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How to Use the Vertical Line Test to Determine if a Relation is a Function

Relation is a set of ordered pairs. Additionally, we learned that a function is a special relation in which each x-value is associated with one and only one y-value. Suppose we had the following relation:

  • {(0,3), (2,5), (4,2), (6,3)}

Does this relation represent a function? Yes, since each x-value is associated with one and only one y-value.

  • 0 » 3 - an x-value of 0 is associated with a y-value of 3
  • 2 » 5 - an x-value of 2 is associated with a y-value of 5
  • 4 » 2 - an x-value of 4 is associated with a y-value of 2
  • 6 » 3 - an x-value of 6 is associated with a y-value of 3

In some cases, an illustration will make things more clear:
Vertical Line Test | The Complete SAT Course - Class 10

Another way to determine if a relation is a function is with the use of a graph. If we plot each ordered pair on the coordinate plane, no vertical line should intersect more than one ordered pair. This test is known as the "vertical line test".
Vertical Line Test | The Complete SAT Course - Class 10

Let's look at another example.

  • {(2,-7), (4,1), (2,3), (-1,-5)}

Does this relation represent a function? No, since an x-value of 2 is associated with more than one y-value: -7 and 3.

2 » -7 and 3 - an x-value of 2 is associated with a y-value of -7 and 3

4 » 1 - an x-value of 4 is associated with a y-value of 1

-1 » -5 - an x-value of -1 is associated with a y-value of -5
Let's again look at an illustration:
Vertical Line Test | The Complete SAT Course - Class 10We can also see this relation is not a function with the use of our vertical line test:
Vertical Line Test | The Complete SAT Course - Class 10We can see from the graph above that our vertical line x = 2, intersects more than one ordered pair: (2,3) and (2,-7). This tells us that the x-value of 2 is associated with more than one y-value. When this occurs, we know our relation is not a function.
In most cases, we will not be dealing with a simple set of four or five ordered pairs. These are easy examples designed to help one understand the concept of a function. What happens if we see an equation such as:

  • y = 3x + 5

Let's think about a few things here. First and foremost, let's think about the domain and range. We know the domain is the set of allowable x-values. Ask yourself the question, is there any restriction on what can be plugged in for x? No, so the domain will be all real numbers.

  • domain: {x |x ∈ ℝ}

The above is read as "the set of all x such that x is a real number".

What about our range? Think about y as an output. We plug in a value for x, multiply by 3 and add 5. Since we can plug in anything we want for x, our input, y can also be any real number. We can make y as big as we would like by increasing the size of x. We can make y as small as we would like by decreasing the size of x.

  • range: {y | y ∈ ℝ}

The above is read as "the set of all y such that y is a real number".
Is this relation a function? To determine this, let's graph our equation and use the vertical line test.
Vertical Line Test | The Complete SAT Course - Class 10It is clear that no vertical line will ever impact the graph in more than one location. This means that each x-value is associated with one and only one y-value.
Our relation: y = 3x + 5 is a function.

Let's look at a few examples.

Example 1: Determine if the graph of the relation represents a function, state the domain and the range.
Vertical Line Test | The Complete SAT Course - Class 10

We can use the vertical line test to determine if we have a function.
Vertical Line Test | The Complete SAT Course - Class 10Since we can draw a vertical line and impact the graph in more than one location, this is not the graph of a function.

The domain can be found from the graph. We can see that the smallest x-value is -6 and the largest is 6.

domain: {x | -6 ≤ x ≤ 6}

Similarly, we can find the range from the graph. We can also see the smallest y-value is -6 and the largest is 6.

range: {y | -6 ≤ y ≤ 6}

Example 2: Determine if the graph of the relation represents a function, state the domain and the range.

We can use the vertical line test to determine if we have a function.
Vertical Line Test | The Complete SAT Course - Class 10

It is clear that no vertical line will ever impact the graph in more than one location. This means that each x-value is associated with one and only one y-value. This is the graph of a function.

The domain and range can both be found from our graph. We can see there is no limit on x-values. Essentially, the domain will contain all real numbers. The range, however, is limited. From the graph, we can see the smallest value for y is 2. Therefore, our range will consist of all real numbers that are greater than or equal to 2.

  • domain: {x |x ∈ ℝ}
  • range: {y | y ≥ 2}

More on Domain

Now that we have a good understanding of how to determine if a relation is a function, let's think a little bit more about the domain of a function. The domain of a function is the set of allowable x-values. There are a few things to watch out for:

  • We can't divide by zero
  • We can't take the square root of a negative number and get a real number
  • When we square a number, the result is non-negative

Example 3: Find the domain for each.
y = 1/x-9

Since we are not allowed to divide by zero, think about the denominator here:
x - 9
We can set this equal to zero and solve:
x - 9 = 0
x = 9
This means x can't be 9. If we let x be 9, our denominator will be zero, and division by zero is not defined.

  • domain: {x | x ≠ 9}

Example 4: Find the domain for each.
Vertical Line Test | The Complete SAT Course - Class 10

Since we can only take the square root of a non-negative number and end up with a real number, we think about what is under the square root symbol:
x - 12
We know that whatever is plugged in for x, the result of subtracting away 12 has to be 0 or larger:
x - 12 ≥ 0
x ≥ 12
This means x can be 12 or any larger value. If we plug in a value that is less than 12, we end up with the square root of a negative.

  • domain: {x | x ≥ 12}
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