Table of Contents |
When we discuss the term relation in mathematics we are referring to mapping how one set of values, called the input or domain, relates to another set of values, called the output or range. There are many examples of relations that you may come across in everyday encounters.
EXAMPLE
One instance may be a list of companies and the number of employees each company has working for them. The input would be companies and the output would be the number of employees.Oftentimes though in math, we try to map one set of numbers to another set of numbers and try to determine if there is a relationship between these numbers so that we can make predictions about future events.
EXAMPLE
Suppose we wanted to determine a relationship between age and height. Below is a sample of the type of data that we might find.| x: Age (years) | 5 | 10 | 15 | 15 | 20 | 25 | 25 | 30 | 35 |
|---|---|---|---|---|---|---|---|---|---|
| y: Height (inches) | 24 | 48 | 70 | 72 | 73 | 74 | 72 | 71 | 70 |
Notice in the table above, we have two values in the domain that plot to the same value in the range. Both inputs 15 and 35 have an output of 70. We also have the same values of the domain, 15, that plot to different values in the range, 70 and 72. This is an important observation to make when distinguishing a relation from a function. Let's take a look at what we mean by the term "function" in mathematics in the next section.
A function is a special type of relation where every element, x, in the domain corresponds to exactly one element in the range. Unlike the previous example where we had the age 15 map to two different heights, 70 and 72, in a function, we cannot have this happen.
EXAMPLE
Consider this function where the volume of water in a container and the container's weight are shown in the table below.| x: Volume of Water (liters) | 0.25 | 0.50 | 0.75 | 1 |
|---|---|---|---|---|
| y: Weight of Water (grams) | 250 | 500 | 750 | 1000 |
Notice in the above example that each value in the domain, x, maps to only one value in the range, y. However, you can have an element, y, in the range that corresponds to more than one element in the domain.
Determine if each relation in the table below is a function.
| Relation | Function? | Explanation | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
Yes | Each x-value in the domain appears only once, which means each x-value has only one y-value in the range. | ||||||||||
|
No | There are two x-values in the domain that are the same, but have different y-values in the range. This is NOT a function because each x-value needs to have only one y-value. | ||||||||||
|
Yes | Even though there are two x-values that have the same y-value, each x-value has only one y-value in the range. |
There are other ways to determine if a given set of data represents a function. One of the most common ways we do this is to graph the data on a coordinate plane and perform a vertical line test. To perform a vertical line test, we take a vertical line (a line that goes up and down) and move it across the entire graph. If the vertical line touches the function at only one point across the entire graph, then it passes the vertical line test.
If a function passes the vertical line test, then we can claim that the graph represents a function.
EXAMPLE
Consider the graph below. Notice that in this graph shown above, several vertical lines were drawn in and no line touches the graph more than once. Therefore, we can conclude that this represents a function.
EXAMPLE
On the other hand, suppose we have a horizontal parabola, as shown below. If we perform the vertical line test on the graph of this relation, is it also a function?
Let’s now take a look at the notation we use to represent functions. Suppose we have the linear relation, 
. Notice that, if we were to graph this line and apply the vertical line test to it, we can confirm that this is a function.
When working with functions we like to write those functions in a notation that lets us know that we are working with a function, and what inputs the outputs of the function are dependent on. In this process we rewrite as 
. Note that here y and 
are the same thing. We read this statement as, “f of x is equal to 2x minus 4”, which means that this function, generically represented by f, is dependent on the input x.
for f multiplied by x. When we write 
we are noting that we are working with functions, not that we are multiplying quantities. When dealing with functions, always means that the output of the function is dependent on the input x.
When solving this above function for a given value of x, instead of saying finding the value of y when 
, as we have done before, we say "find 
". This is read as "f of 
". While the notation is slightly different, the process we use is generally the same. Let’s look at an example:
EXAMPLE
Solve the equation and function when 
.
|
|
Problem Statement |
|---|---|---|
|
|
Substitute in
|
|
|
Simplify |
|
|
Our solution |
for the quantity "a", simply substitute all x variables in the function with "a" and simplify.
Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License
