Comprehensive Guide to Functions for ACT Math

Understanding Function Definition and Notation

At its core, a function is a relationship between inputs and outputs where every input ($x$) has exactly one distinct output ($y$ or $f(x)$). Think of a function as a machine: you drop an $x$ value in, and the machine follows a rule to produce a result.

The Vertical Line Test

To determine if a graph represents a function, visualize a vertical line moving across the graph. If the line touches the graph at more than one point at any single location, it is not a function.

Function Notation and Substitution

The notation $f(x)$ is read "f of x." It does not mean $f$ times $x$. It identifies the rule being applied.

• Evaluation: If $f(x) = 2x^2 - 3$, to find $f(4)$, simply replace every $x$ with 4.
  f(4) = 2(4)^2 - 3 = 2(16) - 3 = 29
• Composition: This is when you plug one function into another, written as $f(g(x))$ or $(f \circ g)(x)$.
  • Step 1: Evaluate the inner function first.
  • Step 2: Use that result as the input for the outer function.

Example:
If $f(x) = x + 5$ and $g(x) = 3x$, find $f(g(2))$.

1. Find $g(2)$: $3(2) = 6$.
2. Find $f(6)$: $6 + 5 = 11$.

Domain and Range

• Domain: All possible input values ($x$). Look left to right on a graph.
• Range: All possible output values ($y$). Look bottom to top on a graph.

Key Restrictions:

1. Denominators: Cannot equal zero ($ \frac{1}{x-2} \rightarrow x \neq 2 $).
2. Even Roots: The inside must be non-negative ($ \sqrt{x} \rightarrow x \ge 0 $).

Linear Functions

Linear functions form straight lines and are the foundation of ACT coordinate geometry.

Slope-Intercept Form

f(x) = mx + b

• $m$ (Slope): The rate of change ($ \frac{\text{rise}}{\text{run}} $).
• $b$ (y-intercept): The value of $f(x)$ when $x=0$.

Parallel vs. Perpendicular

• Parallel lines have the same slope.
• Perpendicular lines have negative reciprocal slopes (e.g., $2$ and $-\frac{1}{2}$).

Polynomial Functions

The most common polynomial on the ACT, aside from linear equations, is the quadratic function.

Quadratic Functions (Parabolas)

Standard form: $f(x) = ax^2 + bx + c$. The graph is a parabola.

Key Features:

• Direction: If $a > 0$, it opens up (minimum). If $a < 0$, it opens down (maximum).
• Vertex: The peak or valley of the graph. The x-coordinate is found using $x = -\frac{b}{2a}$.
• roots/Zeros/Solutions: The points where the graph crosses the x-axis ($f(x)=0$).

Higher-Degree Polynomials

For functions like $x^3, x^4$, etc., focus on End Behavior.

• Odd Degree (e.g., $x^3$): Ends point in opposite directions.
• Even Degree (e.g., $x^2, x^4$): Ends point in the same direction.

Radical Functions

Radical functions involve roots, typically square roots: $f(x) = \sqrt{x}$.

• Shape: Look like a "swimmer's arm" or half of a sideways parabola.
• Domain Constraint: Remember, for real numbers, you cannot take the square root of a negative number. If $f(x) = \sqrt{x-3}$, you must set $x-3 \ge 0$, so the domain is $x \ge 3$.

Piecewise Functions

Piecewise functions look scary but are simple if you follow instructions. The function is defined by different rules for different intervals of $x$.

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

Strategy:
To evaluate piecewise functions, look at the condition (the "if" part) first.

• To find $f(-3)$: Since $-3 < 0$, use the top rule: $2(-3) = -6$.
• To find $f(2)$: Since $2 \ge 0$, use the bottom rule: $2^2 + 1 = 5$.

Exponential and Logarithmic Functions

These two are inverses of each other. If you reflect an exponential graph over the line $y=x$, you get a logarithmic graph.

Exponential Functions

f(x) = a \cdot b^x

• $a$: Initial value (y-intercept).
• $b$: Growth factor. If $b > 1$, it is growth; if $0 < b < 1$, it is decay.
• Asymptote: Horizontal line the graph approaches but never touches (usually $y=0$).

Logarithmic Functions

f(x) = \log_b(x)

• Definition: A logarithm is an exponent. $y = \log_b(x)$ is usually rewritten as $b^y = x$.
• Domain: You cannot take the log of a negative number or zero. The argument must be $> 0$.

Function Transformations and Translations

The ACT loves asking how changing an equation moves the graph. Use the "Inside/Outside" mnemonic.

The Rules

Given a parent function $f(x)$:

1. Vertical Shifts (Outside the function): Follows logic.

   • $f(x) + k$: Shift Up $k$ units.
   • $f(x) - k$: Shift Down $k$ units.

2. Horizontal Shifts (Inside the function): Opposite of what you expect.

   • $f(x - h)$: Shift Right $h$ units.
   • $f(x + h)$: Shift Left $h$ units.

3. Reflections:

   • $-f(x)$: Reflection over the x-axis (values become negative).
   • $f(-x)$: Reflection over the y-axis.

Example:
$g(x) = (x-2)^2 + 3$
This is the graph of $x^2$ moved Right 2 and Up 3.

Analyzing Graphs and Key Features

You must be able to extract information directly from a graph without an equation.

• Intercepts: Where the line crosses axes.
  • x-intercept: Where $y=0$.
  • y-intercept: Where $x=0$.
• Increasing/Decreasing:
  • Read the graph from Left to Right.
  • If the pen goes up, it's increasing. If it goes down, it's decreasing.
• Asymptotes: Dotted lines that the function approaches very closely but (usually) never touches. Common in rational and exponential functions.

Common Mistakes & Pitfalls

1. The "Inside Lie": Students often shift $f(x+2)$ to the right. Remember, anything grouped directly with $x$ is the opposite. $x+2$ is a shift Left.
2. Confusing Notation: $f^{-1}(x)$ means the Inverse Function, NOT the reciprocal $\frac{1}{f(x)}$. They are completely different concepts.
3. Undefined Values: Forgetting that division by zero is impossible. If $f(x) = \frac{1}{x}$, the domain is all real numbers except 0.
4. Composition Order: $f(g(x)) \neq g(f(x))$. The order matters immensely. Always start from the innermost parentheses.
5. Exponents as Coefficients: In $f(x) = 2x^2$, only the $x$ is squared, not the 2. If the problem meant $(2x)^2$, it would be written with parentheses.