Question

which statement best describes the relationship shown in the equation?
$x^2 + y^2 = 10$
a. this is not a functional relationship.
b. this is a functional relationship.
c. it cannot be determined if this is or is not a functional relationship.

Explanation:

To determine if a relation is a function, we use the vertical line test. The equation \(x^{2}+y^{2} = 10\) represents a circle (with radius \(\sqrt{10}\) and center at the origin). A vertical line can intersect a circle at more than one point. By the definition of a function (each input \(x\) has exactly one output \(y\)), since there are \(x\)-values (e.g., \(x = 0\) gives \(y=\pm\sqrt{10}\)) that correspond to multiple \(y\)-values, this relation is not a function.

Answer:

A. This is not a functional relationship.