0. Functions

Intro to Functions & Their Graphs

This section introduces the foundational concepts of relations and functions, which are essential for understanding Business Calculus. It explains how to distinguish between general relations and functions, and provides a graphical method for identifying functions.

Relations and Functions

• Relation: A relation is a connection between two sets of values, typically called inputs and outputs. Graphically, relations are represented as ordered pairs .

• Function: A function is a special type of relation where each input has at most one output. In other words, for every value of , there is only one corresponding value of .

Visual Representation

• Relations and functions can be visualized using graphs or mapping diagrams (ovals showing inputs and outputs).

• Examples:

  • If a single input (x-value) is paired with more than one output (y-value), the relation is not a function.

  • If every input is paired with only one output, the relation is a function.

Examples

• Consider the set of points :

  • Each input has only one output .

  • This is a function.

• Consider the set of points :

  • The input is paired with both and .

  • This is not a function.

Vertical Line Test

• The Vertical Line Test is a graphical 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 represent a function.

• If every vertical line intersects the graph at most once, the graph does represent a function.

Summary Table: Function vs. Not a Function

Key Formulas and Definitions

• Ordered Pair:

• Function Notation: denotes the output when the input is .

Example

• Given , for , .

• Graphically, the parabola passes the vertical line test and is a function.

Additional info: The vertical line test is a standard method in calculus and algebra to quickly assess whether a graph represents a function. This foundational concept is critical for later topics such as limits, derivatives, and integrals.