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 elements in the domain are often called inputs, while the elements in the range are called outputs.

• Domain: The set of all possible input values for the function.

• Range: The set of all possible output values.

• Rule: The correspondence that assigns each input to an output.

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

Function Notation and Basic Concepts

The input of a function is called the independent variable, while the output is called the dependent variable. Functions are often denoted by letters such as f, g, or h. The notation f(x) means "the value of the function f at x".

• Example: If , then for , .

Graphs of Functions

The graph of a function is the set of all ordered pairs . This provides a visual representation of the function.

• Example: The graph of consists of all points .

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 a function.

• Theorem: If any vertical line intersects a graph more than once, it is not a function.

• Example: Parabolas that open sideways fail the vertical line test and are not functions of .

Domain and Range from Graphs

The domain of a function is the set of all input values for which the function is defined, and the range is the set of all output values. These can often be determined from the graph.

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

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

• Arrow: The graph extends indefinitely in the direction of the arrow.

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: A mobile data plan cost function:

  x (GB used)	C(x) (Cost)

• Graph: The graph of a piecewise function shows the different rules applied to different intervals.

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 .

• Definition:

• Example: If and , then .

Relations and Functions

A relation is a set of ordered pairs. A function is a relation in which each input is associated with exactly one output.

• Example Table:

  ID	Score
  753456	74.3
  631245	86.1
  589021	92.4
  478263	97.3

  Assigning each ID to a score is a function, but swapping scores and IDs may not be a function if scores repeat.

Inverse Functions

An inverse function reverses the correspondence of a function. If and are two functions such that and for every in the domains, then is the inverse of , denoted .

• Example: and are inverses of each other.

• Finding the inverse: Solve for and interchange and .

Complex Numbers

Definition and Notation

Complex numbers are numbers of the form , where and are real numbers and is the imaginary unit defined by .

• Imaginary unit:

• Standard form:

• Principal square root:

Operations with Complex Numbers

• Addition/Subtraction:

• Multiplication:

• Division: Use complex conjugates to rationalize denominators.

• Complex conjugate: The conjugate of is .

Example: Express in the form by multiplying numerator and denominator by the conjugate of the denominator.

Quadratic Functions

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 called a parabola.

Vertex Form and Properties

• Vertex form:

• Vertex: The point is the vertex of the parabola.

• Axis of symmetry:

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

Theorem: The vertex of is at .

Applications of Quadratic Functions

• Maximum/Minimum: The vertex gives the maximum or minimum value of the quadratic function.

• Example: The height of an arrow follows a parabolic path: .

• Optimization: Quadratic functions are used to find maximum or minimum values in real-world problems.

Summary Table: Key Function Types

Additional info:

• Some examples and explanations have been expanded for clarity and completeness.

• Tables have been recreated to illustrate function assignments and piecewise definitions.

• All equations are provided in LaTeX format for mathematical precision.