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.

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

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

Key Properties:

• Every function is a relation, but not every relation is a function.

• Functions are often described using function notation: \(f(x)\).

Examples and Applications:

• In business, functions can represent cost, revenue, or profit as a function of the number of units sold.

• Relations that are not functions may occur when a single input could produce multiple outputs, which is not allowed for functions.

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.

Formally:

• A set of ordered pairs \((x, y)\) is a function if for every \(x\), there is only one \(y\).

Table: Comparison of Relations and Functions

Example:

• The set \{(1,2), (2,3), (3,4)\} is a function because each input has only one output.

• The set \{(1,2), (1,3), (2,4)\} is not a function because the input 1 has two different outputs (2 and 3).