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) on the coordinate plane.

• Function: A function is a special type of relation where 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 often described using function notation: f(x).

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 maps to two different outputs (2 and 3).

Graphical Representation and the Vertical Line Test

To determine if a graph represents a function, use the Vertical Line Test:

• If any vertical line drawn through the graph intersects it 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 = 1 fails the vertical line test and is not a function.

Additional info: Understanding the distinction between relations and functions is crucial for later topics in calculus, such as limits, derivatives, and integrals, where functions are the primary objects of study.