Use the image to answer the question.

Question

An illustration shows a coordinate plane with four quadrants. The x and y axes range from negative 6 to 6 in increments of 1. A semicircle is graphed on the plane. The semicircle passes through points plotted at coordinates left parenthesis 0 comma 6 right parenthesis, left parenthesis 6 comma 0 right parenthesis, and left parenthesis 0 comma negative 6 right parenthesis.

The graph of a relation is given. Apply the vertical line test to determine if this relation is a function.

(1 point)
Responses

No, the given relation is not a function.
No, the given relation is not a function.

There is not enough information to determine whether this relation is a function.
There is not enough information to determine whether this relation is a function.

Yes, the given relation is a function.
Yes, the given relation is a function.

The relation is a half-circle.

Answer 1

To determine whether the given relation (the semicircle) is a function, we can apply the vertical line test. According to this test, if any vertical line intersects the graph of the relation more than once, then the relation is not a function.

In this case, the semicircle is defined by points above the x-axis and will be intersected by vertical lines in the upper half of the coordinate plane. Since for some x-values (specifically those corresponding to points within the semicircle's horizontal extent) there are two y-values (the upper and lower edges, for the negative y-values and positive y-values you would encounter in a full circle), a vertical line would indeed cross the semicircle at two points.

Given that the semicircle passes through points like (0, 6) and (0, -6), it can be inferred that for certain x-coordinates, multiple y coordinates exist. Thus, we conclude:

No, the given relation is not a function.