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, but no x-value is repeated with a different y-value, so this is also a function.

• However, if an x-value is paired with more than one y-value, the relation is not a function.

Vertical Line Test

The Vertical Line Test is a graphical method to determine if a curve is a function. If any vertical line drawn through the graph intersects the curve at more than one point, the graph does not represent a function.

• Step 1: Draw or imagine vertical lines (parallel to the y-axis) across the graph.

• Step 2: If any vertical line crosses the graph more than once, the graph fails the test and is not a function.

• Step 3: If every vertical line crosses the graph at most once, the graph passes the test and is 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 = 1, fails the vertical line test and is not a function.

Summary Table:

Additional info: Understanding the distinction between relations and functions is crucial for later topics in business calculus, such as modeling cost, revenue, and profit functions, and for applying calculus techniques to real-world business problems.