Precalculus

2.

Functions and Their Graphs

2.1.

Fundamentals of Functions

2.1.1.

Definition of a Function

2.1.1.1.

Relation vs Function

2.1.1.2.

Input and Output

2.1.1.3.

Domain and Codomain

2.1.1.4.

2.1.2.

Function Notation

2.1.2.1.

2.1.2.2.

Evaluating Functions at Specific Values

2.1.2.3.

Evaluating Functions with Expressions

2.1.2.4.

Difference Quotient

2.1.3.

Domain and Range

2.1.3.1.

Finding Domain Algebraically

2.1.3.2.

Finding Range Algebraically

2.1.3.3.

Finding Range Graphically

2.1.3.4.

Restrictions from Square Roots

2.1.3.5.

Restrictions from Denominators

2.1.4.

The Vertical Line Test

2.1.4.1.

Identifying Functions from Graphs

2.1.4.2.

2.2.

Graphing Functions

2.2.1.

Plotting Points Method

2.2.2.

2.2.2.1.

2.2.2.2.

2.2.3.

2.2.3.1.

y-axis Symmetry

2.2.3.2.

x-axis Symmetry

2.2.3.3.

Origin Symmetry

2.2.4.

Graphing by Hand vs Technology

2.3.

Properties of Functions

2.3.1.

Increasing and Decreasing Functions

2.3.1.1.

Identifying Intervals from Graphs

2.3.1.2.

Algebraic Tests

2.3.2.

Local and Absolute Extrema

2.3.2.1.

Local Maximum and Minimum

2.3.2.2.

Absolute Maximum and Minimum

2.3.2.3.

Identifying Extrema Graphically

2.3.3.

Even and Odd Functions

2.3.3.1.

Algebraic Tests for Evenness and Oddness

2.3.3.2.

Graphical Characteristics

2.3.4.

Average Rate of Change

2.3.4.1.

Definition and Formula

2.3.4.2.

Geometric Interpretation

2.3.5.

2.3.5.1.

Continuous Functions

2.3.5.2.

Discontinuous Functions

2.3.5.3.

Types of Discontinuities

2.4.

A Library of Parent Functions

2.4.1.

Constant Functions

2.4.2.

Linear Functions

2.4.3.

Identity Function

2.4.4.

Squaring Function

2.4.5.

Cubing Function

2.4.6.

Square Root Function

2.4.7.

Cube Root Function

2.4.8.

Reciprocal Function

2.4.9.

Absolute Value Function

2.4.10.

Greatest Integer Function

2.4.11.

Piecewise-Defined Functions

2.4.11.1.

Definition and Notation

2.4.11.2.

Graphing Piecewise Functions

2.4.11.3.

Evaluating Piecewise Functions

2.5.

Transformations of Functions

2.5.1.

Vertical Shifts

2.5.2.

Horizontal Shifts

2.5.3.

2.5.3.1.

Reflection across the x-axis

2.5.3.2.

Reflection across the y-axis

2.5.4.

Vertical Stretching and Shrinking

2.5.5.

Horizontal Stretching and Shrinking

2.5.6.

Combining Transformations

2.5.6.1.

Order of Operations

2.5.6.2.

Multiple Transformations

2.6.

Operations on Functions

2.6.1.

Algebra of Functions

2.6.1.1.

Sum of Functions

2.6.1.2.

Difference of Functions

2.6.1.3.

Product of Functions

2.6.1.4.

Quotient of Functions

2.6.1.5.

Domain of Combined Functions

2.6.2.

Composition of Functions

2.6.2.1.

Definition and Notation

2.6.2.2.

Evaluating Composite Functions

2.6.2.3.

Domain of Composite Functions

2.6.2.4.

Decomposing Functions

2.7.

Inverse Functions

2.7.1.

Definition of Inverse Functions

2.7.2.

One-to-One Functions

2.7.3.

The Horizontal Line Test

2.7.4.

Finding Inverse Functions Algebraically

2.7.5.

Verifying Inverse Functions

2.7.6.

Graphs of Inverse Functions

2.7.6.1.

Reflection across y = x

2.7.7.

Domain and Range of Inverse Functions

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