Functions & Formulas

Master the art of manipulating and translating functions to conquer the ACT Math section!

Function Fundamentals

What is a Function?

A function is a relation that assigns exactly one output to each input. Think of a function as a machine that takes an input, performs an operation, and produces an output.

\[ f(x) = \text{output when the input is } x \]

Key ACT Concept: Function Notation

The ACT frequently tests your understanding of function notation. Remember:

If $f(x) = 2x + 3$, then $f(5) = 2(5) + 3 = 13$
If $g(x) = x^2 - 4$, then $g(-2) = (-2)^2 - 4 = 4 - 4 = 0$

Example:

If $f(x) = 3x^2 - 2x + 5$, find $f(2)$.

Step 1: Substitute $x = 2$ into the function.

\[ f(2) = 3(2)^2 - 2(2) + 5 \] \[ f(2) = 3(4) - 4 + 5 \] \[ f(2) = 12 - 4 + 5 \] \[ f(2) = 13 \]

Domain and Range

The domain of a function is the set of all possible input values. The range is the set of all possible output values.

Key ACT Concept: Finding Domain Restrictions

Common domain restrictions occur when:

The denominator of a fraction equals zero
The expression under a square root is negative

Example:

Find the domain of $f(x) = \frac{x+3}{x-2}$.

Step 1: Identify when the denominator equals zero.

\[ x - 2 = 0 \] \[ x = 2 \]

Step 2: Express the domain.

The domain is all real numbers except $x = 2$, which we can write as $\{x | x \neq 2\}$ or $(-\infty, 2) \cup (2, \infty)$.

Linear Functions

Linear functions have the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Key ACT Concept: Interpreting Slope

The slope $m$ represents the rate of change of the function:

Positive slope: Function increases as $x$ increases
Negative slope: Function decreases as $x$ increases
Zero slope: Function is constant

Example:

A car rental company charges $40 per day plus $0.25 per mile. Write a function that gives the cost $C(m)$ of renting a car for one day and driving $m$ miles.

Step 1: Identify the fixed cost and the variable cost.

Fixed cost = $40

Variable cost = $0.25 per mile

Step 2: Write the function.

\[ C(m) = 40 + 0.25m \]

Step 3: Interpret the function.

The function $C(m) = 40 + 0.25m$ gives the cost in dollars for renting a car for one day and driving $m$ miles. The slope of 0.25 means the cost increases by $0.25 for each additional mile driven.

Quadratic Functions

Quadratic functions have the form $f(x) = ax^2 + bx + c$, where $a \neq 0$. The graph of a quadratic function is a parabola.

Key ACT Concept: Vertex Form

The vertex form of a quadratic function is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

If $a > 0$, the parabola opens upward and has a minimum value of $k$ at $x = h$
If $a < 0$, the parabola opens downward and has a maximum value of $k$ at $x = h$

Example:

Find the vertex of the quadratic function $f(x) = 2x^2 - 12x + 19$.

Step 1: Convert to vertex form by completing the square.

\[ f(x) = 2x^2 - 12x + 19 \] \[ f(x) = 2(x^2 - 6x) + 19 \] \[ f(x) = 2(x^2 - 6x + 9 - 9) + 19 \] \[ f(x) = 2(x - 3)^2 - 18 + 19 \] \[ f(x) = 2(x - 3)^2 + 1 \]

Step 2: Identify the vertex.

The vertex form is $f(x) = 2(x - 3)^2 + 1$, so the vertex is at $(3, 1)$.

Function Transformations

Function transformations allow us to shift, stretch, compress, or reflect the graph of a function.

Key ACT Concept: Common Transformations

Given a function $f(x)$:

$f(x) + k$: Shifts the graph up $k$ units
$f(x) - k$: Shifts the graph down $k$ units
$f(x - h)$: Shifts the graph right $h$ units
$f(x + h)$: Shifts the graph left $h$ units
$a \cdot f(x)$: Vertically stretches (if $a > 1$) or compresses (if $0 < a < 1$) the graph
$-f(x)$: Reflects the graph across the x-axis
$f(-x)$: Reflects the graph across the y-axis

Example:

Describe the transformation from $f(x) = x^2$ to $g(x) = -2(x - 3)^2 + 4$.

Step 1: Identify each transformation.

$-2$: Vertically stretch by a factor of 2 and reflect across the x-axis
$(x - 3)$: Shift right 3 units
$+ 4$: Shift up 4 units

Step 2: Describe the complete transformation.

The function $g(x)$ is obtained from $f(x) = x^2$ by:

Stretching vertically by a factor of 2
Reflecting across the x-axis
Shifting right 3 units
Shifting up 4 units

The result is a downward-facing parabola with vertex at $(3, 4)$.

Function Operations

Functions can be combined using arithmetic operations (addition, subtraction, multiplication, division) and composition.

Key ACT Concept: Function Composition

The composition of functions $f$ and $g$ is denoted as $(f \circ g)(x) = f(g(x))$.

To find $f(g(x))$, substitute $g(x)$ for the input variable in $f$.

Example:

If $f(x) = 2x + 3$ and $g(x) = x^2 - 1$, find $(f \circ g)(2)$.

Step 1: Find $g(2)$.

\[ g(2) = 2^2 - 1 = 4 - 1 = 3 \]

Step 2: Substitute $g(2)$ into $f$.

\[ (f \circ g)(2) = f(g(2)) = f(3) = 2(3) + 3 = 6 + 3 = 9 \]

Exponential and Logarithmic Functions

Exponential functions have the form $f(x) = a \cdot b^x$, where $a$ and $b$ are constants and $b > 0, b \neq 1$.

Logarithmic functions have the form $f(x) = \log_b(x)$, where $b$ is the base and $b > 0, b \neq 1$.

Key ACT Concept: Properties of Logarithms

$\log_b(xy) = \log_b(x) + \log_b(y)$
$\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$
$\log_b(x^n) = n \cdot \log_b(x)$
$b^{\log_b(x)} = x$ for $x > 0$
$\log_b(b^x) = x$

Example:

Solve the equation $2^{x+1} = 16$.

Step 1: Rewrite 16 as a power of 2.

\[ 16 = 2^4 \]

Step 2: Set the exponents equal.

\[ 2^{x+1} = 2^4 \] \[ x + 1 = 4 \] \[ x = 3 \]

Ready to Practice?

Now that you've learned the fundamentals of functions, test your knowledge with practice problems!

Practice Problems

Test your understanding with these practice problems. Select a difficulty level to begin.

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Functions Challenge

Put your function skills to the test with these timed challenges. Complete them to earn badges and level up your character!

Function Notation Challenge

100 XP

Evaluate 5 function expressions in 5 minutes.

5 minutes

5 problems

Function Transformations Challenge

150 XP

Identify 5 function transformations in 7 minutes.

7 minutes

5 problems

Function Operations Challenge

200 XP

Solve 5 function composition problems in 10 minutes.

10 minutes

5 problems

Functions ACT Test Simulation

Take a timed test that simulates the function questions you'll see on the ACT. This will help you identify your strengths and areas for improvement.

Test Details

Time Limit: 15 minutes
Questions: 10 function questions in ACT format
Topics Covered: Function notation, domain and range, transformations, and operations
Scoring: Instant results with detailed explanations

Ready to Test Your Skills?

This simulation will help you prepare for the actual ACT math section.