Function Definition Lesson

In this lesson, we will introduce the concept of a function. Functions will be very important to us throughout our study of Algebra and higher-level math. Let’s begin by thinking about a relation. A relation is any set of ordered pairs. A few examples of relations:
{(2,6), (1,9)}
{(3,5), (2,1), (8,4)}
{(9,1), (-3,4), (12,17), (9,4)}
Each set of ordered pairs above represents a relation. We can also think about a relation with an example scenario. Let’s suppose we go shopping at the local grocery store. The store sells mixed nuts for $1.50 per ounce. Let's say for the purposes of our simple example, we can only buy in increments of 1 ounce. Additionally, we can purchase a maximum of 4 ounces. If we think about the relationship mathematically, we can let y be the total cost in dollars and x be the number of ounces purchased. We can write this as the following equation: Note that the x-values here are limited to 0,1,2,3, and 4. We can't purchase less than 0 ounces, so negative values are out. In addition to this, we can't purchase more than 4 ounces. Lastly, we have to buy in increments of 1 ounce.
We can show the ordered pairs that satisfy our given scenario:

Number of Ounces (x)	Total Cost (y)	(x,y)
0	0	(0,0)
1	1.5	(1,1.5)
2	3	(2,3)
3	4.5	(3,4.5)
4	6	(4,6)

The set of ordered pairs in our table represent a relation:
{(0,0), (1,1.5), (2,3), (3,4.5), (4,6)}
The set or list of all first components (x-values) in the ordered pairs of a relation is called the domain. In our above example, the domain would be:
domain: {0,1,2,3,4}
The set or list of all second components (y-values) in the ordered pairs of a relation is called the range. In our above example, the range would be:
range: {0,1.5,3,4.5,6}
Let's take a look at an example.
Example 1: Find the domain and range for each relation.
{(0,2), (3,8), (9,4), (-2,-5)}
The domain is the set of x-values:
domain: {0,3,9,-2}
The range is the set of y-values:
range: {2,8,4,-5}

How to Determine if a Relation is a Function

A function is a special type of relation where each first component or x-value corresponds to exactly one second component or y-value. The definition of a function may seem a bit confusing at first, but once we practice a few examples, it will seem very simple. Let's look at an example of a function:
{(4,3), (2,6), (8,-5), (-1,-9)}
What makes this relation a function? Each x-value is associated with or linked up to one y-value. Let's now look at an example of a relation that is not a function:
{(-1,7), (3,4), (-1,8), (2,5)}
What makes this relation fall in the category of not being a function? We can see that our x-value of (-1) is linked up with two different y-values: 7, and 8. This fails our definition of function. Each x-value is not associated with one y-value. Let's look at a few examples.
Example 2: Determine if each relation represents a function.
{(-4,-3), (2,7), (1,1), (0,3)}
Each x-value is associated with one y-value:
-4 » -3 : (The x-value of -4 is associated with a y-value of -3)
2 » 7 : (The x-value of 2 is associated with a y-value of 7)
1 » 1 : (The x-value of 1 is associated with a y-value of 1)
0 » 3 : (The x-value of 0 is associated with a y-value of 3)
This relation is a function.
Example 3: Determine if each relation represents a function.
{(-5,-1), (3,4), (-5,11), (2,9)}
Each x-value is not associated with one y-value. We can see that the x-value of (-5) is associated with two different y-values (-1 and 11):
-5 » -1 & 11 : (The x-value of -5 is associated with a y-value of -1 and 11)
3 » 4 : (The x-value of 3 is associated with a y-value of 4)
2 » 9 : (The x-value of 2 is associated with a y-value of 9)
This relation is not a function.
A common trap question is to show duplicate y-values. We may have a function where each y-value corresponds to more than one x-value. This may seem very confusing, so let's explain this in an example. Suppose we see the following relation:
{(3,7), (2,9), (-1,7), (5,3)}
Many students will stop and report that this relation is not a function. We may think this since the y-value of 7 is linked up with an x-value of (-1) and 3. This is actually allowed in a function. Recall the definition states that for each x-value there is one y-value.
3 » 7 : (The x-value of 3 is associated with a y-value of 7)
2 » 9 : (The x-value of 2 is associated with a y-value of 9)
-1 » 7 : (The x-value of -1 is associated with a y-value of 7)
5 » 3 : (The x-value of 5 is associated with a y-value of 3)
Each x-value is associated with one y-value. If we asked what is the value of y, when x is a given value, we would have a clear answer.
In our earlier example of a non-function:
{(-5,-1), (3,4), (-5,11), (2,9)}
We think more deeply here about having a clear association. What is the value of y when x is (-5). We could say -1 or 11. There is no clear association between the x-value of (-5) and one y-value. This is the concept of a function. If we choose an x-value, we must get a unique y-value as the output. Let's look at one more example.
Example 4: Determine if each relation represents a function.
{(3,9), (-11,-4), (2,9), (-1,-1)}
3 » 9 : (The x-value of 3 is associated with a y-value of 9)
-11 » -4 : (The x-value of -11 is associated with a y-value of -4)
2 » 9 : (The x-value of 2 is associated with a y-value of 9)
-1 » -1 : (The x-value of -1 is associated with a y-value of -1)
For each x-value, there is one associated y-value. It is okay that an x-value of 3 corresponds to a y-value of 9, while an x-value of 2 also corresponds to a y-value of 9. If we ask what is y when x is 3, we have a clear answer of 9. If we ask what is y when x is 2, we have a clear answer of 9. In a function, each x-value can be linked up to or associated with one y-value. A function is allowed to have a y-value that is linked up to multiple x-values.
This relation is a function.

Vertical Line Test

When studying functions, we will often come across the vertical line test. The vertical line test tells us that no vertical line will impact the graph of a function in more than one location. Let's look at an example.
Example 5: Use the vertical line test to determine if the given relation represents a function.
{(-5,9),(-5,5),(6,3),(8,-6)} We can see from our graph that the x-value of (-5) corresponds to two different y-values (9 and 5). Let's graph the vertical line:
x = -5
Our vertical line impacts the graph at two different points (-5,9) and (-5,5). This relation fails the vertical line test, therefore, we do not have a function.

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