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 has at most one output. In other words, for every value of x, there is only one corresponding value of y.

Key Points:

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

• Functions are fundamental in modeling real-world business scenarios, such as cost, revenue, and profit calculations.

Example: Consider the following sets of ordered pairs:

• Set 1: {(-2,2), (0,1), (1,4), (3,-2)} – Each input (x-value) is paired with only one output (y-value), so this is a function.

• Set 2: {(-4,2), (0,1), (1,2), (3,4), (-2,-1)} – If any input is paired with more than one output, 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 sideways parabola (e.g., x = y^2) fails the vertical line test and is not a function.

Summary Table: Relation vs. Function

Formula:

• General 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 functions are the primary objects of study.