1.14: Vertical Line Test
Suppose each student in your class were represented by a point, with the \begin{align*}x\end{align*} -coordinate being the student's year in school and the \begin{align*}y\end{align*} -coordinate being the student's age. Would the set of points represent a function? How would you know? What if you graphed the set of points on a Cartesian plane? Is there any sort of test you could use to find out if you had a function? Upon finishing this Concept, you'll be able to answer these questions and determine whether or not any relation is a function.
Watch This
Watch this YouTube video giving step-by-step instructions of the vertical line test. CK-12 Basic Algebra: Vertical Line Test (3:11)
Guidance
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.
This usually means that each \begin{align*}x-\end{align*} value has only one \begin{align*}y-\end{align*} value assigned to it. But, not all functions involve \begin{align*}x\end{align*} and \begin{align*}y\end{align*} .
Example A
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.
Example B
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)
Solution:
a) To determine whether this relation is a function, we must use the definition of a function. Each \begin{align*}x-\end{align*} coordinate can have ONLY one \begin{align*}y-\end{align*} coordinate. However, since the \begin{align*}x-\end{align*} coordinate of 3 has two \begin{align*}y-\end{align*} coordinates, 4 and 5, this relation is NOT a function.
b) Applying the definition of a function, each \begin{align*}x-\end{align*} coordinate has only one \begin{align*}y-\end{align*} coordinate. Therefore, this relation is a function.
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.
Theorem:
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.
Example C
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.
Guided Practice
Determine if the graphed relation is a function.
Solution:
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.
Practice
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)
- \begin{align*}&\text{Age} && 20 && 25 && 25 && 30 && 35\\ &\text{Number of jobs by that age} && 3 && 4 && 7 && 4 && 2\end{align*}
- \begin{align*}&&x && -4 && -3 && -2 && -1 && 0\\ &&y && 16 && 9 && 4 && 1 && 0\end{align*}
In 5 and 6, write a function rule for the graphed relation.
In 7-8, determine whether the graphed relation is a function.
Mixed Review
- A theme park charges $12 entry to visitors. Find the money taken if 1296 people visit the park.
- A group of students are in a room. After 25 students leave, it is found that \begin{align*}\frac{2}{3}\end{align*} of the original group are left in the room. How many students were in the room at the start?
- Evaluate the expression \begin{align*}\frac{x^2+9}{y+2}\end{align*} when \begin{align*}y = 3\end{align*} and \begin{align*}x=4\end{align*} .
- The amount of rubber needed to make a playground ball is found by the formula \begin{align*}A = 4 \pi r^2\end{align*} , where \begin{align*}r=radius\end{align*} . Determine the amount of material needed to make a ball with a 7-inch radius.
flow chart
A flow chart is a graphical representation of a relation between two variables, connecting input and output values with lines or arrows.Relation
A relation is any set of ordered pairs . A relation can have more than one output for a given input.vertical line test
The vertical line test is a visual method of determining whether a graph of a relation is a function. The test determines that the relation is a function when no vertical line intersects the graph at more than one point.Function
A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for .Image Attributions
Here you'll learn how to use techniques such as a flow chart or the vertical line test to decide if a relation is a function.
Concept Nodes:
flow chart
A flow chart is a graphical representation of a relation between two variables, connecting input and output values with lines or arrows.Relation
A relation is any set of ordered pairs . A relation can have more than one output for a given input.vertical line test
The vertical line test is a visual method of determining whether a graph of a relation is a function. The test determines that the relation is a function when no vertical line intersects the graph at more than one point.Function
A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for .