Quick Reference Checklist

• Functions and Graphs (Sections 1.1–1.3)

• Linear Functions and Transformations (Sections 1.4–1.6)

• Function Operations and Composition (Section 1.7)

• Inverse Functions (Section 1.8)

Functions and Graphs

Key Concepts

• Function: Each input (x) has exactly one output (y).

• Domain: All possible x-values for which the function is defined.

• Range: All possible y-values the function can take.

Visual Memory Aid: Vertical Line Test: If a vertical line touches the graph more than once, it is not a function.

Step-by-Step Problem Breakdown

Reading Graphs

1. Find Domain: Look at the leftmost and rightmost points. Write in interval notation: [start, end] or (start, end).

2. Find Range: Look at the lowest and highest points. Write in interval notation.

• Increasing: Graph goes up from left to right.

• Decreasing: Graph goes down from left to right.

• Constant: Graph is flat (horizontal line).

Function Evaluation

• To find f(x): Find x on the graph, draw a vertical line up/down, read the y-value.

• To find x when f(x) = a: Find y = a on the graph, draw a horizontal line left/right, read the x-value(s).

Algebraic Function Operations

• Write out the expressions, distribute coefficients, combine like terms, and simplify.

• Example: If and , then

Piecewise Functions

• Functions defined by different expressions over different intervals.

• Example:

• To evaluate, check which interval x falls into and use the corresponding formula.

Linear Functions and Transformations

Key Formulas

• Point-Slope Form:

• Slope-Intercept Form:

• Slope Formula:

Step-by-Step Problem Breakdown

Finding Line Equations

1. Find slope using two points:

2. Use point-slope or slope-intercept form to write the equation.

Perpendicular Lines

• Perpendicular slopes are negative reciprocals: If , then

Average Rate of Change

• Formula:

• Example: For data, ppm/month

Function Transformations

• Shift horizontally:

• Height (vertical):

• Vertical (y-axis) stretch:

• Expansion (horizontal):

Memory Device: "SHAVE" (Shift, Height, Axis, Vertical, Expansion)

Function Operations and Composition

Key Operations

• where

Step-by-Step Problem Breakdown

Basic Operations

• For and ,

• Domain considerations: : all real numbers except where denominator is zero; : ; combined domain: , values making denominator zero.

Function Composition

• Example: If and , then

Decomposing Functions

• Given , identify "inside" and "outside" functions: Inside: , Outside:

Inverse Functions

Key Concepts

• One-to-one: Passes horizontal line test

• Inverse exists: Only if function is one-to-one

• Inverse notation:

Step-by-Step Problem Breakdown

Finding Inverse Functions

1. Replace with

2. Swap and

3. Solve for

4. Write as

• Example: 1. 2. Swap: 3. Solve: 4.

Domains and Ranges

• Original function: Domain: All real numbers except Range: All real numbers except

• Inverse function: Domain: All real numbers except Range: All real numbers except

Verifying Inverses

• Show and

Table: Summary of Function Types and Properties

Additional info:

• Some context and examples were expanded for clarity and completeness.

• All formulas and equations are provided in LaTeX format for mathematical accuracy.