.

Quadratics (f(x) = ax^2 + bx + c)

• Shape: Parabola (U-shape).

• Vertex: The peak or valley. The x-coordinate of the vertex is at x = -\frac{b}{2a}.

• Zeros/Roots: The points where the graph crosses the x-axis. Found by setting f(x) = 0 and factoring or using the Quadratic Formula:
  x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

End Behavior

• If the degree is odd (like x^3), the ends point in opposite directions.

• If the degree is even (like x^2, x^4), the ends point in the same direction.

Exam Focus

• Why it matters: Quadratics are heavily weighted. Expect questions on factoring and finding roots.

• Typical question patterns:

  • "What are the real solutions to the equation x^2 - 5x + 6 = 0?"

  • "Which of the following is a factor of the polynomial 2x^2 + 4x?"

• Common mistakes: Forgetting that a "solution," "root," "zero," and "x-intercept" all refer to the same concept algebraically.

Radical Functions

Functions involving roots, such as f(x) = \sqrt{x} or g(x) = \sqrt[3]{x+2}.

• Domain Restriction: You cannot take the square root (or any even root) of a negative number in the real number system.

  • For f(x) = \sqrt{x-3}, the domain is x-3 \ge 0, so x \ge 3.

Exam Focus

• Typical question patterns: Determining the domain of a radical function.

• Common mistakes: Solving equations like \sqrt{x} = -3. In the real number system, a principal square root cannot result in a negative number, so there is no solution.

Piecewise Functions

Piecewise functions are defined by different rules for different parts of the domain.

f(x) = \begin{cases} 2x & \text{if } x < 0 \ x^2 + 1 & \text{if } x \ge 0 \end{cases}

To evaluate a piecewise function, you must first check which condition the input x meets, then use the corresponding formula.

Exam Focus

• Typical question patterns: "Given the function above, find f(3)." (Since 3 \ge 0, use x^2+1).

• Common mistakes: Plugging the input into both equations. You must pick only the one that satisfies the condition.

Exponential and Logarithmic Functions

Exponential Functions

Form: f(x) = a \cdot b^x

• If b > 1, it represents exponential growth.

• If 0 < b < 1, it represents exponential decay.

• Asymptote: Usually the x-axis (y=0) unless vertically shifted.

Logarithmic Functions

Logarithms are the inverse of exponents.
y = \log_b(x) \iff b^y = x

Crucial Properties:

1. Product Rule: \logb(xy) = \logb(x) + \log_b(y)

2. Quotient Rule: \logb(\frac{x}{y}) = \logb(x) - \log_b(y)

3. Power Rule: \logb(x^n) = n \cdot \logb(x)

Exam Focus

• Why it matters: The ACT usually includes 1-2 logarithm questions, often late in the test (questions 40-60).

• Typical question patterns:

  • Converting between log and exponential form.

  • Simplifying an expression like 2\log(x) + \log(y) into a single log.

• Common mistakes: Thinking \log(x+y) = \log(x) + \log(y). This is false. There is no rule for the log of a sum.

Function Transformations and Translations

Understanding how changing an equation moves the graph allows you to solve visual problems without a calculator.

Given a parent function f(x), the transformed function is:
g(x) = a \cdot f(x - h) + k

1. Horizontal Shift (h): Occurs inside the parentheses. It moves opposite the sign.

   • f(x - 3) moves Right 3.

   • f(x + 2) moves Left 2.

2. Vertical Shift (k): Occurs outside the parentheses. It moves with the sign.

   • f(x) + 4 moves Up 4.

   • f(x) - 1 moves Down 1.

3. Reflections:

   • -f(x): Reflection over the x-axis (flips upside down).

   • f(-x): Reflection over the y-axis (flips left/right).

Exam Focus

• Why it matters: This is a shortcut technique. Instead of plotting points, you can recognize the shift instantly.

• Typical question patterns:

  • "The graph of y = x^2 is shifted 3 units to the right and 2 units down. What is the new equation?" (Answer: y = (x-3)^2 - 2).

• Common mistakes: Confusing the direction of the horizontal shift. Remember: Inside is Opposite.

Analyzing Graphs and Key Features

Even without a specific equation, you must be able to identify key features from a drawing.

• Intercepts: Points where the graph crosses axes (x-intercepts are where y=0; y-intercepts are where x=0).

• Extrema: Maximums (peaks) and Minimums (valleys).

• Increasing/Decreasing: Read the graph from left to right. Is the y value going up or down?

• Asymptotes: A line that a graph approaches but never touches.

Exam Focus

• Typical question patterns: A graph is provided, and you are asked to identify the equation or specific properties (e.g., "How many distinct real zeros does this function have?").

• Common mistakes: Assuming a graph touches an asymptote just because the line looks close. It never touches.

Quick Review Checklist

• Can you apply the Vertical Line Test to identify a function?

• Can you find f(g(2)) given equations for f(x) and g(x)?

• Do you know the domain restrictions for fractions (denominator \neq 0) and radicals (\text{inside} \ge 0)?

• Can you identify the slope and y-intercept in y = -3x + 5?

• Can you solve a quadratic equation by factoring?

• Do you know that f(x-2) shifts a graph to the right?

• Can you convert \log_2(8) = 3 into exponential form (2^3 = 8)?

• Can you identify the zeros of a function by looking at its graph?

Final Exam Pitfalls

1. The "Inside is Opposite" Trap: Students often shift f(x+3) to the right. Correct approach: x+3 inside the function means a shift of 3 units to the left.

2. Undefined Values: When asked for the domain, students forget to exclude values that make the denominator zero. Correct approach: Set denominator = 0, solve, and exclude those values.

3. Function Notation Confusion: Mistaking f(x) for multiplication (i.e., f times x). Correct approach: Treat f as the name of the rule, and the parentheses as the container for the input.

4. Logarithm Invalid Inputs: Trying to take the log of a negative number or zero. Correct approach: The argument inside a log function must be strictly greater than zero.

5. Composite Order: Calculating g(f(x)) when asked for f(g(x)). Correct approach: Always start with the innermost parentheses and work your way out.

═══════════════════════════════════════

GPT 5.2 Pro

═══════════════════════════════════════

What You Need to Know

• A function pairs each input x with exactly one output f(x)—most ACT questions test whether you can evaluate, interpret, and connect equations to graphs.

• Master the “big moves”: find domain/range, intercepts, rate of change, and how transformations change a graph.

• Many function problems are really algebra in disguise—solve for inputs given outputs, interpret parameters (like slope or growth factor), and compare multiple representations (table, graph, equation).

Curriculum anchor (what this is based on): ACT’s published Math test specifications/standards place Functions within Preparing for Higher Math (along with algebra, number & quantity, etc.). ACT emphasizes interpreting and analyzing functions from equations, graphs, and tables, including key features and basic transformations. ACT does not publish a fixed number of “functions-only” questions, so focus on skills that appear repeatedly across many item types.

Function Definition and Notation

A function is a rule that assigns each input in the domain to one output in the range.

Notation you must read fluently

• Function notation: f(x) means “the output of function f when the input is x.”

• Output/value: f(3)=7 means input 3 gives output 7.

• Equation form: y=f(x).

Determine whether a relation is a function

• From ordered pairs/table: each x must map to only one y.

• From a graph: use the vertical line test—any vertical line intersects at most once.

Example (evaluate)

Given f(x)=3x^2-2x+1, find f(-2).

1. Substitute x=-2: f(-2)=3(-2)^2-2(-2)+1

2. Compute: =3\cdot 4+4+1=12+4+1=17

Exam Focus

• Why it matters: Function notation and “is it a function?” checks are foundational and show up as quick points across many question types.

• Typical question patterns:

  • Evaluate f(a) or solve f(x)=k for x.

  • Decide if a table/graph/relation is a function.

  • Interpret f(a+b) vs. f(a)+f(b).

• Common mistakes:

  • Treating f(x) as f\cdot x (it’s notation, not multiplication).

  • Forgetting order of operations when substituting negatives.

  • Assuming any graph is a function without using the vertical line test.

Linear Functions

A linear function has constant rate of change (slope) and can be written as:
y=mx+b
where m is slope and b is the y-intercept.

Key skills

• Slope from two points \left(x1,y1\right) and \left(x2,y2\right):
  m=\frac{y2-y1}{x2-x1}

• Point-slope form:
  y-y1=m(x-x1)

• Parallel lines: same slope. Perpendicular lines: slopes multiply to -1 (negative reciprocals) when both are non-vertical.

Example (build a line)

Find the equation of the line through \left(2,-1\right) with slope 3.

1. Point-slope: y-(-1)=3(x-2)

2. Simplify: y+1=3x-6

3. Solve for y: y=3x-7

Application idea: If m is “dollars per hour,” then b is “starting fee.” ACT often tests interpreting m and b in context.

Exam Focus

• Why it matters: Linear models are among the most frequent function types on ACT Math and underpin rate-of-change reasoning.

• Typical question patterns:

  • Find m or b from a graph, two points, or a table.

  • Compare two linear functions (which grows faster? where do they intersect?).

  • Interpret slope/intercepts in a word problem.

• Common mistakes:

  • Flipping the slope ratio (mixing up \Delta y and \Delta x).

  • Confusing b with an x-intercept.

  • Forgetting vertical lines have undefined slope (equation x=c).

Polynomial Functions

A polynomial function has form:
f(x)=anx^n+a{n-1}x^{n-1}+\cdots+a1x+a0
where n is a nonnegative integer.

Key features ACT likes

• Degree: highest power n.

• Zeros/roots: values of x where f(x)=0 (often found by factoring).

• End behavior depends on leading term a_nx^n.

• Factored form helps identify zeros:
  f(x)=a(x-r1)(x-r2)\cdots

Example (factor to find zeros)

Solve x^2-5x+6=0.

1. Factor: x^2-5x+6=(x-2)(x-3)

2. Set factors to zero: x-2=0 or x-3=0

3. Solutions: x=2, x=3

Graph connection: For a quadratic, the y-intercept is f(0), and the axis of symmetry is halfway between the zeros (if they exist).

Exam Focus

• Why it matters: Polynomials (especially quadratics) drive many equation-solving and graph-feature questions.

• Typical question patterns:

  • Factor and solve; match a polynomial to a graph by zeros and end behavior.

  • Use a given point to find a missing coefficient.

  • Identify number of real zeros from a graph.

• Common mistakes:

  • Sign errors when factoring or expanding.

  • Thinking every quadratic factors over integers.

  • Mixing up f(0) (the y-intercept) with zeros (the x-intercepts).

Radical Functions

A radical function involves square roots or other roots, such as:
f(x)=\sqrt{x-1}

Domain restrictions (real numbers)

• For an even root (like a square root), require the radicand to be nonnegative:
  x-1\ge 0 \Rightarrow x\ge 1

• For an odd root, domain is typically all real numbers.

Example (domain + evaluation)

For f(x)=\sqrt{2x-3}:

1. Domain: 2x-3\ge 0 \Rightarrow x\ge \frac{3}{2}

2. Evaluate f(\tfrac{5}{2}): f(\tfrac{5}{2})=\sqrt{2\cdot\tfrac{5}{2}-3}=\sqrt{5-3}=\sqrt{2}

Application idea: Radicals often model geometry (distance formulas, diagonals) and physics (some growth relationships), though ACT usually focuses on algebra/domain.

Exam Focus

• Why it matters: Radical expressions appear in function evaluation, solving equations, and domain/range questions.

• Typical question patterns:

  • Find the domain of a radical function.

  • Solve simple radical equations (often by isolating the radical then squaring).

  • Identify transformations of \sqrt{x}.

• Common mistakes:

  • Forgetting domain restrictions after simplifying.

  • Squaring both sides and not checking for extraneous solutions.

  • Treating \sqrt{a+b} as \sqrt{a}+\sqrt{b} (not valid in general).

Piecewise Functions

A piecewise function is defined by different formulas on different intervals, e.g.
f(x)=\begin{cases}
2x+1 & x<0\
x^2 & x\ge 0
\end{cases}

How to work with them

• Determine which rule applies by checking the input against the interval condition.

• Watch endpoint symbols: <,\le,>,\ge—they control whether the endpoint is included.

Example (evaluate)

Using the function above:

• f(-3)=2(-3)+1=-5 because -3<0.

• f(2)=2^2=4 because 2\ge 0.

Exam Focus

• Why it matters: Piecewise definitions test careful reading and are common in graph-to-equation matching.

• Typical question patterns:

  • Evaluate f(a) for positive/negative/boundary values.

  • Identify where the function is continuous or has a jump.

  • Match a piecewise rule to a graph with different “pieces.”

• Common mistakes:

  • Using the wrong piece (not checking the condition).

  • Mishandling boundary points (open vs. closed circle conceptually).

  • Assuming formulas “blend” smoothly at the breakpoint without checking values.

Exponential and Logarithmic Functions

An exponential function has the variable in the exponent:
f(x)=a\cdot b^x
Common interpretation: b>1 growth, 0<b<1 decay.

A logarithmic function is the inverse of an exponential:
y=\log_b(x) \iff b^y=x
(typically with b>0, b\ne 1, and x>0).

Key properties to remember

• If b^x=b^y (same base, valid base), then x=y.

• Log/exponential inverse relationship is the main tool for solving.

Example (solve an exponential equation)

Solve 2^{x+1}=16.

1. Rewrite 16 as a power of 2: 16=2^4

2. Set exponents equal: x+1=4

3. x=3

Application idea: Exponentials model repeated percent change (interest, depreciation, population). ACT may ask you to interpret b as a growth factor like “multiply by 1.05 each year.”

Exam Focus

• Why it matters: Exponential growth/decay and basic logs show up in advanced algebra items and modeling.

• Typical question patterns:

  • Solve equations by rewriting to a common base.

  • Interpret parameters a and b in context.

  • Recognize that logs invert exponentials (convert between forms).

• Common mistakes:

  • Treating b^{x+y} as b^x+b^y (should be b^{x+y}=b^x\cdot b^y).

  • Forgetting log domain: input must satisfy x>0.

  • Mixing bases without rewriting or using the inverse definition.

Function Transformations and Translations

Transformations describe how a graph changes from a base function f(x) to a new function g(x).

Master template

g(x)=a\,f\left(b(x-h)\right)+k

• h: horizontal shift (right if h>0).

• k: vertical shift (up if k>0).

• a: vertical stretch/compression; reflect over x-axis if a<0.

• b: horizontal compression/stretch; reflect over y-axis if b<0.

Quick examples

• g(x)=f(x)+3 shifts up 3.

• g(x)=f(x-2) shifts right 2.

• g(x)=-f(x) reflects over the x-axis.

• g(x)=f(-x) reflects over the y-axis.

Exam Focus

• Why it matters: Transformation questions are fast if you recognize patterns, and they connect equations directly to graph changes.

• Typical question patterns:

  • Identify shifts/reflections from an equation like f(x-4)+1.

  • Determine a transformed function given a base graph.

  • Compare two functions and describe the transformation.

• Common mistakes:

  • Reversing horizontal shift direction: f(x-h) shifts right, not left.

  • Confusing vertical scaling a with horizontal scaling b.

  • Forgetting reflections occur when coefficients are negative.

Analyzing Graphs and Key Features of Functions

ACT often gives a graph and asks for key features—treat it like reading a story about inputs and outputs.

Features to identify

• Intercepts

  • y-intercept: value at x=0 is f(0).

  • x-intercepts: where f(x)=0.

• Increasing/decreasing: where outputs rise/fall as x increases.

• Maximum/minimum: highest/lowest points on an interval.

• Average rate of change from x=a to x=b:
  \frac{f(b)-f(a)}{b-a}

• Asymptotes (common with rational/exponential): a line the graph approaches.

• Discontinuities: holes/jumps where the function value is missing or changes suddenly.

Example (rate of change)

If f(2)=5 and f(6)=17, average rate of change from 2 to 6 is:
\frac{17-5}{6-2}=\frac{12}{4}=3

Application idea: Average rate of change is “average speed” in motion contexts or “cost per item” over an interval.

Exam Focus

• Why it matters: Graph interpretation is a high-yield skill tested across many function types and representations.

• Typical question patterns:

  • Read intercepts, turning points, or intervals where f(x)>0 or f(x)<0.

  • Match a graph to an equation using key features (intercepts, end behavior, asymptotes).

  • Compute average rate of change from a graph/table.

• Common mistakes:

  • Mixing up x- and y-intercepts.

  • Reading scales incorrectly (especially when tick marks skip values).

  • Confusing “where f(x) is increasing” with “where f(x) is positive.”

Quick Review Checklist

• Can you explain what makes a relation a function (and apply the vertical line test)?

• Can you evaluate f(a) correctly from a formula, table, or graph?

• Can you find slope and write a linear equation from points, a table, or a graph?

• Can you factor a quadratic and connect its zeros to x-intercepts?

• Do you know how to find the domain of a radical function using an inequality?

• Can you evaluate a piecewise function at boundary points using < vs. \le correctly?

• Can you solve basic exponential equations by rewriting to a common base?

• Can you describe transformations in g(x)=a\,f(b(x-h))+k and predict the shift/reflection?

• Can you compute average rate of change \frac{f(b)-f(a)}{b-a} from values or a graph?

Final Exam Pitfalls

1. Horizontal shift reversal: For f(x-h), shift right by h—don’t shift left.

2. Intercept confusion: y-intercept is f(0); zeros are where f(x)=0 (the x-intercepts).

3. Wrong piece in piecewise functions: Always check which interval the input satisfies before evaluating.

4. Domain errors with radicals/logs: Enforce \text{radicand}\ge 0 for even roots and \text{log input}>0 for logs.

5. Rate-of-change mix-ups: Use \frac{f(b)-f(a)}{b-a}—don’t swap numerator/denominator or reverse subtraction inconsistently.

═══════════════════════════════════════

Claude Opus 4.6

═══════════════════════════════════════

What You Need to Know

• Functions account for a significant portion of the ACT Math section — roughly 12–15% of questions fall under the "Functions" reporting category (about 7–9 questions out of 60). You must be fluent in function notation, graph interpretation, and the behavior of common function types.

• You need to recognize and work with multiple function families: linear, polynomial, radical, piecewise, exponential, and logarithmic. The ACT tests your ability to evaluate, interpret, and transform these functions — not just memorize formulas.

• Graph analysis is essential. Many ACT function questions present a graph and ask you to identify key features — intercepts, domain, range, maxima/minima, or the effect of transformations. Train yourself to read graphs quickly and accurately.

• Function notation is more than just substitution. The ACT uses notation like f(g(x)), f(x + h), and f^{-1}(x) to test whether you truly understand what the notation communicates.

Function Definition and Notation

A function is a rule that assigns exactly one output to each input. If f is a function, then f(x) represents the output when the input is x.

Key Concepts

• Domain: the set of all valid inputs (x-values)

• Range: the set of all possible outputs (y-values or f(x)-values)

• Evaluating a function: replace every instance of x with the given value. For example, if f(x) = 3x^2 - 2, then f(4) = 3(4)^2 - 2 = 46.

• Composition of functions: f(g(x)) means "plug g(x) into f." Work from the inside out.

• Inverse functions: f^{-1}(x) "undoes" f. If f(a) = b, then f^{-1}(b) = a.

Example

Given f(x) = 2x + 1 and g(x) = x^2, find f(g(3)).

1. Evaluate the inner function: g(3) = 3^2 = 9

2. Substitute into the outer function: f(9) = 2(9) + 1 = 19

Exam Focus

• Why it matters: Nearly every function question requires correct use of notation — misreading f(x+2) versus f(x) + 2 is a classic trap.

• Typical question patterns:

  • Evaluate f(a) for a given expression

  • Find f(g(x)) or g(f(x)) and simplify

  • Use a table of values to determine f^{-1}(k)

• Common mistakes:

  • Confusing f(x + 2) (shift the input) with f(x) + 2 (shift the output)

  • Composing functions in the wrong order — f(g(x)) \neq g(f(x)) in general

Linear Functions

A linear function has the form:

f(x) = mx + b

where m is the slope (rate of change) and b is the y-intercept.

Essential Formulas

• Parallel lines have the same slope.

• Perpendicular lines have slopes that are negative reciprocals: m1 \cdot m2 = -1.

Exam Focus

• Why it matters: Linear functions are the most frequently tested function type on the ACT — they appear in both straightforward algebra questions and word problems.

• Typical question patterns:

  • Find the equation of a line given two points or a point and slope

  • Determine whether two lines are parallel, perpendicular, or neither

  • Interpret slope and y-intercept in a real-world context (e.g., cost per item, starting value)

• Common mistakes:

  • Flipping the slope formula (subtracting coordinates in inconsistent order)

  • Forgetting that the perpendicular slope is the negative reciprocal, not just the reciprocal

Polynomial Functions

Polynomial functions have the general form:

f(x) = anx^n + a{n-1}x^{n-1} + \cdots + a1x + a0

Key Facts

• The degree of the polynomial is the highest power of x.

• A polynomial of degree n has at most n real zeros (x-intercepts) and at most n - 1 turning points.

• Quadratic functions (n = 2): f(x) = ax^2 + bx + c. The vertex is at x = -\dfrac{b}{2a}.

• End behavior: for large |x|, the leading term anx^n dominates. If n is even and an > 0, both ends go up; if n is odd and a_n > 0, left end goes down and right end goes up.

Exam Focus

• Why it matters: Quadratics are heavily tested; higher-degree polynomials appear occasionally in graph-matching or zeros-counting questions.

• Typical question patterns:

  • Find the vertex, axis of symmetry, or zeros of a quadratic

  • Determine how many x-intercepts a polynomial has from its graph

  • Factor or use the quadratic formula: x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

• Common mistakes:

  • Sign errors in the quadratic formula — especially with -b

  • Confusing the number of zeros with the degree (a degree-3 polynomial can have only 1 real zero)

Radical Functions

A radical function involves a root, most commonly a square root:

f(x) = \sqrt{x}

Key Properties

• The domain of f(x) = \sqrt{x} is x \geq 0 (the expression under the radical must be non-negative for even roots).

• The range of f(x) = \sqrt{x} is f(x) \geq 0.

• Cube root functions f(x) = \sqrt[3]{x} have domain and range of all real numbers.

Exam Focus

• Why it matters: The ACT commonly asks domain questions involving radicals and tests your ability to solve radical equations.

• Typical question patterns:

  • Find the domain of f(x) = \sqrt{3x - 6} → set 3x - 6 \geq 0, so x \geq 2

  • Solve radical equations by squaring both sides

• Common mistakes:

  • Forgetting to check for extraneous solutions after squaring both sides of a radical equation

  • Allowing negative values inside an even root

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain.

Example

f(x) = \begin{cases} 2x + 1 & \text{if } x < 0 \ x^2 & \text{if } x \geq 0 \end{cases}

To evaluate f(-3): since -3 < 0, use 2(-3) + 1 = -5.

To evaluate f(4): since 4 \geq 0, use 4^2 = 16.

Exam Focus

• Why it matters: Piecewise questions test whether you can match the correct rule to the correct interval — a skill the ACT values because it combines reading comprehension with algebra.

• Typical question patterns:

  • Evaluate a piecewise function at a specific value

  • Identify which graph matches a piecewise definition

  • Determine where a piecewise function is discontinuous

• Common mistakes:

  • Using the wrong piece for the given x-value — pay close attention to strict inequalities (<) versus inclusive ones (\leq)

  • Misreading open vs. closed circles on graphs

Exponential and Logarithmic Functions

Exponential Functions

f(x) = a \cdot b^x

• If b > 1: exponential growth

• If 0 < b < 1: exponential decay

• The y-intercept is always a (when x = 0).

• The horizontal asymptote is y = 0 (for the basic form).

Logarithmic Functions

A logarithm is the inverse of an exponential: \log_b(y) = x means b^x = y.

Memory aid: "A logarithm answers the question: what exponent do I need?"

Exam Focus

• Why it matters: Exponential growth/decay appears in real-world ACT word problems (population, interest, depreciation). Logarithm questions are less frequent but do appear — especially on higher-difficulty items.

• Typical question patterns:

  • Identify growth vs. decay from an equation or table

  • Solve for x in 2^x = 32 or \log_3(x) = 4

  • Apply compound interest: A = P\left(1 + \dfrac{r}{n}\right)^{nt}

• Common mistakes:

  • Confusing the base and the exponent when converting between exponential and logarithmic form

  • Misapplying log rules — e.g., \log(a + b) \neq \log(a) + \log(b)

Function Transformations and Translations

Starting from a parent function f(x), transformations follow predictable rules:

Memory aid: Horizontal transformations are counterintuitive — they do the opposite of what the sign suggests. "Inside is opposite."

Exam Focus

• Why it matters: Transformation questions are among the most reliable ways the ACT tests conceptual understanding of functions. They appear across all difficulty levels.

• Typical question patterns:

  • Given f(x), describe or graph f(x - 3) + 2

  • Identify the transformation from a graph comparison

  • Determine the new vertex of a parabola after a transformation

• Common mistakes:

  • Shifting horizontally in the wrong direction — f(x - 3) shifts right, not left

  • Applying vertical and horizontal stretches/compressions incorrectly when combined

Analyzing Graphs and Key Features of Functions

The ACT frequently provides graphs and asks you to extract information.

Key Features to Identify

• x-intercepts (zeros): where f(x) = 0 — the graph crosses or touches the x-axis

• y-intercept: the value of f(0) — where the graph crosses the y-axis

• Domain and range: read from the graph's horizontal and vertical extent

• Increasing/decreasing intervals: where the graph goes up (left to right) vs. down

• Maximum and minimum values: the highest and lowest points (local or absolute)

• Asymptotes: lines the graph approaches but never reaches (common in rational, exponential, and logarithmic functions)

• Symmetry: even functions are symmetric about the y-axis (f(-x) = f(x)); odd functions have rotational symmetry about the origin (f(-x) = -f(x))

Exam Focus

• Why it matters: Graph-reading questions are common on the ACT and span all difficulty levels. They reward students who can quickly and accurately extract information visually.

• Typical question patterns:

  • "Over which interval is f(x) decreasing?"

  • "What is the range of the function shown?"

  • "How many real solutions does f(x) = 3 have?" (count where y = 3 intersects the graph)

• Common mistakes:

  • Confusing domain with range

  • Reading the wrong axis — e.g., giving an x-value when the question asks for a y-value

  • Forgetting that a maximum value is a y-coordinate, not an x-coordinate

Quick Review Checklist

• Can you evaluate f(a) for any function type, including piecewise functions?

• Can you compute f(g(x)) and g(f(x)) and recognize they are generally different?

• Do you know how to find the slope between two points and write the equation of a line?

• Can you find the vertex and zeros of a quadratic function?

• Do you know the domain restrictions for radical functions with even roots?

• Can you evaluate a piecewise function by selecting the correct piece?

• Do you know how to convert between exponential and logarithmic form?

• Can you describe the effect of each transformation in a \cdot f(x - h) + k?

• Can you identify intercepts, domain, range, and increasing/decreasing intervals from a graph?

• Do you know that horizontal shifts work opposite to the sign inside the function?

Final Exam Pitfalls

1. Composing functions in the wrong order. f(g(x)) means apply g first, then f. Always work inside-out. Double-check which function is the "inner" one.

2. Shifting horizontally in the wrong direction. f(x - 3) shifts the graph right 3, not left. Remember: inside changes are opposite.

3. Forgetting to check for extraneous solutions in radical equations. After squaring both sides, always substitute your answer back into the original equation to verify.

4. Misapplying logarithm rules. \log(a + b) cannot be simplified — it is not \log(a) + \log(b). The product rule applies to \log(a \cdot b).

5. Confusing maximum/minimum values with their locations. If a question asks "What is the maximum value of f(x)?", give the y-coordinate of the peak, not the x-coordinate.

6. Using the wrong piece of a piecewise function. Before evaluating, carefully check whether the given x-value satisfies the condition with a strict inequality (<) or an inclusive one (\leq). A single boundary value can change the answer entirely.