Model Comparison: Functions

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

Gemini 3 Pro

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

What You Need to Know

• Notation is key: Understanding $f(x)$ notation is fundamental; remember that $x$ is the input and the entire expression $f(x)$ represents the output ($y$-value).
• Graphs tell the story: You must be able to link an equation to the shape of its graph (e.g., $y=x^2$ is a parabola, $y=2^x$ is a curve approaching an asymptote) without a graphing calculator.
• Transformations are predictable: Shifts, stretches, and reflections follow strict rules that apply to all function types, not just one.
• Domain restrictions: Always check for division by zero and negative numbers under even roots.

Function Definition and Notation

The Core Concept

A function is a relation where every input ($x$) produces exactly one output ($y$). Visually, a graph represents a function if it passes the Vertical Line Test: no vertical line can intersect the graph at more than one point.

Function Notation

Standard notation is $f(x)$, read as "$f$ of $x$."

• $x$ = Input (Independent Variable)
• $f(x)$ = Output (Dependent Variable, usually $y$)

Example:
If $f(x) = 3x^2 - 4$, find $f(-2)$.
$f(-2) = 3(-2)^2 - 4$
$f(-2) = 3(4) - 4$
$f(-2) = 8$

Domain and Range

• Domain: The set of all possible inputs ($x$-values) for which the function is defined. Watch out for denominators (cannot be zero) and even roots (radicand must be non-negative).
• Range: The set of all resulting outputs ($y$-values).

Composite Functions

Composition involves plugging one function into another. Notation includes $f(g(x))$ or $(f \circ g)(x)$.

Example:
If $f(x) = 2x + 1$ and $g(x) = x^2$, find $f(g(3))$.

1. Find $g(3)$: $3^2 = 9$
2. Plug result into $f$: $f(9) = 2(9) + 1 = 19$

Exam Focus

• Why it matters: Functions form the backbone of the ACT Math section (Intermediate Algebra). You will see 4-6 questions specifically testing notation and evaluation.
• Typical question patterns:
  • "Given $f(x) = …$, what is $f(a+b)$?"
  • "For which value of $x$ is $f(x)$ undefined?" (Check denominators).
  • Nested functions: $g(h(f(2)))$.
• Common mistakes:
  • Confusing $f(2)$ (plugging in 2) with $2f(x)$ (multiplying the result by 2).
  • Assuming the domain is always "all real numbers" without checking for restrictions.

Linear Functions

Slope-Intercept Form

The most common form on the ACT is $f(x) = mx + b$.

• $m$ = Slope (Rate of change, $\frac{\text{rise}}{\text{run}}$, $\frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$)
• $b$ = y-intercept (Value of $f(x)$ when $x=0$)

Parallel and Perpendicular Lines

• Parallel Lines: Have the same slope ($m<em>1 = m</em>2$).
• Perpendicular Lines: Have negative reciprocal slopes ($m<em>1 = -\frac{1}{m</em>2}$). Their product is $-1$.

Exam Focus

• Why it matters: Linear functions are the easiest points to gain but also easy to lose if you rush algebraic manipulation.
• Typical question patterns:
  • Word problems interpreting slope as a "rate" (e.g., cost per mile).
  • Finding the equation of a line given two points.
• Common mistakes:
  • Flipping $x$ and $y$ in the slope formula.
  • Forgetting that a vertical line ($x = k$) has an undefined slope and is not a function.

Polynomial Functions

Quadratic Functions

A polynomial of degree 2, typically written as $f(x) = ax^2 + bx + c$. The graph is a parabola.

• Vertex: The peak or valley of the parabola. The $x$-coordinate is at $x = -\frac{b}{2a}$. Plug this $x$ back in to find $y$.
• Roots/Zeros: The $x$-intercepts where $f(x) = 0$. Solve by factoring or using the Quadratic Formula:
  $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Higher-Degree Polynomials

• Degree: The highest exponent determines the shape.
  • Odd degree (like $x^3$): Ends go in opposite directions.
  • Even degree (like $x^4$): Ends go in the same direction.
• Leading Coefficient:
  • Positive: Right side goes up.
  • Negative: Right side goes down.

Exam Focus

• Why it matters: Quadratics appear frequently. You must be comfortable factoring and finding the vertex quickly.
• Typical question patterns:
  • "What are the solutions to…" (Find roots).
  • "What is the maximum height…" (Find the vertex $y$-coordinate).
• Common mistakes:
  • Forgetting the $2a$ in the denominator of the quadratic formula.
  • Sign errors when calculating the discriminant $b^2 - 4ac$.

Radical Functions

Basics

The parent function is $f(x) = \sqrt{x}$.

• Graph shape: Half a parabola on its side, starting from $(0,0)$ and extending to the right.
• Domain: For square roots (and other even roots), the argument must be $\ge 0$.

Example:
Find the domain of $f(x) = \sqrt{2x - 6}$.
Set $2x - 6 \ge 0$ $\rightarrow$ $2x \ge 6$ $\rightarrow$ $x \ge 3$.

Exam Focus

• Why it matters: These questions usually test domain restrictions.
• Typical question patterns:
  • "For which values of $x$ is the function real-valued?"
• Common mistakes:
  • Thinking square roots can output negative numbers (the principal root function $\sqrt{x}$ is always non-negative).

Piecewise Functions

Definition

A function defined by different formulas for different intervals of the domain.

Example:
$f(x) = \begin{cases} 2x &amp; \text{if } x &lt; 0 \ x^2 &amp; \text{if } x \ge 0 \end{cases}$

To evaluate $f(-3)$, check the condition: -3 < 0, so use the top rule: $2(-3) = -6$.
To evaluate $f(5)$, check the condition: $5 \ge 0$, so use the bottom rule: $5^2 = 25$.

Exam Focus

• Why it matters: Students often panic when they see the bracket notation, but the math is simple substitution.
• Typical question patterns:
  • "Given the piecewise function below, what is $f(4)$?"
  • Graphs with open circles ($<$ or $>$) vs. closed dots ($\le$ or $\ge$) at the boundaries.
• Common mistakes:
  • Plugging the $x$-value into both equations instead of selecting the correct one based on the inequality.

Exponential and Logarithmic Functions

Exponential Functions

Form: $f(x) = a \cdot b^x$ where b > 0 and $b \neq 1$.

• Growth: If b > 1, the graph rises rapidly to the right.
• Decay: If 0 < b < 1, the graph falls to the right, approaching zero.
• Asymptote: The graph approaches but never touches the horizontal line (usually $y=0$ unless shifted).

Logarithmic Functions

Logarithms are the inverses of exponentials: $y = \log_b(x)$ is equivalent to $b^y = x$.

• Domain: You can only take the log of a positive number (x > 0).
• Key Properties:
  • Product Rule: $\log<em>b(xy) = \log</em>b(x) + \log_b(y)$
  • Quotient Rule: $\log<em>b(\frac{x}{y}) = \log</em>b(x) - \log_b(y)$
  • Power Rule: $\log<em>b(x^n) = n \cdot \log</em>b(x)$

Exam Focus

• Why it matters: The ACT loves testing the conversion between log form and exponent form.
• Typical question patterns:
  • "If $5^x = 12$, write $x$ in terms of logs." ($x = \log_5(12)$).
  • Simplifying log expressions using properties.
  • Compound interest problems: $A = P(1 + r)^t$.
• Common mistakes:
  • Thinking $\log(a+b) = \log a + \log b$ (This is FALSE).
  • Forgetting that $\log x$ (with no base) implies base 10, and $\ln x$ implies base $e$.

Function Transformations and Translations

The Golden Rules of Shifting

If you know what $y=f(x)$ looks like, you can graph variations by identifying $h$ and $k$ in $y = a \cdot f(x - h) + k$.

Note: Horizontal shifts are counter-intuitive. $x - 3$ moves Right, $x + 3$ moves Left.

Exam Focus

• Why it matters: This allows you to solve graphing problems without actually plotting points.
• Typical question patterns:
  • "The graph of $y=x^2$ is shifted 3 units right and 2 units down. What is the new equation?" (Answer: $y = (x-3)^2 - 2$).
• Common mistakes:
  • Moving left for $f(x-2)$ instead of right.

Analyzing Graphs and Key Features

Intercepts

• $y$-intercept: Set $x=0$ and solve for $y$. Every function (that is defined at 0) has exactly one.
• $x$-intercepts (Zeros): Set $y=0$ and solve for $x$. A function can have none, one, or many.

Intervals

• Increasing: As $x$ gets bigger (move right), $y$ goes up.
• Decreasing: As $x$ gets bigger (move right), $y$ goes down.

Asymptotes

Lines that a graph approaches but doesn't touch.

• Vertical Asymptotes: Occur where the denominator is zero (rational functions).
• Horizontal Asymptotes: Occur in exponential functions or rational functions based on end behavior.

Exam Focus

• Why it matters: ACT questions often show a graph and ask you to identify the equation, or vice-versa.
• Typical question patterns:
  • Matching an equation to a graph based on roots and intercepts.
  • Identifying the range of a function from its picture.
• Common mistakes:
  • Confusing the $x$-coordinate of the vertex with the axis of symmetry.

Quick Review Checklist

• Can you evaluate a composite function like $f(g(2))$ from the inside out?
• Do you know the difference between a domain (inputs) and a range (outputs)?
• Can you identify the slope and y-intercept instantly from $y = mx + b$?
• Do you know how to find the vertex of a parabola given in standard form $y = ax^2 + bx + c$?
• Can you rewrite a logarithmic equation $y = \log_b(x)$ as an exponential equation $b^y = x$?
• Do you know that $f(x-5)$ shifts a graph to the right, not the left?
• Can you determine the domain of a radical function like $\sqrt{x-4}$?
• Do you know that parallel lines have equal slopes and perpendicular lines have negative reciprocal slopes?

Final Exam Pitfalls

1. The "Undefined" Trap:

   • Mistake: Assuming a function exists everywhere.
   • Correction: Always check for division by zero ($\frac{1}{0}$) or square roots of negative numbers. These values are excluded from the domain.

2. The Parentheses Error:

   • Mistake: Typing $-3^2$ into a calculator instead of $(-3)^2$.
   • Correction: $-3^2$ is $-9$; $(-3)^2$ is $9$. Function inputs often act like they are in parentheses.

3. The Horizontal Shift Switch:

   • Mistake: Seeing $y = (x+4)^2$ and thinking the graph moves 4 units to the positive side.
   • Correction: Remember "inside is opposite." $x+4$ moves the graph to the negative side (Left 4).

4. Slope Confusion:

   • Mistake: Calculating slope as $\frac{x<em>2 - x</em>1}{y<em>2 - y</em>1}$.
   • Correction: It is always Rise over Run ($\frac{y}{x}$). Think "$y$ goes to the sky."

5. f(x) is just y:

   • Mistake: Getting confused by the notation $f(x)$ and trying to solve for $f$.
   • Correction: Mentally replace $f(x)$ with $y$ whenever you see it. It is just a label for the output variable.

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

GPT 5.2 Pro

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

What You Need to Know

• A function assigns each input in its domain to exactly one output—ACT questions often test domain/range, evaluating $f(x)$, and interpreting graphs.
• Most function problems reduce to spotting key features (intercepts, vertex/turning points, end behavior, asymptotes) and using them to match equations ↔ graphs.
• Transformations (shifts, stretches, reflections) let you rewrite and sketch graphs quickly—frequently tested via “how does the graph change?” questions.
• Expect functions to appear throughout ACT Math (especially the ACT’s “Preparing for Higher Math” skills), often embedded inside algebra, modeling, and graph interpretation.

Function Definition and Notation

A function is a relation where every input has exactly one output.

Key terms

• Domain: all allowable inputs $x$.
• Range: all possible outputs $y$.
• Function notation: $f(x)$ means the output when the input is $x$.

Notation reference

Evaluating and solving with functions

• Evaluate: if $f(x)=2x-3$, then $f(5)=2(5)-3=7$.
• Solve: if $f(x)=7$, solve $2x-3=7$.

Domain basics (common ACT patterns)

• Fractions: denominator cannot be $0$.
• Radicals (even root): radicand must be $\ge 0$.
• Logs: argument must be >0.

Worked example

Given $f(x)=\dfrac{x+1}{x-2}$, find the domain.
1) Denominator restriction: $x-2 \ne 0$
2) So $x \ne 2$
Domain: all real $x$ except $2$.

Exam Focus

• Why it matters: Function notation and domain/range are core ACT “language” used across algebra and graph questions.
• Typical question patterns:
  • “If $f(x)=\dots$, find $f(\text{number})$” or “solve $f(x)=\text{number}$.”
  • Identify whether a relation is a function from a table/graph.
  • Find domain restrictions from a formula.
• Common mistakes:
  • Treating $f(x)$ as $f\cdot x$—it’s a single symbol.
  • Forgetting domain restrictions (especially denominators and logs).
  • Assuming a graph is a function without checking the “one output per input” idea.

Linear Functions

A linear function has constant rate of change (slope) and graphs as a line.

Forms you should recognize

• Slope-intercept: $y=mx+b$ (slope $m$, $y$-intercept $b$)
• Point-slope: $y-y<em>1=m(x-x</em>1)$

Slope and intercepts

• Slope from two points $(x<em>1,y</em>1)$ and $(x<em>2,y</em>2)$:
  $m=\dfrac{y<em>2-y</em>1}{x<em>2-x</em>1}$
• $x$-intercept: set $y=0$.
• $y$-intercept: set $x=0$.

Worked example

Line through $(2,5)$ with slope $-3$.
1) Use point-slope: $y-5=-3(x-2)$
2) Simplify: $y-5=-3x+6$
3) $y=-3x+11$

Real-world link

Linear models fit constant-rate situations: hourly pay, constant-speed travel, steady growth/decay per unit.

Exam Focus

• Why it matters: Linear functions are among the most frequent function types on ACT due to quick algebra/graph interpretation.
• Typical question patterns:
  • Find slope/intercepts from an equation or two points.
  • Match a line’s equation to its graph (sign of slope, intercept size).
  • Interpret slope as a rate (units).
• Common mistakes:
  • Mixing up $x$- vs $y$-intercepts (remember: intercept is where the other variable is $0$).
  • Sign errors when computing slope.
  • Forgetting to distribute in $y-y<em>1=m(x-x</em>1)$.

Polynomial Functions

A polynomial function is $f(x)=a<em>nx^n+a</em>{n-1}x^{n-1}+\dots+a<em>1x+a</em>0$ with nonnegative integer exponents.

Key graph features

• Degree: highest exponent $n$.
• Leading coefficient $a_n$ controls end behavior.
• Zeros/roots: where $f(x)=0$ (the $x$-intercepts).
• Multiplicity idea (ACT-level): repeated factors may make the graph “touch” and turn instead of crossing.

Factored form helps

If $f(x)=(x-1)(x+2)$, zeros are $x=1$ and $x=-2$.

Worked example

Find zeros of $f(x)=x^2-5x+6$.
1) Factor: $x^2-5x+6=(x-2)(x-3)$
2) Set equal to $0$: $(x-2)(x-3)=0$
3) Solutions: $x=2$ or $x=3$

Exam Focus

• Why it matters: Polynomials appear in equation solving, graph matching, and modeling (especially quadratics).
• Typical question patterns:
  • Factor to find zeros/intercepts.
  • Use degree/leading coefficient to predict end behavior.
  • Compare graphs of polynomials with different degrees.
• Common mistakes:
  • Sign errors when factoring (e.g., constant term).
  • Confusing zeros with $y$-intercept (compute $f(0)$ for that).
  • Ignoring end behavior clues when choosing a matching graph.

Radical Functions

A radical function involves a root, commonly $f(x)=\sqrt{x}$ or $f(x)=\sqrt{ax+b}$.

Domain and shape

• For even roots: require $\text{radicand} \ge 0$.
• Graph of $y=\sqrt{x}$ starts at $(0,0)$ and increases, concave down.

Worked example

Find the domain of $f(x)=\sqrt{2x-8}$.
1) Require radicand $\ge 0$: $2x-8 \ge 0$
2) Solve: $2x \ge 8$ so $x \ge 4$
Domain: $x \ge 4$.

Real-world link

Square-root models show up in geometry (side lengths from area), physics (some distance–time relationships), and statistics (standard deviation formulas).

Exam Focus

• Why it matters: ACT often tests domain restrictions and graph shifts of radicals.
• Typical question patterns:
  • Find domain or evaluate at a given $x$.
  • Solve equations with radicals (often by squaring both sides).
  • Identify a shifted/scaled square-root graph.
• Common mistakes:
  • Forgetting the radicand constraint for domain.
  • Introducing extraneous solutions after squaring—always check in the original equation.
  • Misapplying transformations (inside vs outside the radical).

Piecewise Functions

A piecewise function defines different rules on different intervals of the domain.

How to evaluate

1) Locate which condition your input satisfies.
2) Use that rule only.

Worked example

Let
$f(x)=\begin{cases}2x+1 &amp; x<0\ x^2 & 0\le x\le 2\ 3 & x>2\end{cases}$
Find $f(-3)$, $f(2)$, and $f(5)$.

• x=-3<0: $f(-3)=2(-3)+1=-5$
• $x=2$ is in $0\le x\le 2$: $f(2)=2^2=4$
• x=5>2: $f(5)=3$

Real-world link

Tax brackets, shipping costs, and utility billing often use piecewise rules.

Exam Focus

• Why it matters: ACT uses piecewise functions to test careful reading and graph interpretation.
• Typical question patterns:
  • Evaluate $f(a)$ from a piecewise definition.
  • Identify jump/closed/open circles on graphs at boundary points.
  • Solve $f(x)=k$ by checking each interval.
• Common mistakes:
  • Using the wrong interval rule.
  • Mishandling endpoints (e.g., confusing < vs $\le$).
  • Forgetting that boundaries may have different left/right behavior.

Exponential and Logarithmic Functions

An exponential function has the variable in the exponent, typically $f(x)=a\cdot b^x$ with b>0 and $b\ne 1$.
A logarithmic function is the inverse type of exponential and is defined where its argument is positive.

High-value facts

• Growth if b>1, decay if 0<b<1.
• Exponential graphs often have a horizontal asymptote (commonly $y=0$ if no vertical shift).
• Log domain: argument must satisfy >0.

Worked example (exponential)

If $f(x)=3\cdot 2^x$, find $f(4)$.
1) Substitute: $f(4)=3\cdot 2^4$
2) Compute: $2^4=16$
3) $f(4)=48$

Real-world link

Population growth, compound interest, depreciation, and half-life are exponential models.

Exam Focus

• Why it matters: These functions appear in modeling and graph-feature questions (growth/decay, asymptotes, domain).
• Typical question patterns:
  • Compare growth vs decay from the base $b$.
  • Evaluate exponential expressions quickly.
  • Determine allowed inputs for logs (argument >0) or identify asymptotes/shifts.
• Common mistakes:
  • Treating $b^x$ like $x^b$—they behave very differently.
  • Ignoring the log domain restriction.
  • Missing the effect of vertical shifts on asymptotes.

Function Transformations and Translations

A transformation changes a base graph’s position or shape.

Common transformations (start from $y=f(x)$)