Functions and Graphs

Definition of a Function

A function is a correspondence from a first set, called the domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range. The set of ordered pairs that defines the function is called the graph of the function.

• Domain: The set of all possible input values (independent variable).

• Range: The set of all possible output values (dependent variable).

• Ordered Pair: Each input-output relationship can be written as .

Example: The rule that assigns to each person in this room their UTC ID is a function, since each person has exactly one UTC ID.

Function Notation and Basic Concepts

Function notation expresses the output of a function for a given input. If is a function and is in its domain, then denotes the output corresponding to .

• Independent Variable: The input value, usually denoted by .

• Dependent Variable: The output value, usually denoted by .

Example: If , then for , .

Evaluating Functions

• To evaluate , substitute for in the function's formula.

• Examples: , , .

Graphs of Functions

The graph of a function is the set of all ordered pairs . It visually represents the relationship between input and output values.

• Example: Graph and in the same coordinate system.

The Vertical Line Test

The Vertical Line Test is a method to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not define as a function of .

• Theorem: If any vertical line intersects a graph in more than one point, the graph does not define a function.

• Example: Use the vertical line test to identify which graphs are functions.

Domain and Range from Graphs

• Closed dot: The graph does not extend beyond this point; the point belongs to the graph.

• Open dot: The graph does not extend beyond this point; the point does not belong to the graph.

• Arrow: The graph extends indefinitely in the direction indicated.

Example: Use the graph of to find , the domain, and the range of .

Piecewise Functions

A piecewise function is defined by two or more equations over specified domains. Each "piece" applies to a different part of the domain.

• Example:

• Application: Modeling cell phone data plans, tax brackets, etc.

Example: Graph

Operations on Functions

Function Composition

The composition of functions and is written as or . The domain of is the set of all such that is in the domain of and is in the domain of .

• Formula:

• Example: If and , then

Relations and Functions

A relation is a set of ordered pairs. A function is a special type of relation where each input has exactly one output.

Example: Assigning scores to student IDs is a function if each ID has only one score.

Inverse Functions

An inverse function reverses the effect of the original function. If and are inverses, then and for every in the domain.

• Notation: denotes the inverse of .

• Example: If , then .

Complex Numbers

Definition and Notation

Complex numbers extend the real numbers to include solutions to equations like . The imaginary unit is defined as , so .

• Any complex number can be written as , where and are real numbers.

• Standard form:

• Principal square root:

Operations with Complex Numbers

• Addition/Subtraction:

• Multiplication:

• Division: Use complex conjugates to rationalize denominators.

Example:

Quadratic Equations with Complex Solutions

Quadratic equations may have complex solutions if the discriminant is negative.

• Example: Solve

Quadratic Functions and Parabolas

Definition and Standard Form

A quadratic function is any function of the form , where are real numbers and . The graph of a quadratic function is a parabola.

Vertex Form and Properties

• Vertex form:

• The vertex is at .

• If , the parabola opens upward; if , it opens downward.

• The axis of symmetry is .

Example: Graph

Finding the Vertex

• For , the vertex is at .

Example: For , find the maximum value and where it occurs.

Applications of Quadratic Functions

• Projectile motion: The height of an object can be modeled by a quadratic function.

• Example: models the height of an arrow.

Optimization Problems

• Quadratic functions can be used to find minimum or maximum values in real-world problems.

• Example: Find the pair of numbers whose difference is 10 and whose product is minimized.

Summary Table: Key Concepts

Additional info: These notes expand on the original material by providing definitions, formulas, and examples for each concept, ensuring a self-contained and comprehensive study guide for Precalculus students.