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 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 as "rules" that assign to each input exactly one output.

Example:

• Relation: \{(-2,2), (1,1), (3,-2)\}

• Function: Each input (x-value) is paired with only one output (y-value).

• Not a Function: If an input is paired with more than one output, such as \{(-2,2), (-2,-1), (1,2), (3,4)\}, it 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.

Example:

• The graph of \(y = x^2\) passes the vertical line test and is a function.

• The graph of a circle \(x^2 + y^2 = r^2\) fails the vertical line test and is not a function.

Summary Table:

Additional info: In business calculus, understanding functions is crucial for modeling relationships between variables such as cost, revenue, and profit.