Relations and Functions

Definitions and Key Concepts

Understanding the foundational concepts of relations and functions is essential in Business Calculus, as these ideas underpin much of mathematical modeling and analysis.

• Relation: A relation is a connection between two sets of values, typically represented as ordered pairs .

• Function: A function is a special type of relation where each input (usually ) has at most one output (usually ).

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Example: The set is a function, but is not, since maps to two different values.

Inputs and Outputs

Functions and relations can be visualized using input-output diagrams, which help clarify whether a relation is a function.

• Inputs (x): The independent variable or domain values.

• Outputs (y): The dependent variable or range values.

Practice: Given , inputs are ; outputs are . This is a function.

Domain and Range

Definitions

The domain and range are fundamental concepts for describing the set of possible inputs and outputs for a function.

• Domain: The set of all allowed input values () for a function.

• Range: The set of all allowed output values () for a function.

Finding Domain: "Squish" the graph horizontally onto the -axis to see which values are covered.

Finding Range: "Squish" the graph vertically onto the -axis to see which values are covered.

Notation

• Interval Notation: Uses brackets and parentheses to describe intervals. For example, means is at least and less than $5$.

• Set Builder Notation: Describes sets using conditions, e.g., .

• Brackets [ ]: Include the endpoint.

• Parentheses ( ): Exclude the endpoint.

Example: For a graph covering from to $5), domain is .

Union of Intervals

When a graph has multiple intervals, use the union symbol to combine them.

Example: means all less than $0x.

Piecewise Functions

Definition and Structure

Piecewise functions are functions defined by different equations over different intervals of the domain. They are useful for modeling situations where a rule changes based on the input value.

• Piecewise Function: A single function made up of two or more equations, each valid for a specific interval of the domain.

• If the values at the boundaries do not match, the function has a jump (discontinuity).

Evaluating Piecewise Functions

To evaluate a piecewise function at a given input, determine which interval the input falls into and use the corresponding equation.

Example:

• Given , to find , use the first case: .

• To find , use the second case: .

Graphing Piecewise Functions

Graph each "piece" over its specified interval, paying attention to open or closed circles at endpoints to indicate whether the endpoint is included.

Example: For , graph each segment over its interval.

Summary Table: Function Properties

Practice Problems

• Determine if a given relation is a function using the vertical line test.

• Find the domain and range of a graph using interval notation.

• Evaluate piecewise functions for specific input values.

• Graph piecewise functions, indicating endpoints clearly.

Additional info: These notes cover foundational topics in functions, domain, range, and piecewise functions, which are essential for Business Calculus and further study in mathematical modeling and analysis.