Which of these pairs of the form (x, y)
could not lie on the graph of a function
of x?
F (1, 1) and (3, 1)
G (1, 1) and (2, 1)
H (1, 1) and (1, 2)
J (1, 1) and (2, 2)
to have a function
you cannot have 2 or more different values of y for the same x
that is, you cannot have 2 or more points on the same vertical line
(this is often called the "vertical line test" )
To determine which pair could not lie on the graph of a function of x, we need to check if there is more than one y-value assigned to the same x-value. If there is more than one y-value for the same x-value, then it violates the definition of a function, as a function should assign a unique y-value for each x-value.
Let's analyze each pair:
F (1, 1) and (3, 1):
In this pair, both points have the same y-value of 1. It means that for both x-values, 1 and 3, the y-value is 1. This pair can lie on the graph of a function.
G (1, 1) and (2, 1):
Similar to pair F, both points have the same y-value of 1. It means that for both x-values, 1 and 2, the y-value is 1. This pair can also lie on the graph of a function.
H (1, 1) and (1, 2):
In this pair, both points have different y-values for the same x-value of 1. It means that for x = 1, the y-value is 1 in the first point but 2 in the second point. This violates the definition of a function, as an x-value cannot have more than one y-value. Therefore, this pair could not lie on the graph of a function.
J (1, 1) and (2, 2):
This pair has different x and y-values. For x = 1, the y-value is 1, and for x = 2, the y-value is 2. Since each x-value has a unique y-value assigned to it, this pair can lie on the graph of a function.
In summary, the pair of points (1, 1) and (1, 2) (pair H) could not lie on the graph of a function of x.
To determine which pairs of the form (x, y) could not lie on the graph of a function of x, we need to check if there is a distinct value of y for every value of x. If there exists a pair of points where the x-values are the same but the y-values are different, it would violate the definition of a function.
Let's analyze each pair:
F (1, 1) and (3, 1):
In this pair, both points have the same y-value (1) even though the x-values (1 and 3) are different. This is allowed in a function because each x-value maps to the same y-value.
G (1, 1) and (2, 1):
Similar to the previous pair, both points have the same y-value (1), and the x-values (1 and 2) are different. This is also allowed in a function because there is a unique y-value for every x-value.
H (1, 1) and (1, 2):
In this pair, both points have the same x-value (1), but the y-values (1 and 2) are different. This violates the definition of a function since the x-value 1 is associated with two different y-values. Therefore, this pair could not lie on the graph of a function of x.
J (1, 1) and (2, 2):
Both points in this pair have different x-values (1 and 2) and different y-values (1 and 2). This is allowed in a function because each x-value maps to a unique y-value.
So, the pair (1, 1) and (1, 2) (pair H) could not lie on the graph of a function of x.