BackIntroduction to Functions and Their Graphs
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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.
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) has at most one output (y-value).
This means that for every x-value, there is only one corresponding y-value.
Key Definitions:
Input: The independent variable, usually denoted as x.
Output: The dependent variable, usually denoted as y.
Examples and Applications:
In business, functions can represent cost, revenue, or profit as a function of the number of units sold.
For example, if C(x) is the cost to produce x items, then C is a function if each x produces only one cost value.
Determining Functions Graphically
To determine if a relation is a function, we use the Vertical Line Test:
Vertical Line Test: 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 is a function.
Examples:
A parabola opening upwards (e.g., y = x2) passes the vertical line test and is a function.
A circle (e.g., x2 + y2 = r2) fails the vertical line test and is not a function.
Graph | Function? | Reason |
|---|---|---|
Parabola (y = x2) | Yes | Each x has only one y |
Circle (x2 + y2 = r2) | No | Some x have two y values |
Summary Table: Function vs. Not a Function
Inputs (x) | Outputs (y) | Function? |
|---|---|---|
Each x has one y | Unique for each x | Function |
Some x have multiple y | Not unique | Not a Function |
Additional info: In business calculus, understanding functions is crucial for modeling and analyzing real-world scenarios such as demand, supply, and cost functions. Mastery of the vertical line test and the distinction between relations and functions forms the basis for more advanced topics like limits, derivatives, and integrals.
