Functions and Their Graphs

Definition of a Function

A function is a relation between two non-empty sets, X (domain) and Y (co-domain), such that each element in X is assigned a unique element in Y. This is denoted as .

• Domain: The set X, containing all possible input values.

• Co-domain: The set Y, containing all possible output values.

• Vertical Line Test: A graph represents a function if every vertical line intersects the graph at most once.

Determining Functions from Relations

• A relation is a function if no input (x-value) is paired with more than one output (y-value).

• Example: The set {(1, 1), (−1, 2), (2, 1), (3, 1), (−3, 3)} is a function because each x-value is unique.

• Example: {(1, 1), (1, 2), (2, 1), (3, 1), (−3, 3)} is not a function because x = 1 is paired with two different y-values.

Evaluating Functions

• To evaluate a function, substitute the given value into the function's formula.

• Example: For , , , .

Difference Quotient

The difference quotient is used to measure the average rate of change of a function:

• Formula:

• Example: For ,

Operations on Functions

• Addition:

• Multiplication:

• Division: ,

• Example: If , , then , ,

Graph of a Function and the Vertical Line Test

• A graph is a function if every vertical line crosses it at most once.

• Example: The graph of passes the vertical line test.

Intercepts and Domain

• x-intercept: Where the graph crosses the x-axis ().

• y-intercept: Where the graph crosses the y-axis ().

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

Properties of Functions

Even and Odd Functions

Functions can be classified based on their symmetry:

• Even Function: for all in the domain. The graph is symmetric with respect to the y-axis. Example: .

• Odd Function: for all in the domain. The graph is symmetric with respect to the origin. Example: .

• Neither: If neither condition holds, the function is neither even nor odd.

Increasing, Decreasing, and Constant Intervals

• A function is increasing on an interval if whenever .

• A function is decreasing on an interval if whenever .

• A function is constant if for all in the interval.

Local and Absolute Extrema

• Local Maximum: is a local maximum if for all near .

• Local Minimum: is a local minimum if for all near .

• Absolute Maximum: is the largest value on the interval.

• Absolute Minimum: is the smallest value on the interval.

Library of Functions and Piece-wise Defined Functions

Common Functions and Their Properties

Piece-wise Defined Functions

A piece-wise function is defined by different expressions on different intervals of the domain.

• Example:

• To evaluate, determine which interval x belongs to and use the corresponding formula.

• Piece-wise functions must have mutually exclusive intervals to be valid functions.

Example: Cellphone Plan as a Piece-wise Function

• This models a plan with a flat rate for the first 120 minutes and a per-minute charge thereafter.

Transformations of Functions

Vertical and Horizontal Shifts

• Vertical Shift: shifts the graph up by units; shifts it down by $k$ units.

• Horizontal Shift: shifts the graph left by units; shifts it right by $k$ units.

Stretching and Compressing

• Vertical Stretch: , , stretches the graph vertically by a factor of .

• Vertical Compression: , , compresses the graph vertically by a factor of .

• Horizontal Stretch: , , stretches the graph horizontally by a factor of .

• Horizontal Compression: , , compresses the graph horizontally by a factor of .

Reflections

• Across y-axis: reflects the graph horizontally.

• Across x-axis: reflects the graph vertically.

Linear and Quadratic Functions

Linear Functions

• A linear function has the form .

• Slope (m): Measures the rate of change; .

• y-intercept (b): The value of .

• The graph is a straight line.

• Increasing if , decreasing if , constant if .

Quadratic Functions

• A quadratic function has the form , .

• Vertex:

• Axis of Symmetry:

• Discriminant:

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

Graphing Quadratic Functions

1. Find the vertex and plot it.

2. Compute the discriminant to determine the number of x-intercepts.

3. Plot intercepts and additional points using symmetry.

Quadratic Models and Applications

• Quadratic functions are used to model maximum/minimum problems, such as area, revenue, and geometric applications.

• Example: The area of a rectangle with perimeter constraints can be expressed as a quadratic function of one side.

Quadratic Inequalities

Solving Quadratic Inequalities

• Graphically: Plot the functions and identify intervals where the inequality holds.

• Analytically: Solve or by finding roots and testing intervals.

• Example: can be rewritten as and solved using the quadratic formula.

Domain of Functions Involving Square Roots

• For , the domain is all such that .

• Solve to find the domain.

Summary Table: Types of Function Transformations

Examples and Applications

• Piece-wise Function Example:

• Quadratic Model Example: The area of a rectangle with perimeter and side is .

• Supply and Demand Example: , ; equilibrium occurs when .

Additional info: Some context and examples were inferred for completeness, such as the general forms of piece-wise functions and quadratic applications.