Introduction to Relations and Functions
Algebra introduces us to crucial concepts like relations and functions, each with its unique characteristics. While the two topics are related, there are some critical distinctions between them. This lesson will dive into relations and functions and how each can be represented. We will also introduce what domain and range are and how to find both the domain and range for relations.
Exploring Relations
A relation is a set of ordered pairs, usually denoted as (input, output). Unlike functions, relations can have multiple outputs for a single input. We represent relations in various ways, such as with lists, tables, mapping diagrams, or graphs.
Let’s take a look at a relation and the ways this can be represented.
Here is a relation with the following coordinate points represented as a list:
Example
{(-8, -4) ; (-5, 3) ; (6, 5) ; (0, 0) ; (8, -5) ; (6, -2)}
In a list, the coordinate points are usually written inside these brackets {}.
Now let’s see this same relation shown in a table, mapping diagram, and graph.
Notice that a single x-value may correspond to multiple y-values, as seen in the representation methods mentioned.
Understanding the Domain and Range of Relations
In mathematical terms, the domain refers to possible x-values, while the range represents possible y-values. Using the example relation depicted above:
Example
{(-8, -4) ; (-5, 3) ; (6, 5) ; (0, 0) ; (8, -5) ; (6, -2)}
Domain: {-8, -5, 0, 6, 8}
Range: {-5, -4, -2, 0, 3, 5}
When writing both the domain and range, we organize these values in increasing order, and we can exclude any duplicates.
Functions
A function is a special kind of relation where each input corresponds to a single output. Not all relations are functions, but all functions are relations.
A common way we can write, or notate, a function is by writing f(x), which we can read as “f of x,” a function of the variable x.
We can think of functions like a machine. You input a number, an equation acts on it, and an output is produced. The input (x-value) is the independent variable, and the output (y-value) is the dependent variable.
Now, let’s take a look at our relation example again:
Example
{(-8, -4) ; (-5, 3) ; (6, 5) ; (0, 0) ; (8, -5) ; and (6, -2)}
In our relation example, because {(6, -2) ; (6, 5)} exist, the relation is NOT a function. Why? Because a single input “6” has two different output values.
For many functions, any x-value can be used, making them continuous functions. Continuous functions have coordinate points connected and there are no breaks in the graph.
Determining Functions Using Tables and Mapping Diagrams
Functions follow a rule: one input yields one output. Tables and mapping diagrams can help us understand relationships and decide if the relation is a function.
Allowed Relationships
One-to-one relationships: One input will go to one output.
Many-to-one relationships: Many inputs may go to one output.
With both of these relationships, we can see that the rule for functions is being followed.
Not Allowed Relationships
One-to-many: One input will go to many outputs.
This relationship between inputs and outputs shows that this relation is NOT a function.
Determining Functions Using Graphs
To determine if a relation is a function using a graph, we can use the Vertical Line Test. To use the Vertical Line Test, we visually create a vertical line, start on the left side of the graph, and move the vertical line to the right side. If a vertical line intersects the graph at only one point at a time, it's a function. More intersections mean it's not a function.
Conclusion
This foundational knowledge prepares us for future lessons on writing the domain and range of continuous functions. If you're ready to explore that topic, click the title below:
