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 ordered pairs (x, y) on the coordinate plane.

• Function: A function is a special type of relation in which each input (x-value) is associated with at most one output (y-value). In other words, no input value maps to more than one output value.

Key Points:

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

• Functions are fundamental in modeling real-world business scenarios where each input (such as time or cost) produces a unique output (such as revenue or profit).

Example: Consider the following sets of points:

• Set 1: {(-2,2), (1,1), (3,-2)} – Each x-value is paired with only one y-value, so this is a function.

• Set 2: {(-4,2), (1,2), (3,4), (-2,-2)} – The x-value 1 is paired with 2, and 3 is paired with 4, but if any x-value repeats with a different y-value, 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 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.

Example:

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

• The graph of a circle, such as x^2 + y^2 = r^2, fails the vertical line test and is not a function.

Summary Table: Relation vs. Function

Formula:

• Function notation:

Additional info: Understanding the distinction between relations and functions is crucial for later topics in calculus, such as limits, derivatives, and integrals, where the behavior of functions is analyzed in detail.