Watch this YouTube video giving step-by-step instructions of the vertical line test. CK-12 Basic Algebra: Vertical Line Test (3:11)
Determining Whether a Relation Is a Function
You saw that a function is a relation between the independent and the dependent variables. It is a rule that uses the values of the independent variable to give the values of the dependent variable. A function rule can be expressed in words, as an equation, as a table of values, and as a graph. All representations are useful and necessary in understanding the relation between the variables.
Definition: A relation is a set of ordered pairs.
Mathematically, a function is a special kind of relation.
Definition: A function is a relation between two variables such that the independent value has EXACTLY one dependent value.
One way to determine whether a relation is a function is to construct a flow chart linking each dependent value to its matching independent value. Consider the relation that shows the heights of all students in a class. The domain is the set of people in the class and the range is the set of heights. Each person in the class cannot be more than one height at the same time. This relation is a function because for each person there is exactly one height that belongs to him or her.
Notice that in a function, a value in the range can belong to more than one element in the domain, so more than one person in the class can have the same height. The opposite is not possible; that is, one person cannot have multiple heights.
Determine if the relation is a function.
a) (1, 3), (–1, –2), (3, 5), (2, 5), (3, 4)
b) (–3, 20), (–5, 25), (–1, 5), (7, 12), (9, 2)
Determining Whether a Graph Is a Function
Suppose all you are given is the graph of the relation. How can you determine whether it is a function?
You could organize the ordered pairs into a table or a flow chart, similar to the student and height situation. This could be a lengthy process, but it is one possible way. A second way is to use the vertical line test. Applying this test gives a quick and effective visual to decide if the graph is a function.
Part A) A relation is a function if there are no vertical lines that intersect the graphed relation in more than one point.
Part B) If a graphed relation does not intersect a vertical line in more than one point, then that relation is a function.
Is this graphed relation a function?
By drawing a vertical line (the red line) through the graph, we can see that the vertical line intersects the circle more than once. Therefore, this graph is NOT a function.
Here is a second example:
No matter where a vertical line is drawn through the graph, there will be only one intersection. Therefore, this graph is a function.
Determine if the graphed relation is a function.
Imagine moving a vertical line across the plane. Do you see anywhere that this vertical line would intersect the graph at more than one place?
There is no place on this graph where a vertical line would intersect the graph at more than one place. Using the vertical line test, we can conclude the relation is a function.
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Functions as Graphs (9:34)
In 1 – 4, determine if the relation is a function.
- (1, 7), (2, 7), (3, 8), (4, 8), (5, 9)
- (1, 1), (1, –1), (4, 2), (4, –2), (9, 3), (9, –3)
AgeNumber of jobs by that age203254257304352
In 5 and 6, write a function rule for the graphed relation.
In 7-8, determine whether the graphed relation is a function.
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