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

The relation is a half-circle.
The relation is a half-circle.

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.

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

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

Answer 1

To apply the vertical line test to determine if the given relation (the semicircle) is a function, we consider whether any vertical line drawn through the graph intersects it in more than one point.

A semicircle, by definition, consists of the upper or lower half of a full circle. In this case, the semicircle passes through the points (0, 6), (6, 0), and (0, -6), which indicates that it is the top half of a circle centered at the origin with a radius of 6.

If we were to draw vertical lines through the coordinate plane, each vertical line would intersect the semicircle at most once (since it's only the upper half of the full circle). Therefore, any vertical line would either intersect the graph at one point or not at all, confirming that for every x-value, there is exactly one unique y-value.

Hence, the conclusion is:

Yes, the given relation is a function.