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.

Question

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

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

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

No, the given relation is not a function.

Answer 1

To determine if the given relation (a semicircle) is a function, we apply the vertical line test. This test states that if a vertical line intersects the graph of the relation at more than one point, then the relation is not a function.

In this case, the semicircle is the upper half of a circle centered at the origin with a radius of 6. The equation of the semicircle can be expressed as \( y = \sqrt{36 - x^2} \) for \( -6 \leq x \leq 6 \).

When we take a vertical line at any x-value within the range of the semicircle (except at the endpoints), it can only intersect the semicircle at one point. This means each input (x-value) corresponds to exactly one output (y-value).

Therefore, since the vertical line test holds true for the semicircle (it never intersects the graph at more than one point), we can conclude:

Yes, the given relation is a function.