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 using the Vertical Line Test.

Relations and Functions

• Relation: A relation is a connection between input and output values.

  • Graphically, relations are represented as ordered pairs (x, y).

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

  • This means that for every value of x, there is only one corresponding value of y.

Examples and Visualizations

• Example 1: Consider the set of points {(-2,2), (1,1), (3,-2)}. Each input (x-value) is paired with only one output (y-value), so this is a function.

• Example 2: Consider the set {(-4,2), (1,2), (3,4), (-2,-1)}. If any input is paired with more than one output, it is not a function.

Inputs and Outputs

• Inputs (x): The set of all possible independent variable values.

• Outputs (y): The set of all possible dependent variable values.

• For a relation to be a function, each input must correspond to only one output.

Vertical Line Test

The Vertical Line Test is a graphical method to determine if a graph represents a function:

• If any vertical line drawn on the graph passes through more than one point, the graph does not represent a function.

• If every vertical line passes through at most one point, the graph does represent a function.

Graphical Examples

• Example: The graph of y = x^2 passes the vertical line test and is a function.

• Example: The graph of a sideways parabola (e.g., x = y^2) fails the vertical line test and is not a function.

Summary Table: Function vs. Not a Function

Key Formula:

• Function notation:

• Ordered pair:

Additional info: The concept of functions is foundational for all subsequent topics in Business Calculus, including limits, derivatives, and integrals.