Question

which statement best describes the relationship shown in the equation?
$y^2 - x = 8$
a. it cannot be determined if this is or is not a functional relationship.
b. this is a functional relationship.
c. this is not a functional relationship.

Explanation:

To determine if a relation is a function, we use the vertical line test (for graphs) or check if each input (x - value) has exactly one output (y - value). For the equation \(y^{2}-x = 8\), we can rewrite it as \(x=y^{2}-8\). If we take a value of \(y\), say \(y = 2\) and \(y=- 2\), when \(y = 2\), \(x=2^{2}-8=4 - 8=-4\); when \(y=-2\), \(x=(-2)^{2}-8 = 4 - 8=-4\). So, for the input \(x=-4\), there are two different outputs (\(y = 2\) and \(y=-2\)). This violates the definition of a function (each input must have exactly one output), so the relation is not a function.

Answer:

C. This is not a functional relationship.