Functions
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Functions are special types of relations where each input is paired with exactly one output. Functions can be expressed as sets of ordered pairs, tables, mappings, or graphs. A function expressed as a graph will pass the vertical line test, meaning a vertical line drawn on the graph will intersect it at only one point.
Functions
- 1. Functions A special type of relation! Image from MathisFun.com
- 2. Let’s review relations! • A relation is a set of ordered pairs. {(1,2), (2,-3), (-2,4), (-1,0)} is an example of a relation. • The x-values in each coordinate make up what is referred to as the domain. {-2, -1, 1, 2} is the domain of the relation. • The y-values in each coordinate make up what is referred to as the range. {-3, 0, 2, 4} is the range of the relation.
- 3. So What Are Functions? • • • • • Functions are relations, so they are sets of ordered pairs! What makes them special is that the x-values don’t repeat! Each x-value is paired with exactly one y-value. There are many ways to express functions. They can be expressed as ordered pairs, in tables, in a graph, or as a mapping.
- 4. Let’s take a closer look at how functions can be expressed. • Let’s use the relation:{(-2,3), (1, 4), (0, 4), (-1, 2)}. This relation is a function because the x-values don’t repeat. It is expressed here as a set of ordered pairs. x -2 We can also use a table to express this function. y 3 -1 2 0 4 1 4
- 5. A Mapping is another way to express functions. You can express a function as a mapping. A mapping is two ovals. The first oval contains the domain (x-values). The second oval contains the range (y-values). For example: -1 0 1 2 -2 1 2 This is an example of a mapping. Each x-value is mapped or paired with exactly one y-value.
- 6. You can also express functions in a graph. Here are some examples of functions in graphs: Images from OCS Algebra IA U4L3 Guided Notes
- 7. Graphs of Functions You can tell if the graph of a relation is a function if it passes something called the Vertical Line Test. The VLT says that if you draw a vertical line through a graph and it intersects the graph in exactly one point, then the graph is a function. This is the graph of a function; it passes the VLT. This is not the graph of a function; it fails the VLT. Images from OCS Algebra IA Unit 4 Lesson 3